References:

[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).

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.

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.

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.

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.

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.

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.

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

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

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.

References:
[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.

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.

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.

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 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.

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.

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.

‘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.

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.

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.

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.

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).

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].

References
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.