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Online Talks

Durham Cathedral

London Mathematical Society Durham Symposium
Numerical Analysis of Multiscale Problems
Monday 5th July - Thursday 15th July 2010
List of abstracts
Assyr Abdulle (EPFL, Lausanne) Saturday 10th July 14:30
A posteriori error analysis for numerical homogenization methods
In this talk we present an a posteriori error analysis for the numerical homogenization of elliptic problems. The discretization scheme relies on macro and micro finite elements, following the framework of the heterogeneous multiscale method. In this multiscale method, the desired macroscopic solution is obtained by a suitable averaging procedure based on microsolution probing the fine-scale structure of the problem. As the macroscopic data (such as the macroscopic diffusion tensor) are not available beforehand, appropriate error indicators have to be defined for designing adaptive methods. We will show that such indicators based only on the available macro- and microsolutions (used to compute the actual macrosolution) can be defined, allowing for a macroscopic mesh refinement strategy which is both reliable and efficient. Numerical experiments illustrating the efficiency and reliability of the adaptive multiscale method will be presented.


[1] A. Abdulle and A. Nonnenmacher, A posteriori error analysis of the heterogeneous multiscale method for homogenization problems, C.R. Math. Acad. Sci. Paris 247 (2009), no. 17-18, 1081--1086.
[2] A. Abdulle and A. Nonnenmacher, Adaptive finite element heterogeneous multiscale method for homogenization problems, to appear in Comput. Methods Appl. Mech. Engrg. (2010).
[3] A. Abdulle, A priori and a posteriori error analysis for numerical homogenization: a unified framework, to appear in Contemporary Applied Mathematics, (2010).

Ben Adcock (University of Cambridge) Poster
Accurate and stable recovery of functions from spectral data
(Joint work with Anders C. Hansen)
We consider the problem of reconstructing a function (defined on some bounded domain) to high accuracy from a finite number of its coefficients with respect to some orthogonal basis. Straightforward expansion in this basis may converge slowly. Yet, as we prove, it is always possible to reconstruct the function in another, more rapidly convergent basis. Such reconstruction technique is stable, and the resultant approximation near-optimal.

A common example of this approach is the reconstruction of an analytic, nonperiodic function from its Fourier coefficients, with numerous applications including image and signal processing. Fourier series converge slowly. Nonetheless, by reconstructing in a polynomial basis we obtain exponential convergence in terms of n (the polynomial degree), or root exponential convergence in m (the number of Fourier coefficients). The procedure can be implemented in O(mn) operations.

Vivi Andasari (Division of Mathematics, University of Dundee) Poster
Mathematical Modelling of Cancer Invasion of Tissue: The Roles of Cell-cell and Cell-matrix Adhesion
(Joint work with M.A.J Chaplain)

Adhesion, which includes cell-to-cell and cell-to-extracellular-matrix adhesion, plays an important role in cancer invasion and metastasis. After undergoing morphological changes malignant and invasive tumour cells, i.e., cancer cells, break away from the primary tumour by loss of cell-cell adhesion, degrade their basement membrane and migrate through the extracellular matrix by enhancement of cell-matrix adhesion. These processes require interactions and signalling cross-talks between proteins and cellular components facilitating the cell adhesion. Although such processes are very complex, the necessity to fully understand the mechanism of cell adhesion is crucial for cancer studies, which may contribute to improving cancer treatment strategies.

Cancer cell migration and invasion of the extracellular matrix involving adhesive interactions between cells mediated by cadherins and between cell and matrix mediated by integrins, are modelled by employing two types of mathematical models: an individual-based approach and a continuum approach. In the individual-based approach, we first develop pathways for cell-cell and cell-matrix adhesion using Ordinary Differential Equations and later incorporate the pathways in the Cellular Potts Model for computational multi-scale modelling. In the continuum approach, we use Partial Differential Equations in which cell adhesion is treated as non-local and formulated by integral terms.

The computational simulation results from the two different mathematical models show that we can predict invasive behaviour of cancer cells from cell adhesion properties. Invasion occurs if we reduce cell-cell adhesion and increase cell-matrix adhesion and vice versa. Changing the cell adhesion properties can affect the spatio-temporal behaviour of cancer cell invasion. These results may lead to broadening our understanding of cancer cell invasion and suggest optimal methods of patient treatment.

Todd Arbogast (University of Texas at Austin) Monday 12th July 10:10
Mixed multiscale methods for heterogeneous elliptic problems
In this three-part series of lectures, we consider a second order elliptic problem with a heterogeneous coefficient written in mixed form (i.e., as a system of two first order equations). Multiscale methods can be viewed in one of three equivalent frameworks: as a Galerkin or finite element method with nonpolynomial basis functions, as a variational multiscale method with standard finite elements, or as a domain decomposition method with restricted degrees of freedom on the interfaces. We treat each case, and discuss the advantages of the approach for devising effective local multiscale methods. Included is recent work on methods that incorporate information from homogenization theory and effective domain decomposition methods.

Todd Arbogast (University of Texas at Austin) Tuesday 13th July 10:10
Mixed multiscale methods for heterogeneous elliptic problems

Todd Arbogast (University of Texas at Austin) Wednesday 14th July 09:15
Mixed multiscale methods for heterogeneous elliptic problems

Anthony Baran (Met Office UK) Friday 9th July 10:40
Electromagnetic and light scattering by atmospheric particulates: How well does theory compare against observation ?
Ubiquitous cirrus (ice crystal clouds) and atmospheric dust are usually found towards the upper and lower parts of the troposphere, respectively. The common feature between these two types of atmospheric occurrences is that they are composed of irregular particles of varying sizes and shapes. The interaction between incident radiation and these two types of particulates has a profound influence on the degree to which the surface of the Earth is either generally warmed (cirrus) or cooled (dust) in a warming climate due to increased carbon dioxide emissions. The radiative importance of cirrus and dust is clear, yet understanding the magnitude of amplification or diminution of the greenhouse effect is poorly understood. The reason for this is that the variability of shapes and sizes is highly significant and so commonly characterizing them is problematic. Moreover, in the case of ice crystals and large dust particles, currently there is no one light scattering method that can be applied to predict their scattering properties over the observed range of size parameter space. In this talk the traditional electromagnetic and light scattering methods that are usually applied to this problem using single ice crystal and dust shapes will be reviewed. More recent attempts to model cirrus and dust scattering using ensembles of shapes will be emphasized. Theoretical and observational requirements needed to further understand the interaction between incident radiation and ice crystals/dust, so that the uncertainty in the magnitude of the warming or cooling is reduced, will also be explored.

Peter Bastian (University of Heidelberg) Saturday 10th July 09:15
Centre for Numerical Algorithms and Intelligent Software Special Lecture: The Distributed and Unified Numerics Environment (DUNE)
DUNE (www.dune-project.org) is a set of C++-based open-source software framework for the grid-based numerical solution of partial differential equations (PDEs). Its main design principles are: (i) separation of data structures and algorithms through abstract interfaces, (ii) use of generic programming techniques for achieving performance and (iii) enabling reuse of existing finite element software through appropriate interface design. DUNE provides support for many different kinds of grids, a flexible linear solver package, is parallel as well as dimension-independent and offers a full simulation workflow using free software. New discretization schemes and PDE models can be integrated with relative ease through reuse of existing components and powerful abstraction mechanisms. In this talk I will give a short overview of the framework and then concentrate on the flexible implemention of various finite element schemes in the "PDELab" module with applications to flow and transport in porous media.

Timo Betcke (University of Reading) and Euan Spence (University of Bath)Monday 12th July 18:45
Coercivity of boundary integral operators in acoustic scattering
Much research effort in recent years has been focused on designing effective numerical methods for high frequency acoustic scattering. The main difficulty is that, as the frequency increases, the solution becomes more oscillatory, leading to a rapid increase of degrees of freedom in conventional methods to maintain accuracy. One way around this difficulty is to use the high frequency asymptotics of the solution of the scattering problem to design approximation spaces which take into account the high oscillation of the solution. Once these hybrid asymptotic-numerical methods have been designed, an interesting question is whether rigorous error bounds can be established which are explicit in the frequency.

One strategy for proving rigorous error bounds for boundary integral methods for these high frequency problems is to seek to prove that the integral operator is coercive. For these high frequency problems one ideally wants to establish coercivity independent of (or at least explicit in) the frequency.

Coercivity has so far been established only for the case of the circle (in 2d) and sphere (3d) using Fourier analysis. This talk will present some new results on proving coercivity for a much wider class of domains, and also on investigating coercivity numerically.

Liliana Borcea (Rice University) Tuesday 6th July 15:30
Source localization in random acoustic waveguides
Mode coupling due to scattering by weak random inhomogeneities in waveguides leads to loss of coherence of wave fields at long distances of propagation. This in turn leads to serious deterioration of coherent source localization methods. I will show with analysis and numerical simulations how such deterioration occurs, and introduce a novel incoherent approach for long range source localization in random waveguides. It is based on a special form of transport theory for the incoherent fluctuations of the wave field. I will show with analysis that the method is statistically stable and will illustrate its performance with numerical simulations. I will also show how it can be used to estimate the correlation function of the random fluctuations of the wave speed.

Dan Brinkman (University of Cambridge) Poster
Numerical modelling of bilayer organic photovoltaic devices
We use a finite element scheme with hybrid discontinuous galerkin elements implemented in ngsolve to model bilayer polymer solar cells. The model depends non-trivially on the width of the polymer-polymer interface, which is 2 orders of magnitude smaller than the device itself. We examine how changing the size and shape of this interface alters the large-scale behaviour of the device.

Phil Browne (University of Bath) Poster
Structural Optimization using a SIMP approach
Structural optimization attempts to solve a material distribution problem. That is, given some design space and loading conditions what is the optimal distribution of the material within this space to achieve a given objective. These problems arise in many situations, and techniques to solve these problems are regularly used in aerospace, automotive and civil engineering industries. This poster will present the basics of using a relaxation approach known as the SIMP method to turn a discrete programming problem into a continuous problem whilst recovering discrete solutions, and also will show the latest work on trying to make the resulting solutions resistant to buckling.

Chris Budd (University of Bath) Wednesday 14th July 15:40
Adaptive methods for multi-scale problems
(Joint work with Emily Walsh (Bath) and JF Williams (SFU))
Many partial differential equations evolve to have solution structures on very small scales. Examples are blow-up in combustion problems and developing weather fronts in meteorology. Resolving such structures is a difficult numerical challenge and usually requires some form of adaptive method. In this talk I will describe an adaptive numerical method based on ideas in optimal transport, which aims to equidistribute a numerical mesh in an optimal manner. I will show that this leads to a very powerful method of adapting a mesh which can resolve evolving structures of the solutions of PDES over an extremely wide range of length scales. Moreover this method is relatively easy to implement, in any number of dimensions.
In this talk I will develop the theory and application of this method, with a special emphasis on the problems of resolving developing storms in meteorology.

Simon Chandler-Wilde (University of Reading) Monday 12th July 17:00
Numerical methods for high frequency scattering
(Joint work with Steve Langdon and Ashley Twigger.)
In this talk we review progress on the development, implementation and numerical analysis of computational methods for high frequency scattering problems. These problems are at least two-scale, with the wavelength of the incident wave very much smaller than a typical dimension of the scattering obstacle. This is a classical problem, and effective asymptotic methods are available for very high frequencies, for at least certain classes of problems, while fast mutipole methods make boundary element methods effective for moderate frequencies. In this talk we discuss aspects of a joint project between Bath and Reading Universities (for details seehttp://people.bath.ac.uk/eas25/HF/), in which the aim is to produce novel boundary integral equation methods at the interface of classical boundary element methods and high frequency asymptotics, which build the oscillatory behaviour at high frequency into the approximation space. This enables, at least for some classes of scatterer, the development of algorithms which can achieve a specified error tolerance with a cost which is bounded independently of the frequency. Proving this rigorously is a challenging exercise in asymptotics uniform in frequency and discretisation parameter, and in the conditioning of oscillatory integral operators.

John Chapman (University of Durham) Poster
The continuous discontinuous Galerkin finite element method for an advection-diffusion equation
(Joint work with Max Jensen, Emmanuil Georgoulis and Andrea Cangiani)
When attempting to solve a prototype advection-diffusion equation using the standard continuous Galerkin finite element method the numerical solution exhibits non physical oscillations. One solution is to use a discontinuous Galerkin finite element method. This however is more computationally intensive.

We present the hypothesis that a Galerkin method that is continuous on the domain away from any boundary or internal layers, and discontinuous in the vicinity of any layers, is stable. We present a proof that this is the case for the discontinuous portion and several numerical experiments, as well as work for the future on the continuous region.

Zhiming Chen (Chinese Academy of Sciences) Saturday 10th July 16:55
Convergence of the uniaxial perfectly matched layer method for time-harmonic scattering problems in two-layered media
We propose a uniaxial perfectly matched layer (PML) method for solving the time-harmonic scattering problems in two-layered media. The exterior region of the scatterer is divided into two half spaces by an infinite plane, on two sides of which the wave number takes different values. We surround the computational domain where the scattering field is interested by a PML layer with the uniaxial medium property. By imposing homogenous boundary condition on the outer boundary of the PML layer, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational domain as either the PML absorbing coefficient or the thickness of the PML layer tends to infinity.

Paul Childs (Schlumberger) Friday 9th July 10:10
Challenges in seismic imaging
Inversion and imaging methods using the full seismic waveforms are computationally challenging. We will review the current practice in the seismic imaging industry, and outline several multiscale challenges where new mathematical algorithms are being sought.

Andrew Cliffe (University of Nottingham) Friday 9th July 17:00
Deep Geological Disposal of Radioactive Waste
The talk will discuss various modelling and computational issue related to the deep geological disposal of radioactive wastes. Particular attention will be paid to uncertainty quantification. The major problems will be described together with some of the outstanding mathematical and computational challenges.

Masoumeh Dashti (University of Warwick) Poster
Bayesian approach to an elliptic inverse problem
(Joint work with Andrew Stuart)
We consider the inverse problem of determining the permeability from the pressure in a Darcy model of flow in a porous medium. Mathematically the problem is to find the diffusion coefficient for a linear uniformly elliptic partial differential equation in divergence form, in a bounded domain in two or three dimensions, from pointwise measurements of the solution in the interior. We adopt a Bayesian approach to the problem. We place a prior Gaussian random field measure on the log permeability, specified through its two point correlation function. We study the regularity of functions drawn from this prior measure, by use of the Karhunen-Loeve expansion. We also study the Lipschitz properties of the observation operator mapping the log permeability to the observations. Assuming that the observations are subject to mean zero noise, and combining the aforementioned regularity and continuity estimates, we show that the posterior measure is well-defined. Furthermore the posterior measure is shown to be Lipschitz continuous with respect to the data in the Hellinger and total variation metrics, giving rise to a form of well-posedness of the inverse problem.

Niall Deakin (University of Dundee) Poster
Mathematical modelling of cancer growth and spread: the role of enzyme degradation of tissue
(Joint work with Mark Chaplain, George Lolas and Alastair Thompson)
There are many steps involved in the growth and spread of cancer - the current work will focus on the local invasion of the host tissue. A crucial aspect of cancer cell growth and development is the process in which they invade locally by the secretion of enzymes involved in proteolysis, namely plasmin and matrix metalloproteinases (MMPs). These overly expressed proteolytic enzymes then proceed to degrade the host tissue allowing the cancer cells to spread throughout the region by active migration and interaction with components of the extracellular matrix such as collagen. We will consider two approaches for modelling the invasion on the macro scale (cell population level). The first mathematical model considers cancer cells and a number of different matrix degrading enzymes (MDEs) from the MMP family and their interaction with and effect on the extracellular matrix (ECM). The second model focuses on the specific role of the urokinase-type plasminogen activation (uPA) system. Both models consist of a system of reaction-diffusion- taxis partial differential equations in an attempt to capture the qualitative dynamics of the migratory response of the cancer cells.

Louis Durlofsky (Stanford University) Saturday 10th July 10:10
Uncertainty quantification for subsurface flow problems using coarse-scale models
Fine-scale features can have a large impact on key subsurface flow quantities such as injection or production rates. Because the geological characteristics of subsurface formations are highly uncertain, multiple realizations are typically simulated in an attempt to capture the impact of geological uncertainty on flow behavior. It is, however, expensive to perform flow simulation on highly resolved models; for this reason a number of upscaling and multiscale procedures have been devised. Most such techniques aim to provide coarse models that reproduce the fine-model response on a realization-by-realization basis. This may not be necessary, however, when the goal is to replicate the statistics of the flow responses of multiple realizations. In this talk, I will present an upscaling approach that entails the statistical assignment of upscaled functions. This approach is more efficient than traditional treatments as it greatly reduces the most time-consuming upscaling computations. I will also describe new procedures for upscaling in the vicinity of injection and production wells. Numerical results demonstrate that, by combining near-well upscaling and statistical assignment of coarse-scale flux functions, coarse models that are well suited for computing ensemble quantities can be efficiently constructed.

Yalchin Efendiev (Texas A & M University) Wednesday 7th July 15:30
Multiscale simulation techniques for high-contrast subsurface flows
The development of numerical algorithms for modeling flow processes in large-scale highly heterogeneous formations is very challenging because the properties of natural geologic porous formations (e.g., permeability) display high variability levels and complex spatial correlation structures, which span a rich hierarchy of length scales. Thus, it is usually necessary to resolve a wide range of length and time scales in order to obtain accurate predictions of the flow, mechanical deformation, and transport processes under investigation. In practice, however, some type of coarsening (or upscaling) of the detailed model is usually performed before the model can be used to simulate complex displacement processes. Many approaches have been developed and applied successfully when a scale separation adequately describes the spatial variability of the subsurface properties (e.g., permeability) that have bounded variations. The quality of these approaches deteriorates for complex heterogeneities, especially when the contrast in the media properties is large, e.g., in the case of fractured porous media. In this talk, I will describe coarse-scale spaces that can be used in upscaling flow equations as well as in domain decomposition methods when media properties have high contrast and are spatially heterogeneous. Numerical results will be presented that show that one can improve the accuracy of multiscale methods and obtain contrast-independent preconditioners.

Bjorn Engquist (University of Texas, Austin) Thursday 8th July 09:15
Fast algorithms for high frequency wave propagation
Boundary integral formulations of high frequency scattering problems are difficult to handle numerically due to the oscillatory nature of the kernel. The number of unknowns (N) in the approximation of the boundary potential or currents must be very large and the standard fast multi-pole method does not give a reduction in the computational complexity of the core matrix vector multiplication in the solution process. A new multi-level method based on directional decomposition can be proved to have near optimal order of complexity: O(NlogN). A random sampling algorithm to further increase the efficiency will also be introduced. The principles behind this technique also apply to preconditioning of numerical approximations of differential equation formulations.

Oliver Ernst (TU Bergakademie Freiberg) Saturday 10th July 16:20
On the Convergence of Generalized Polynomial Chaos Expansions
A number of approaches for discretizing partial differential equations with random data are based on generalized polynomial chaos expansions of random variables. These constitute generalizations of the polynomial chaos expansions introduced by Norbert Wiener to expansions in polynomials orthogonal with respect to non-Gaussian probability measures. We present conditions on such measures which imply mean-square convergence of generalized polynomial chaos expansions to the correct limit and complement these with illustrative examples.

Leonardo Figueroa (University of Oxford) Poster
Separated representation approximation of a high-dimensional Fokker–Planck PDE for dilute polymers.
(Joint work with Endre Süli)

The evolution of the configuration of polymer molecules in a viscous incompressible solvent is naturally modelled by a system of Langevin-type stochastic differential equations. The associated probability density function satisfies a high-dimensional Fokker–Planck equation on the space of all possible polymer chain configurations. One of the key difficulties, apart from high dimensionality, is the fact that the nonlinear spring-laws featuring in the model introduce degeneracies in the coefficients of the Fokker–Planck equation.

We consider an algorithm based on an SVD-like separated representation strategy, which exploits the tensor-product structure of the space of polymer chain-configurations and approximates the probability density function with a sum of products of functions defined on the low-dimensional space of admissible configurations for a single spring. We establish the convergence of the proposed algorithm.

Martin Gander (University of Geneva) Monday 12th July 18:10
Why it is difficult to solve Helmholtz problems with classical iterative methods
In contrast to the positive definite Helmholtz equation, the deceivingly similar looking indefinite Helmholtz equation is difficult to solve using classical iterative methods; in particular, both classical domain decomposition and multigrid methods fail to converge. I will show in my presentation where the problems lie, and present remedies for domain decomposition methods, and also some new insight for constructing an efficient multigrid method for the indefinite Helmholtz equation.

Uduak George (University of Sussex) Poster
Modelling and simulation of cell membrane dynamics.
The study of cell membrane dynamics is part of a more general study of cell behaviour. Understanding the dynamics of the cell membrane is vital as it plays a critical role in many biological processes such as wound healing, embryogenesis, immune response and pathological processes such as formation of primary and secondary tumours. We will show a model that is able to describe the shapes, expansions, contractions, protrusions and retraction motions of the cell.

Mike Giles (University of Oxford) Thursday 8th July 17:40
Multilevel Monte Carlo for elliptic SPDEs
(Joint work with Rob Scheichl and Aretha Teckentrup at Bath, and Andrew Cliffe at Nottingham.)
Elliptic SPDEs, in which the log-diffusivity is a stochastic field, arise in the modelling of oil reservoirs and nuclear waste repositories. In this talk I will discuss how the multilevel Monte Carlo method, which was recently developed for financial Monte Carlo applications, can be used in this context to efficiently estimate the expected value of various output functionals.

Andrew Gordon (University of Manchester) Poster
Solving stochastic collocation systems with algebraic multigrid
Stochastic collocation methods facilitate the numerical solution of partial differential equations (PDEs) with random data and give rise to long sequences of similar discrete linear systems. When elliptic PDEs with random diffusion coefficients are discretized with mixed finite element methods in the physical domain, the resulting collocation systems can be solved iteratively with the minimal residual (MINRES) method and algebraic multigrid (AMG) can be used as a key components for a highly robust preconditioner. When considered individually, the stochastic collocation systems are trivial to solve, however, the challenge lies in exploiting the systems' similarities to recycle information and minimize the cost of solving the entire sequence.

In this poster, we consider full tensor and sparse grid stochastic collocation schemes applied to a model stochastic elliptic problem and discretize in physical space using lowest order Raviart-Thomas mixed finite elements. We propose an efficient solver for the resulting sequence of linear systems and show, in particular, that it is feasible to use finely-tuned AMG preconditioning for each system if key set-up information is reused. Crucially, this preconditioning strategy is robust with respect to variations in the discretization and statistical parameters for both stochastically linear and nonlinear diffusion coefficients.

Viet Ha Hoang (NTU, Singapore) Wednesday 14th July 14:30
Sparse Tensor Galerkin Approximations for Parametric and Random Hyperbolic PDEs
(Joint with Christoph Schwab)
We consider stochastic wave equations whose coefficients depend on countably many random variables on [-1.1]. The problem is cast into the form of a parametric wave equation which depends on a countably many parameters. This equation is approximated by Galerkin projection onto polynomial spaces of finite dimensions in the parameter space. We establish uniform stability with respect to the support of the resulting coupled hyperbolic system, and analyticity of the solution with respect to the countably many parameters. We also establish regularity for the solution of the parametric deterministic system.

Thomas Hou (Caltech) Saturday 10th July 15:05
Model Reduction via a Multi-scale Random Basis Method
Uncertainty arises in many complex real-world problems of scientific and engineering interests. Many of these problems involve multiple scales in both space and time, which may vary in several orders. The presence of randomness further complicates multi-scale problems because its effect may span many scales and grow with time through nonlinear interactions. Earlier methods, such as the worst-case analysis, sensitivity analysis and etc, tend to be over-pessimistic or unreliable when the problems become complicated. Wiener Chaos Expansion methods and their variants proposed in recent years show some promising features but still suffer from the curse of dimensionality.

In this talk, we propose a multiscale stochastic method which consists of two parts, offline and online computations. In the offline computation, a set of nonlinear stochastic bases are constructed using the Karhunen-Loeve (KL) expansion and the Monte Carlo simulations. In the online computation, the stochatic solution is expanded in terms of the nonlinear stochastic bases constructed offline, resulting in a sparse representation of the stochastic solution. By solving a small set of coupled PDEs for the coefficients of the expansion, we obtain an efficient numerical method to compute the solution of SPDEs in the online step. We have applied this method to some elliptic problems with random coefficients. Our numerical results confirm that the proposed method indeed offers an efficient computational method for solving stochastic PDEs. It also provides an effective reduced model as a result of our method. Our method is semi-non-intrusive in the sense that certified legacy codes can be used with minimum changes in the offline computation.

Arieh Iserles (University of Cambridge) Monday 12th July 17:35
Asymptotic–numerical multiscale expansions
(Joint work with Marissa Condon and Alfredo Deanho)
In this talk we present an introduction to a methodology for the solution of ODEs, DAEs and DDEs with highly oscillatory forcing. The asymptotic expansion of such equations involves two hierarchies of scales, which are derived explicitly by solving non-oscillatory problems.

Patrick Jenny (ETH, Zürich) Tuesday 6th July 10:10
Transported probability density function (PDF) methods for multi-scale and uncertainty problems - part 1
An introduction to the basic ideas of PDF modeling with the necessary mathematical background is provided in this first part of the short course.

Further objective of the lecture is to identify potential targets which can benefit from this attractive approach, e.g. turbulent reactive flow.

Patrick Jenny (ETH, Zürich) Wednesday 7th July 10:10
Transported probability density function (PDF) methods for multi-scale and uncertainty problems - part 2
As an illustrative example it is shown how the PDF approach can be employed to model non-equilibrium gas flow.

This allows to emphasize strength and limitations and at the same time efficient solution methods are discussed

Patrick Jenny (ETH, Zürich) Friday 9th July 09:15
Transported probability density function (PDF) methods for multi-scale and uncertainty problems - part 3
It is shown how PDF modeling can be employed to deal with uncertainty in sub-surface transport.

On one hand it is explained how simple stochastic, microscopic "rules" lead to a closure at the Darcy scale, which is only possible by honoring arbitrary joint distributions and spatial correlations.

Second, a PDF approach to assess uncertainty of tracer transport is presented, which is not limited to small variances like e.g. perturbation methods.

Peter Jimack (University of Leeds) Friday 9th July 16:30
Numerical Models for the Simulation of Elastohydrodynamic Lubrication Problems
This talk will describe joint research undertaken as part of a collaboration between academia and industry that is funded as part of the EU FW6 Transfer of Knowledge scheme. The focus of the research is the efficient, accurate and reliable numerical simulation of lubricated contacts in which the applied load is sufficiently large to lead to elastic deformation in the contacting elements (hence elastohydrodynamic lubrication). One aspect of these problems that is of significant industrial importance is the behaviour of the lubricant and the contacting elements when their surfaces are not smooth. This roughness is often at a much smaller length-scale than the contact region and so poses significant computational challenges. The talk will provide an overview of the numerical techniques that may be used and present a selection of simulation results.

Jesper Karlsson (KAUST, Saudi Arabia) Poster
A Computable Weak Error Expansion for the Tau-Leap Method
This work develops novel error expansions with computable leading order terms for the global weak error in the tau-leap discretization of pure jump processes arising in kinetic Monte Carlo models. Accurate computable a posteriori error approximations are the basis for adaptive algorithms; a fundamental tool for numerical simulation of both deterministic and stochastic dynamical systems. These pure jump processes are simulated either by the tau-leap method, or by exact simulation, also referred to as dynamic Monte Carlo, the Gillespie algorithm or the Stochastic simulation algorithm. Two types of estimates are presented: an a priori estimate for the relative error that gives a comparison between the work for the two methods depending on the propensity regime, and an a posteriori estimate with computable leading order term.

Tatiana Kim (University of Bath) Poster
Hybrid numerical-asymptotic boundary integral method for solving high-frequency acoustic scattering problems.
In the paper [1] by Dominguez, Graham and Smyshlyaev, a numerical method is presented for solving high- frequency acoustic scattering problems in two dimensions, where the incident wave is a plane wave and the boundary of the scatterer is smooth and convex. The problem is formulated for the surface current, i.e. the normal derivative of the total wavefield on the boundary of the scatterer, using a combined potential boundary integral approach, that results in a one-dimensional boundary integral equation. A novel Galerkin scheme is proposed that incorporates known asymptotic behavior of the solution on the boundary into the approximation space and the error of this Galerkin discretization is obtained. The key feature of this hybrid method is that the degrees of freedom must grow only slightly greater than k1/9 in order to maintain the accuracy as k grows. However, the Galerkin discretization of the boundary integral equation, leads to a system of linear equations with coefficients that are highly-oscillatory double integrals, that, in practice, can not be computed exactly. We propose a novel numerical technique for computing these integrals with the number of quadrature points required to maintain the accuracy, independent of the wavenumber.

[1] V.Dominguez, I.G.Graham, V.P.Smyshlyaev, A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering. Numer. Math. 106 (2007), pp. 471-510.
[2] T.Kim, V.Dominguez, I.G.Graham, V.P.Smyshlyaev, Resent progress on hybrid numerical asymptotic boundary integral methods for high-frequency scattering problems, Proceedings of UKBIM7 (2009), pp. 15-23.

John King (University of Nottingham) Tuesday 6th July 17:00
Multiscale modelling of cell populations
Intercellular signalling processes in populations of biological cells can lead to neighbours adopting different fates. Homogenisation approaches suited to describing the tissue-scale properties of such discrete systems will be outlined, together with some of their implications.

Frances Kuo (University of New South Wales) Thursday 8th July 18:15
Lifting the curse of dimensionality - quasi Monte Carlo methods for high dimensional integration
High dimensional problems, that is, problems with a very large number of variables, are coming to play an ever more important role in applications. These include, for example, option pricing problems in mathematical finance, maximum likelihood problems in statistics, and porous flow problems in computational physics. High dimensional problems pose immense challenges for practical computation, because of a nearly inevitable tendency for the costs of computation to increase exponentially with dimension: this is the celebrated "curse of dimensionality". In this talk I will give an introduction to "quasi-Monte Carlo methods" for tackling high dimensional integrals, with a focus on "lattice rules", and discuss the challenges that we face while attempting to lift the curse of dimensionality.

Seong Lee (Chevron Energy Technology Company ) Friday 9th July 14:30
Adaptive Multiscale Finite Volume Method for Multiphase Flow in a Heterogeneous Reservoir
Recent advances in multiscale methods show great promise in efficiently simulating a high-resolution model for highly heterogeneous media. We propose numerical, adaptive prolongation and restriction operators of flow and transport equations that will greatly improve numerical efficiency over the conventional finite difference reservoir simulation. We also discuss iterative methods to control numerical errors in MSFV simulation and devise adaptive, numerical strategy that yields high computational efficiency within acceptable error tolerance.

Ben Leimkuhler (University of Edinburgh) Tuesday 6th July 17:35
Simplified modelling of energetic interactions using thermal baths, with application to a fluid vortex system.
Using the thermodynamic concept of a reservoir, we investigate a computational model for interaction with unresolved degrees of freedom (a thermal bath) [1]. We assume that a finite restricted system can be modelled by a generalized canonical ensemble, described by a density which is a smooth function of the energy of the restricted system. A generalized stochastic-dynamic thermostat [2] enables modelling of a restricted resolved dynamics embedded within a larger energetic bath, while leaving the desired equilibrium distribution invariant. To illustrate the method, we apply these techniques in the setting of a simplified point vortex flow on a disc, in which a modified Gibbs distribution (modelling a finite, rather than infinite, bath of weak vortices) provides a regularizing formulation for restricted system dynamics.

Although our method does not provide a proper dynamical closure, it is very straightforward to implement in a wide range of situations and can provide realistic averages. Numerical experiments, effectively replacing many vortices by a few artificial degrees of freedom, are in excellent agreement with the two-scale simulations that have appeared in the literature [3].

[1] Dubinkina, S., Frank, J. and Leimkuhler, B., Simplified modelling of energetic interactions with a thermal bath, with application to a fluid vortex system, preprint, 2010.
[2] Leimkuhler, B., Generalized Bulgac-Kusnezov methods for sampling of the Gibbs-Boltzmann measure, Physical Review E 026703, 2010.
[3] Bühler, O., Statistical mechanics of strong and weak point vortices in a cylinder, Physics of Fluids, 14, 2139-2149, 2001.

Qifeng Liao (University of Manchester) Poster
Effective Error Estimators for Low Order Elements
This poster focuses on a posteriori error estimation for (bi-)linear and (bi-)quadratic elements. At first, the simple diffusion problem is tested for introducing the methodology we adopted for doing error estimation, which is based on solving local Poisson problems. Next, this methodology is applied to dealing with classical mixed approximations of incompressible flow problems. Computational results suggest that our error estimators are cost-effective, both from the perspective of accurate estimation of the global error and for the purpose of selecting elements for refinement within a contemporary self-adaptive refinement algorithm.

Ping Lin (University of Dundee) Wednesday 7th July 17:35
Quasicontinuum methods for crystalline materials with simple and complex lattice structure
Many scientific systems such as materials may be modeled by a large number of particles (or atoms) where any particle interacting with any other through, for example, a pair potential energy. The equillibrium configuration is a minimizer of the total energy of the system. The computational cost is extremely high since the number of particles (or atoms) is usually huge. An approximate sparse representation of the system is necessary to reduce the computational cost. Recently in material research much attention has been paid to a so-called quasicontinuum (QC) approximation which may be seen as an approximate representation of the accurate atomistic model. We will study some QC methods for cyrstalline materials with simple and complex lattice structures and estimate the error of QC methods. Part of the talk is based on joint work with A Abdulle and A Shapeev.

Mitchell Luskin (University of Minnesota) Tuesday 6th July 11:40
Hybrid Atomistic-to-Continuum Coupling Methods
Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long ranged elastic fields with a much larger region that cannot be computed atomistically. Materials scientists have proposed many methods to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, a mathematical structure has been given to the description and formulation of atomistic-to-continuum coupling methods, and corresponding mathematical analysis has clarified the relation between the various methods and the sources of error.

I will present three tutorial lectures covering the relation between atomistic and continuum models and the formulation and analysis of coupling methods with a focus on the quasicontinuum method. The development of coupling methods for crystalline materials that are reliable and accurate for configurations near the onset of lattice instabilities such as dislocation formation has been particularly challenging. I will present theory developed with Matthew Dobson and Christoph Ortner to assess currently utilized methods and to propose more reliable, accurate, and efficient methods.

Mitchell Luskin (University of Minnesota) Wednesday 7th July 09:15
Hybrid Atomistic-to-Continuum Coupling Methods

Mitchell Luskin (University of Minnesota) Thursday 8th July 11:40
Hybrid Atomistic-to-Continuum Coupling Methods

Mitchell Luskin (University of Minnesota) Thursday 31st December :
Hybrid Atomistic-to-Continuum Coupling Methods

Roland Masson (Institute Francais du Petrole) Friday 9th July 12:15
Finite volume schemes for multiphase porous media flows
(Joint work with Leo Agelas, Daniele Di Pietro, Robert Eymard, Cindy Guichard, Roland)
This talk focuses on cell centered finite volume schemes with applications to multiphase porous media flows. We shall first motivate the choice of cell centered finite volume schemes for reservoir simulation, CO2 sequestration and basin modeling and enhance the difficulties to be overcome from the points of view of the models, of the geometry and of the properties of the porous media. Unfortunately there is not yet a cell centered finite volume scheme on practical meshes combining all the desired properties, mainly, linear fluxes, coercivity, compact stencil, parallelism, exactness on piecewise linear solutions for cellwise constant diffusion tensors, ... We will hence discuss the pros and cons for our applications of some finite volume schemes chosen among MPFA (MultiPoint Flux Approximation) schemes based on flux construction such as the O,L,G schemes and among SUSHI type schemes based on discrete variational formulations.

Markus Melenk (TU Vienna) Monday 12th July 11:40
Helmholtz problems at large wavenumbers
(Joint work with S. Sauter (Zurich) and M. Loehndorf (Vienna))
Time-harmonic wave propagation problems are often modelled with the Helmholtz equation.This setting arises, for example, in acoustic or electromagnetic scattering. When numerically solving Helmholtz problems, several issues arise, in particular in the case of large wavenumbers k.Firstly, a decision has to be made whether a volume-based method (such as FEM) is used or a boundary integral equation formulation (i.e., BEM) is employed. Secondly, given the highly oscillatory nature of the solution, it may be of interest to employ special, problem-adapted ansatz functions in the numerical method instead of the standard piecewise polynomial based ones. Such function could be obtained, for example, from asymptotic methods. Thirdly, besides the approximation properties of the ansatz spaces, the stability of the numerical method has to be considered.

This series of talks will survey several methods currently employed for Helmholtz problems. In particular, we will discuss methods that take the highly oscillatory nature of the solution into account. The primary focus of the talks, however, will be on recent results concerning the stability of discretizations. Here, we will restrict our attention to the setting of standard piecewise polynomial ansatz spaces. We discuss standard high order finite element discretizations (hp-FEM) as well high order boundary integral equation approaches (hp-BEM). In the latter case, we focus on the so-called Brakhage-Werner or Burton-Miller formulations. For both hp-FEM and hp-BEM, we show k-independent stability of the discretization under the following two assumptions:

  • (scale resolution condition):the mesh spacing h and the approximation order p satisfy the condition that kh/p is sufficiently small and p > C log k.
  • (well-posedness of the continuous problem): the solution operator for the continuous problem grows at most polynomially in the wavenumber k.
The stability analysis rests on suitably defined adjoint problems and how the solutions of these adjoint problems can be approximated from the ansatz spaces. Thus, the stability analysis is reduced to an approximation theoretic problem, which, in the present context of piecewise polynomial approximation, can in turn be answered by a suitable k-explicit regularity theory for Helmholtz problems. This regularity theory takes the form of an additive splitting of the solution into a part with finite Sobolev regularity and an analytic part. The essential point of the regularity theory is that the stability constants in the estimates for both parts can be controlled explicitly in the wavenumber k.

Markus Melenk (TU Vienna) Tuesday 13th July 09:15
Helmholtz problems at large wavenumbers

Markus Melenk (TU Vienna) Wednesday 14th July 10:10
Helmholtz problems at large wavenumbers

Ray Millward (University of Bath) Poster
A new adaptive multiscale finite element method with applications to high contrast interface problems.
This new adaptive multiscale method extends the work of Durlofsky, Efendiev and Ginting to introduce a multiscale method where the shape of the basis functions adapt to the underlying pde. The method avoids the need for technical local boundary conditions when performing local solves and still allows a coarse global solve. The method has been applied to high contrast interface problems where the loss of regularity at the interface reduces the rate of convergence, however, the adaptive method converges as if the interface wasn't there. The method also doesn't require the mesh to fit the interface and through the adaptive process the mesh remains constant. Examples will be presented which relate to structural optimization and the resulting high contrast problem that arises.

Peter Monk (University of Delaware) Monday 12th July 15:30
The solution of time harmonic wave equations using complete families of elementary solutions
This presentation is devoted to discussing plane wave methods for approximating the time-harmonic wave equation paying particular attention to the Ultra Weak Variational Formulation (UWVF). This method is essentially an upwind Discontinuous Galerkin (DG) method in which the approximating basis functions are special traces of solutions of the underlying wave equation. In the classical UWVF, due to Cessenat and Despres, sums of plane wave solutions are used element by element to approximate the global solution. For these basis functions, convergence analysis and considerable computational experience shows that, under mesh refinement, the method exhibits a high order of convergence depending on the number of plane wave used on each element. Convergence can also be achieved by increasing the number of basis functions on a fixed mesh (or a combination of the two strategies). However ill-conditioning arising from the plane wave basis can ultimately destroy convergence. This is particularly a problem near a reentrant corner where we expect to need to refine the mesh.

The presentation will start with a summary of the UWVF and some typical analytical and numerical results for the Hemholtz equation. It may be that different basis functions need to be used in different parts of the domain. I shall present some numerical results investigating convergence on an L-shaped domain using singular Bessel functions near the corner. An alternative, that also extends to 3D, is to use polynomial basis functions on small elements. Using mixed finite element methods, we can view the UWVF as a hybridization strategy and I shall also present theoretical and numerical results for this approach.

Although neither the Bessel function or the plane wave UWVF are free of dispersion error (pollution error) they can provide a method that can use large elements and small number of degrees of freedom per wavelength to approximate the solution. Extensions to Maxwell's equations and elasticity will be briefly discussed. Perhaps the main open problems are how to improve on the bi-conjugate gradient method that is currently used to solve the linear system, and how to adaptively refine the approximation scheme.

Frédéric Nataf (Université Pierre et Marie Curie) Tuesday 13th July 16:45
Coarse grid correction for domain decomposition methods for problems with high heterogeneities.
We present an automatic construction of an adapted coarse grid for problems with highly discontinuous coefficients. The method is very robust with respect to the size of the jumps and the decomposition (automatic or manual partitioner).

Richard Norton (University of Oxford) Poster
Evolution of Microstructure
A simple model problem for the emergence and evolution of microstructure based on a double well potential is considered. Analytical issues of interest are the existence and stability of rest points while numerical issues include how the error analysis of FEM depends on a regularization parameter.

Jill Ogilvy (BAE Systems (Operations) Limited) Friday 9th July 11:45
Modelling electromagnetic performance of large structures
The talk will provide an overview of some of the interests and capabilities of BAE SYSTEMS in the modelling of electromagnetic phenomena for radar applications. General methods will be outlined, together with some examples of model predictions. Some implications for multi-scale modelling will be addressed.

Christoph Ortner (University of Oxford) Wednesday 7th July 18:10
Atomistic/continuum coupling schemes for solids.
Low energy equilibria of crystalline materials are typically characterised by localized defects that interact with their environment through long-range elastic fields. By coupling atomistic models of the defects with continuum models for the elastic far field one can, in principle, obtain models with near-atomistic accuracy at significantly reduced computational cost. However, several pitfalls need to be overcome to find a reliable coupling mechanism. In this talk I will discuss some selected possible mechanisms and their analysis.

Tim Payne (Met Office UK) Friday 9th July 15:00
The assimilation of data into atmospheric models, and the use of linearisations optimised for finite perturbations.
Atmospheric forecast models attempt to represent spatial scales from tens to millions of metres, and temporal scales from seconds to many hours.

All major weather prediction centres currently assimilate data into their forecasting model by four dimensional variational data assimilation. This finds the atmospheric state (or "analysis") which "best fits" both the prior information (or "background", a short forecast from a previous analysis) and recent observations. It does this by minimising a cost function which simultaneously penalises the departure of the analysis from the background, and the departures of the forecast from the analysis to the observations distributed in time. To make the problem manageable the latter is done using a linear model which predicts the evolution of the analysis-background increments to the observation times.

Conventionally this linear model is taken to be the first derivative of the forecast model, which is appropriate for infinitesimal increments but is a poor predictor of the true evolution of finite-sized increments. In this talk we show that if the pdf of the increments is known we may construct better linearisations, and show how the use of this type of linearisation can improve the assimilation of data and thereby the model forecast.

Gibin Powathil (University of Dundee) Poster
Modelling the Spatial Distribution of Chronic Tumour Hypoxia: Implications for Experimental and Clinical Studies
Tumour hypoxia (i.e. a lack of oxygen) is considered to be an important prognostic factor in tumor progression, possibly affecting the aggressiveness of tumours as well as the metastatic and invasive potential of cancer cells. It is usually measured by direct (invasive) measurements of tumour oxygenation tension using needle electrodes or through quantification of intrinsic or extrinsic biomarkers. An alternative approach to estimate tumour hypoxia is through theoretical computational simulations that incorporate knowledge of various measurable parameters supplemented by non-invasive imaging of tumour vasculature. The method developed here illustrates an alternative way to estimate tumour hypoxia and provides guidance in planning accurate and effective therapeutic strategies and invasive estimation techniques.

The main purpose of this study is to model and quantify hypoxia using a known spatial distribution of tumour vasculature, obtained through available imaging techniques, and study the effects of hypoxia on radiation response. The results of theoretical analysis of estimated hypoxia, quantified by finding the percentage of area and through the electrode sampling method, show a reasonable agreement with biomarker stained hypoxic proportions obtained through biopsies. In addition, the estimated hypoxic proportions are used to study the effect of radiation, by using a modified linear quadratic model.

Catherine Powell (University of Manchester) Thursday 8th July 17:05
Recycling Techniques for Solving Stochastic Collocation Systems
Recently there have been attempts to compare the computational costs of solving elliptic PDEs with random coefficients via stochastic Galerkin and stochastic collocation techniques. Fair computational comparisons can only be made if the best possible solvers are used for the linear systems in question. In the case of stochastic collocation, the linear systems are not only decoupled (which is seen as the major advantage) but also (for some model problems) highly similar. In this talk, we discuss a number of ways in which this similarity can be exploited to gain computational savings.

Olof Runborg (KTH Stockholm) Wednesday 7th July 17:00
A Multiscale Method for the Wave Equation in Heterogeneous Medium
We consider the wave equation in a medium with a rapidly varying speed of propagation. We construct a multiscale scheme based on the heterogeneous multiscale method, which can compute the correct coarse behavior of wave pulses traveling in the medium, at a computational cost essentially independent of the size of the small scale variations. This is verified by theoretical results and numerical examples. We also consider the case when waves travel over long time in heterogeneous medium, where dispersion effects are introduced which are not captured by standard homogenization.

Ruth Sabariego (University of Liege) Poster
Multiscale computational modelling in electromagnetism
Almost all problems in science and engineering are multiscale (in space and/or time) and multiphysical. The interactions in the microscale may significantly influence the solution of the macroscale problem, and should not be disregarded in the numerical model. However, resolving the full problem at the microscopic level with classical numerical methods is prohibitively expensive if not impossible. Dedicated multiscale techniques that take advantage of the separation of scales prove indispensable. Material synthesis is an example of an emerging technology that urgently needs efficient multiscale methods for numerically determining the effective properties of engineered materials, i.e. their constitutive law. Artificially tailored materials exhibit exceptional macroscopic properties that are directly linked to their microstructural complexity. The ability to simulate numerically the properties of novel materials is an invaluable help for design optimization and can avoid expensive and time-consuming trial and error tests. Our team aims at developing efficient numerical techniques for solving multiscale electromagnetic problems. The developments already successfully applied in mechanical and thermal analyses will be adapted to the particularities of electromagnetism. For electrostatic, magnetostatic or magnetodynamic problems, the electromagnetic multiscale methods will be related to the techniques recently proposed for heat transfer problems. An important and open challenge lies in solving full wave problems, in particular in the presence of internal resonances.

Marcus Sarkis (Worcester Polytechnic Institute ) Tuesday 13th July 15:30
Infinite-dimensional stochastic Darcy equations, finite-dimensional Petrov-Galerkin approximations and a priori error estimates.
(Joint work with Juan Galvis (Texas AM))
In this talk we consider a stochastic Darcy's pressure equation with random log-normal permeability and random right-hand side. To accommodate the lack of ellipticity and continuity, and singular right-hand sides, we introduce an appropriate representation of the permeability stochastic fields and infinite-dimensional norms and spaces. We then introduce new continuous and discrete weak formulations based on a Petrov-Galerkin strategy and present inf-sup conditions, well-posedness, a priori error estimations and numerical experiments.

Daniela Schlueter (University of Dundee) Poster
Multi-scale mathematical modelling of cancer cell invasion: The role of cell-cell and cell-matrix adhesion

‘Cancer’ is an umbrella term for about 200 different diseases and with its diversity it is one of the main causes of death in the world. The malignancy of almost all types of tumours is determined by the ability of cancer cells to invade the surrounding tissues and then to form secondary tumours (metastases) at distant sites in the body. These metastases are responsible for ~90% of cancer deaths. In order to advance in cancer treatment strategies, it is therefore of high importance to understand the processes involved in cancer cell invasion. A crucial aspect of cancer cell invasion is the role of cell adhesion, both cell-cell and cell-matrix.

We focus on understanding the first steps leading to cancer invasion and try to identify key processes that allow the detachment of individual cells or small cell clusters from the main tumour mass and their local invasion. For this we use an individual force-based multi-scale approach to model physical properties of the cells and intra- and inter-cellular protein pathways involved in tumour growth, cell-cell and cell-matrix adhesion. The key pathways include those of E-cadherin and beta-catenin.

Using computational simulations, with our model we can investigate the spatio-temporal distribution of E-cadherin and beta-catenin levels in individual cancer cells and predict what implications this has for the adhesion of the cancer cells to each other and to the extracellular matrix. By examining the cell-matrix interactions with our model we can also highlight the importance of the microenvironment in tumour progression and how cell-matrix interactions can lead to more aggressive tumours.

Christoph Schwab (ETH Zürich) Saturday 10th July 11:40
Sparse Tensor Discretizations of PDEs with stochastic and multiscale data
We report on recent work on the numerical analysis of several discretization schemes for PDEs with random inputs.

The three lectures will address
1. Sparse adaptive tensor FEM for elliptic and parabolic PDEs with random loadings.
2. Sparse adaptive gpc FEM for elliptic and parabolic PDEs with random coefficients.
3. Sparse adaptive tensor FEM for elliptic problems with multiple scales and random coefficients.

We shall survey recent mathematical results and algorithmic developments on the numerical analysis of deterministic and adaptive sparse tensor discretizations of PDEs with random inputs.

We review in 1. and 2. recent, sharp mathematical results on the convergence rates of MC methods as well as of adaptive spectral methods of "generalized polynomial chaos" type for these problems and derive guidelines for implementation and complexity bounds. In 3., we apply sparse tensor techniques to obtain efficient solutions of k-scale elliptic homogenization problems, possibly with random coefficients by combining with 1. and 2.

Christoph Schwab (ETH Zürich) Monday 12th July 09:15
Sparse Tensor Discretizations of PDEs with stochastic and multiscale data

Christoph Schwab (ETH Zürich) Tuesday 13th July 11:40
Sparse Tensor Discretizations of PDEs with stochastic and multiscale data

Andrew Stuart (Warwick University) Tuesday 6th July 09:15
Multiscale modelling and inverse problems
Joint work with Greg Pavliotis, Imperial College

The need to blend observational data and mathematical models arises in many applications and leads naturally to inverse problems. Parameters which are functions, such as constitutive tensors, initial conditions and forcing can be estimated on the basis of observed data. The resulting inverse problems are often ill-posed and some form of regularization is required. When the function being estimated has a multiscale structure a number of natural questions arise, in particular: (i) how should the data be used, and the regularization chosen, if only an averaged or homogenized solution to the inverse problem is required; (ii) how should the data be used, and the regularization chosen, if the details of the multiscale structure are important. We will devote three lectures to a development of the mathematics required to address these questions. We adopt a probabilistic approach to the inverse problems, based on the Bayesian viewpoint, and show how the choice of prior measure is intimately related to answering questions (i) and (ii). The ideas will be illustrated throughout in the context of simple models for groundwater flow.

The lectures will be pedagogical in style and accesible to an audience with basic knowledge of differential equations and probability.

Background material on the Bayesian approach to inverse problems can be found in:

Inverse Problems: A Bayesian Perspective AM Stuart, Acta Numerica 19 (2010).

Background material on multiscale methodology can be found in:

Multiscale Methods: Averaging and Homogenization, GA Pavliotis and AM Stuart, Springer-Verlag, 2008. An excerpt from the book may be found at: Multiscale Methods: Averaging and Homogenization.

Andrew Stuart (Warwick University) Wednesday 7th July 11:40
Multiscale modelling and inverse problems

Andrew Stuart (Warwick University) Thursday 8th July 10:10
Multiscale modelling and inverse problems

Marc Sturrock (University of Dundee) Poster
Mathematical modelling of the p53-mdm2 oscillatory system
(Joint work with A.J. Terry, D.P. Xirodimas, A.M. Thompson and M.A.J. Chaplain)
The p53 network is arguably the most important pathway involved in preventing the initiation of cancer. The p53 transcription factor is responsible for the regulation of DNA repair, cellular senescence and apoptosis. Mutations that inactivate p53 function have been detected in more than 50% of human cancers and even tumours with wild type p53 have defects in upstream regulators or downstream effectors of p53. A vital negative regulator of p53 function in cells is the Mdm2 oncogene product. Mdm2 protein enhances p53 degradation in both the nucleus and cytoplasm via ubiquitination. Mdm2 is also a target gene for p53. This creates a negative feedback loop which provides tight regulation of p53 function in cells. Experiments have been performed to measure the dynamics of fluorescently tagged p53 and Mdm2 over several days in individual living cells. Some cells exhibited undamped oscillations for at least 3 days (more than 10 peaks).

Building on previous mathematical modelling approaches, we derive a system of partial differential equations (PDEs) to capture the evolution in space and time of the concentrations of variables in the p53-Mdm2 system. Through computational simulations we show that our reaction-diffusion model is able to produce sustained oscillations both spatially and temporally, reflecting experimental evidence well and providing further insight than previous models. The simulations of our models also allow us to calculate a diffusion coefficient range for which the model exhibits oscillatory dynamics.

Endre Süli (University of Oxford) Wednesday 14th July 11:40
Existence, equilibration and approximation of global weak solutions to kinetic models of dilute polymers
We establish the existence of global-in-time weak solutions to a general class of coupled microscopic-macroscopic FENE-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two and three space dimensions, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side of the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational.

With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian, we prove the existence of global-in-time weak solutions to the coupled Navier-Stokes-Fokker-Planck system, satisfying the initial condition, such that the velocity belongs to the classical Leray space and the probability density function has bounded relative entropy and square integrable Fisher information over any time interval. The key analytical tool in our proof is Dubinskii's compactness theorem in seminormed sets. It is also shown using the Csisza´r-Kullback inequality that, in the absence of a body force, the global weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient. We also discuss briefly computational difficulties associated with the numerical approximation of the high-dimensional Fokker-Planck equation with unbounded drift featuring in the model.

The talk is based on joint work with John W. Barrett (Department of Mathematics, Imperial College London).

Aretha Teckentrup (University of Bath) Poster
Multilevel Monte Carlo methods for elliptic PDEs with random coefficients
When solving partial differential equations (PDEs) with random coefficients numerically, one is usually interested in finding the expected value of a certain statistic of the solution. A common way to obtain estimates is to use Monte Carlo methods combined with spatial discretisations of the PDE on sufficiently fine grids. However, standard Monte Carlo methods have a rather slow rate of convergence with respect to the number of samples used, and individual samples of the solution are usually costly to compute numerically. In this talk we introduce the multilevel Monte Carlo method, with the aim of achieving the same accuracy of standard Monte Carlo at a much lower computational cost. The method exploits the linearity of expectation, by expressing the quantity of interest on a fine spatial grid in terms of the same quantity on a coarser grid and some “correction” terms. It has been extensively studied in the context of stochastic differential equations in the area of financial mathematics by Mike Giles and co-authors. We will give an outline of the method applied to elliptic PDEs with random coefficients, and also show some numerical results on the reduction of the computational cost resulting from it. The efficiency of the multilevel method is assessed by comparing it to standard Monte Carlo.

Raul Tempone (KAUST, Saudi Arabia) Thursday 8th July 16:30
Towards automatic global error control: computable weak error expansion for the tau-leap method
This work develops novel error expansions with computable leading order terms for the global weak error in the tau-leap discretization of pure jump processes arising in kinetic Monte Carlo models. Accurate computable a posteriori error approximations are the basis for adaptive algorithms; a fundamental tool for numerical simulation of both deterministic and stochastic dynamical systems. These pure jump processes are simulated either by the tau-leap method, or by exact simulation, also referred to as dynamic Monte Carlo, the Gillespie algorithm or the Stochastic simulation algorithm. Two types of estimates are presented: an emph{a priori} estimate for the relative error that gives a comparison between the work for the two methods depending on the propensity regime, and an emph{a posteriori} estimate with computable leading order term. Numerical examples show good agreement with the theory.

Alan Terry (University of Dundee) Poster
A mathematical model of the NF-kB negative feedback loop
Many stressful, inflammatory, and innate immune responses are regulated by the NF-kB signal transduction pathway. Deregulation of this pathway has been observed in numerous types of human cancer. A negative feedback loop is central to the mechanism by which NF-kB proteins signal. Experiments have shown that this feedback loop can cause oscillations in NF-kB activity. Various target genes of NF-kB are only transcribed after a certain number of such oscillations have occurred. Here we describe the mechanism by which the NF-kB pathway functions and we capture its essence in a mathematical model. Simulations of our model are consistent with experimental results in that they demonstrate oscillatory dynamics. Non-dimensionalisation of our model allows us to estimate the diffusion rates for the key proteins in the NF-kB pathway. Given that we can estimate these diffusion rates, we hope that experimentalists will feel inspired to measure them.

Dumitru Trucu (University of Dundee) Poster
The bio-heat equation, which is widely accepted as a mathematical model that describes the heat transfer process within the human body tissue, has important applications in many biomedical investigations. Among other theoretical aspects that we are concerned with regard to this equation, the perfusion coefficient, denoted with Pf, receives a particularly important interest because of its physical meaning, as the rate at which a unit of blood travels through a unit of tissue in a unit of time. Our analysis is placed in the non-steady state case and is focused on the inverse problems that are concerned with the perfusion coefficient, when Pf is considered as being either constant, time-dependent, space- and time- dependent, or temperature- dependent. This inverse analysis allows us to accurately recover of the perfusion information from temperature and heat flux measurements taken in minimally invasive regions.

Richard Tsai (University of Texas - Austin) Tuesday 6th July 18:10
A continuum-pore scale coupling algorithm for flow in porous media
(Joint work with Bjorn Engquist, Jay Chu, and Masa Prodonovic.)
We present our results in simulating flows in porous media by a coupled continuum and network model. In reservoir simulations, total flow rate and the oil cut are import macroscopic qualities. They can be computed by knowing macroscopic pressure and saturation. Typical simulations assume that the flux is a linear function of the pressure gradient whose coefficient stays constant. However, in reality, there is no general way to determine the effective flux or the permeability field at the macroscopic level. In particular, these functions may be nonlinear function of the macroscopic quantities. We propose to use some detailed microscopic network models to estimate the macroscopic flux. Furthermore, the macro-quantities (macroscopic pressure, velocity, or flux) from macroscopic simulations are used to determine whether updating the microscopic configurations is needed, e.g. the opening of the throats or growing of the fractures. The updated microscopic configurations then are used to update our estimate on the flux at the macroscopic level.

Grigory Vilensky (University College London) Friday 9th July 16:00
Mathematical modelling of anomalous absorption of ultrasound in human tissue
The work examines possible formulations of the problem of nonlinear ultrasound wave propagation in soft biological tissue from the standpoint of high intensity focused ultrasound applications for medical treatment.

The proposed work has been written with the needs of the applied mathematical and modelling communities in mind and brings together essential information about the available experimental results, complementing these with the practical approach to modelling. It aims to provide the means for theoretical understanding of the physics of the interaction of ultrasound with tissue.

Central to this is the problem of anomalous absorption of sound energy by tissue. It occurs as a result of excitation of internal degrees of freedom of molecular motion by the sound field known in the literature as molecular relaxation processes. The work discusses the phenomenological theory of anomalous absorption of ultrasound in tissue and also proposes a statistical model to describe the underlying physical mechanism.

The proposed model treats the phenomenon as a mixture of random variables, each of these characterised by its own probability density function. In its main features the theory is similar to the related traditional methodologies used in reliability theory, [1], and statistical radio physics, [2].

1. Gnedenko, B.V., Beliaev, Yu.K, Soloviev, A.D. 1965 Mathematical Methods of reliability Theory. Moscow. “Nauka”. 524 p.
2. Rytov, S.M. 1976 Introduction Into Statistical Radiophysics. Pt. 1. Random Processes. GRFML “Nauka”. 494 p.

Holger Wendland (University of Oxford) Wednesday 14th July 15:05
Multiscale Radial Basis Functions
We study a multiscale scheme for the approximation of Sobolev functions on bounded domains. Our method employs scattered data sites and compactly supported radial basis functions of varying support radii at scattered data sites. The actual multiscale approximation is constructed by a sequence of residual corrections, where different support radii are employed to accommodate different scales. Convergence theorems for the scheme are proven, and it is shown that the condition numbers of the linear systems at each level are independent of the level, thereby establishing for the first time a mathematical theory for multiscale approximation with scaled versions of a single compactly supported radial basis function at scattered data points on a bounded domain. This work is based upon earlier work with Ian Sloan and Thong Le Gia (University of New South Wales, Australia)

Xiao-Hui Wu (ExxxonMobil) Tuesday 13th July 17:55
Approaches toward Reliable Reservoir Performance Prediction
A survey of emerging approaches toward reliable reservoir performance prediction will be presented.

Ludmil Zikatanov (Penn State) Tuesday 13th July 17:20
Decompositions of Discontinuous Galerkin Finite Element Spaces and Preconditioning
(Based on joint works with Blanca Ayuso de Dios from Centre de Recerca Matematica (CRM), Spain)
We introduce a natural decomposition of the discontinuous Galerkin Finite Element spaces. For the lowest order case this decomposition is a direct sum of of the Crouzeix-Raviart non-conforming finite element space and a subspace that contains functions discontinuous at interior faces. We will also indicate how to construct such decompositions for higher order elements. Based on these decompositions we develop iterative and preconditioning techniques for the solution of the linear systems resulting from several discontinuous Galerkin (DG) Interior Penalty (IP) discretizations of elliptic problems. We analyze the convergence properties of these algorithms for both symmetric and non-symmetric IP schemes. We also present numerical examples confirming the theoretical results. Further extension to problems with jumps in the coefficients will also be discussed.