Model Order Reduction

Lecture 1: We will cover the construction of POD reduced order models for nonlinear PDE systems. The approximation of the nonlinearities is performed with DEIM and/or QDEIM. Emphasis will be taken on the choice of the inner product for the basis construction and the treatment of spatially adaptively generated snapshots. Furthermore, error analysis will be sketched.

In this basic introductory lecture we are concerned with a class of model reduction techniques for parametric partial differential equations, the so-called Reduced Basis (RB) methods. These allow to obtain low-dimensional parametric models for various complex applications, enabling accurate and rapid numerical simulations. Important aspects are output estimation, basis generation and certification of the simulation results by reliable and efficient a posteriori error control. The latter also requires accurate estimation of stability constants. The main terminology, ideas and assumptions will be explained for the case of linear stationary elliptic, as well as nonlinear problems. Reproducable experiments will illustrate the theoretical findings. We will mainly follow the first part of a recent book chapter: Haasdonk, B.: Reduced Basis Methods for Parametrized PDEs - A Tutorial Introduction for Stationary and Instationary Problems, IANS, University of Stuttgart, Germany, 2014. To appear in P. Benner, A. Cohen, M. Ohlberger, K. Willcox: "Model Reduction and Approximation: Theory and Algorithms", SIAM, Philadelphia, 2017. The revised version (in a non-SIAM format) is available online as SimTech preprint: http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=938

In recent years, model order reduction has been recognized to be a powerful tool in analysis, simulation and optimization of complex technical processes in many applications including integrated circuit design, microsystem technology and computational fluid dynamics. A general idea of model reduction is to approximate a large-scale dynamical system by a reduced-order model that captures the input-output behavior of the original system to a required accuracy and also preserves essential physical properties. In this talk, we will consider model reduction of linear time-invariant systems using balanced truncation techniques. These techniques guarantee the preservation of system properties like stability, passivity and contractivity and provide computable error bounds. We will review the recent developments on numerical methods for large-scale matrix equations that play a crucial role in balanced truncation. An extension of balancing-related model reduction methods to differential-algebraic equations, second-order systems and coupled systems will also be discussed. The performance of these methods will be demonstrated on several problems from different application areas.

A major challenge in the application of sampling methods in Bayesian inverse problems is the typically large computational cost associated with solving the forward problem. To overcome this issue, we consider using a Gaussian process emulator to approximate the forward map. This results in an approximation to the solution of the Bayesian inverse problem, and more precisely in an approximate posterior distribution. In this talk, we analyse the error in the approximate posterior distribution, and show that the approximate posterior distribution tends to the true posterior as the accuracy of the Gaussian process emulator increases.

In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the more established Newton-Kleinman iteration. In spite of convincing numerical experiments, a systematic matrix analysis of this class of methods is still lacking. We derive new relations for the approximate solution, the residual and the error matrices, giving new insights into the role of the matrix $A-BB^*X$ and of its approximations in the numerical procedure and in the model reduction. In the context of linear-quadratic regulator problems, we show that the Riccati approximate solution is related to the optimal value of the reduced cost functional, thus completely justifying the projection method from a model order reduction point of view.

Eigenvalue analysis is a well-established tool for stability analysis of dynamical systems. However, there are situations where eigenvalues miss some important features of physical models. For example, in models of incompressible fluid dynamics, there are examples where linear stability analysis predicts stability but transient simulations exhibit significant growth of infinitesimal perturbations. In this study, we show that an approach similar to pseudo-spectral analysis can be performed inexpensively using stochastic collocation methods and the results can be used to provide quantitive information about the nature and probability of instability.

We will discuss model reduction (and data-driven reduced-order modeling) using rational interpolation. We will first introduce interpolatory model reduction via projection for linear dynamical systems using the concept of transfer function, followed by a brief summary of rational interpolation from data. Then, we will illustrate how to choose the interpolation points to satisfy a system-theoretic optimality criterion, leading to optimal $H_2$ approximation via Iterative Rational Krylov Algorithm. The recent advances in this direction will be discussed. Finally, we will show how these concepts can be extended to structured linear dynamical systems and to nonlinear dynamical systems. The theory will be illustrated via various numerical examples.

In recent years data-driven science developments have become important. This holds also for model reduction of linear and nonlinear systems. In this talk we will discuss a versatile tool for addressing such problems, namely, the Loewner framework. The presentation will be accompanied by several linear and nonlinear examples.

In this basic introductory lecture we are concerned with a class of model reduction techniques for parametric partial differential equations, the so-called Reduced Basis (RB) methods. These allow to obtain low-dimensional parametric models for various complex applications, enabling accurate and rapid numerical simulations. Important aspects are output estimation, basis generation and certification of the simulation results by reliable and efficient a posteriori error control. The latter also requires accurate estimation of stability constants. The main terminology, ideas and assumptions will be explained for the case of linear stationary elliptic, as well as nonlinear problems. Reproducable experiments will illustrate the theoretical findings. We will mainly follow the first part of a recent book chapter: Haasdonk, B.: Reduced Basis Methods for Parametrized PDEs - A Tutorial Introduction for Stationary and Instationary Problems, IANS, University of Stuttgart, Germany, 2014. To appear in P. Benner, A. Cohen, M. Ohlberger, K. Willcox: "Model Reduction and Approximation: Theory and Algorithms", SIAM, Philadelphia, 2017. The revised version (in a non-SIAM format) is available online as SimTech preprint: http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=938

Lecture 2: This talk deals with the use of POD models in optimization with PDE constraints. Emphasis is put on the variational discretization of the controls, which is perfectly tailored to the use of POD models for the state approximation. In addition we introduce a new snapshot location procedure for POD in optimal control. A priori and a posteriori error analysis will be sketched.

We will discuss model reduction (and data-driven reduced-order modeling) using rational interpolation. We will first introduce interpolatory model reduction via projection for linear dynamical systems using the concept of transfer function, followed by a brief summary of rational interpolation from data. Then, we will illustrate how to choose the interpolation points to satisfy a system-theoretic optimality criterion, leading to optimal $H_2$ approximation via Iterative Rational Krylov Algorithm. The recent advances in this direction will be discussed. Finally, we will show how these concepts can be extended to structured linear dynamical systems and to nonlinear dynamical systems. The theory will be illustrated via various numerical examples.

In recent years, model order reduction has been recognized to be a powerful tool in analysis, simulation and optimization of complex technical processes in many applications including integrated circuit design, microsystem technology and computational fluid dynamics. A general idea of model reduction is to approximate a large-scale dynamical system by a reduced-order model that captures the input-output behavior of the original system to a required accuracy and also preserves essential physical properties. In this talk, we will consider model reduction of linear time-invariant systems using balanced truncation techniques. These techniques guarantee the preservation of system properties like stability, passivity and contractivity and provide computable error bounds. We will review the recent developments on numerical methods for large-scale matrix equations that play a crucial role in balanced truncation. An extension of balancing-related model reduction methods to differential-algebraic equations, second-order systems and coupled systems will also be discussed. The performance of these methods will be demonstrated on several problems from different application areas.

Starting with a linear system we explore the creation of a surrogate model as a combination of reduced order modelling techniques, data assimilation and dynamic mode decomposition.

We will explain why Krylov methods (moment matching) are popular methods for acoustics and vibrations. We will show how these fit in a class of design optimization problems and problems with frequency dependent system matrices.

We will discuss model reduction (and data-driven reduced-order modeling) using rational interpolation. We will first introduce interpolatory model reduction via projection for linear dynamical systems using the concept of transfer function, followed by a brief summary of rational interpolation from data. Then, we will illustrate how to choose the interpolation points to satisfy a system-theoretic optimality criterion, leading to optimal $H_2$ approximation via Iterative Rational Krylov Algorithm. The recent advances in this direction will be discussed. Finally, we will show how these concepts can be extended to structured linear dynamical systems and to nonlinear dynamical systems. The theory will be illustrated via various numerical examples.

In recent years, model order reduction has been recognized to be a powerful tool in analysis, simulation and optimization of complex technical processes in many applications including integrated circuit design, microsystem technology and computational fluid dynamics. A general idea of model reduction is to approximate a large-scale dynamical system by a reduced-order model that captures the input-output behavior of the original system to a required accuracy and also preserves essential physical properties. In this talk, we will consider model reduction of linear time-invariant systems using balanced truncation techniques. These techniques guarantee the preservation of system properties like stability, passivity and contractivity and provide computable error bounds. We will review the recent developments on numerical methods for large-scale matrix equations that play a crucial role in balanced truncation. An extension of balancing-related model reduction methods to differential-algebraic equations, second-order systems and coupled systems will also be discussed. The performance of these methods will be demonstrated on several problems from different application areas.

Lecture 3: We propose a simulation-based model order reduction (MOR) approach for nonlinear PDE systems in networks. We show how proper orthogonal decomposition (POD) can be used to reduce the dimension of the PDE systems if snapshots of the full network simulation are available. Furthermore we discuss residual-based sampling combined with the MIT greedy approach to adaptively construct POD models which are valid over certain parameter ranges. As application we consider parametric MOR of semiconductors in integrated circuits with frequency as parameter, and if time allows also POD-MOR for robust optimal control of induction machines.

In this basic introductory lecture we are concerned with a class of model reduction techniques for parametric partial differential equations, the so-called Reduced Basis (RB) methods. These allow to obtain low-dimensional parametric models for various complex applications, enabling accurate and rapid numerical simulations. Important aspects are output estimation, basis generation and certification of the simulation results by reliable and efficient a posteriori error control. The latter also requires accurate estimation of stability constants. The main terminology, ideas and assumptions will be explained for the case of linear stationary elliptic, as well as nonlinear problems. Reproducable experiments will illustrate the theoretical findings. We will mainly follow the first part of a recent book chapter: Haasdonk, B.: Reduced Basis Methods for Parametrized PDEs - A Tutorial Introduction for Stationary and Instationary Problems, IANS, University of Stuttgart, Germany, 2014. To appear in P. Benner, A. Cohen, M. Ohlberger, K. Willcox: "Model Reduction and Approximation: Theory and Algorithms", SIAM, Philadelphia, 2017. The revised version (in a non-SIAM format) is available online as SimTech preprint: http://www.simtech.uni-stuttgart.de/publikationen/prints.php?ID=938

Parabolic problems are nowadays well-understood in terms of model reduction. This situation changes in case of transport-dominated of wave-type problems. We introduce possible variational formulations allowing for sharp error bounds for the RBM.

We will give an overview of MOR from an industrial point of view. By discussing requirements and context that are typical for industrial software, and with numerical results, we will explain why certain methods are more successfully adopted than others. Furthermore, we will present a number of open industrial MOR challenges.

One important issue in the development of complex technical system is the usage of rapid to evaluate substructures/ surrogate models for system level simulations. For safety critical simulations it is important to know the error introduced by model order reduction (MOR), used to create the surrogate models, to trust the simulations. Typically, a-priori error estimates, e.g., the sum of neglected singular values in balanced truncation, are used. They deliver upper bounds for the error, independent of the applied excitation. Whereas, the error estimates from the reduced basis community deliver a-posteriori error bounds, which account for the current excitation. Those error bounds use the residual between the reduced and original model to deliver the error bound. In this work, we apply the error bound to a simple elastic multibody system (EMBS) described in the floating frame of reference formulation. For the simulation of elastic bodies in a multibody system framework MOR is essential. Furthermore, the MOR procedure is made in a modular fashion. First, only the linear second order FE model is reduced to calculate the ansatz spaces used to calculate the nonlinear ODE which describes one elastic body expressed in the floating frame of reference formulation. In the final step, multiple reduced elastic bodies are assembled into one large elastic multibody system. Our error estimation strategy follows the same modular approach: First we estimate the error due to the approximation of the linear elasticity. In the next step, the solution data from the solve of the reduced EMBS body is used to approximate the error of the nonlinear ODE. We show the results by applying the error estimator to a two-link flexible manipulator in 2-D and 3-D.

We describe a variational method to assimilate multiple data streams into the terrestrial ecosystem carbon cycle model DALEC2. Ecological and dynamical constraints are employed to constrain unresolved components of an otherwise ill-posed problem. Using an adjoint method we study a linear approximation of the inverse problem: firstly we perform a sensitivity analysis of the different outputs under consideration, and secondly we use the concept of resolution matrices to diagnose the nature of the ill-posedness and evaluate regularisation strategies based on model reduction.

The inverse problem of data assimilation (DA) can be treated either by sequential techniques based on optimal filtering methods or by variational techniques that aim to solve an optimization problem subject to a set of dynamical system equations. For the very large inverse problems arising in the prediction of atmosphere and ocean circulations and other environmental systems, these methods are too costly to be used for real-time forecasting and approximations are needed for computational efficiency. In this talk we will give a brief overview of data assimilation methods currently applied in practice, describe the use of model reduction techniques within DA and discuss some current research issues.

Numerically simulating the Earths weather and climate poses a number of unique problems. Primarily atmospheric modelling is an all scales problem, from large planetary waves and features such as the Indian monsoon through meso-scale phenomena such as tropical cyclones down to the dynamics of a single cloud and the microphysics of precipitation. There is no spectral gap and all scales must be represented in the model. Compounding this, in order to produce a timely weather forecast, each simulation has only short time window in which to be run and so any model needs a careful balance of complexity, accuracy and performance. This talk will discuss some of the issues and approaches in developing numerical methods for atmospheric modelling including the hierarchy of approximations and reductions to simpler models that are made to study particular features of interest such as wave propagation and adjustment towards balance.

In this talk, I will present the multiscale model reduction methods for high-contrast heterogeneous flow problems, which has a wide range of applications, e.g., reservoir simulation, porous media and material science. The physical parameters are multiple scales, with the values varying over several orders of magnitude. To be specific, I will present numerical results within the Generalized Multiscale Finite Element Methods (GMsFEM). Besides, I will introduce the adaptive and randomized techniques to improve its efficiency. Furthermore, error estimate is briefly discussed.

We consider a variant of balanced truncation model order reduction that tries to restrict the reduction process to limited time- or frequency intervals. These restrictions lead to different Lyapunov equations which also include matrix functions. We discuss the singular value decay of the solutions and their numerical approximation by rational Krylov subspace methods. Numerical experiments show that, under some conditions, these restricted balanced truncation variants lead to a superior system approximation in the targeted time- or frequency regions.

In this talk I will present a new reduced order model (ROM) Hessian approximation for large-scale linear-quadratic optimal control problems where the optimal control is the initial value. Such problems arise in parameter identification, where the parameters to be identified appear in the initial data of the underlying instationary PDE, and as subproblems in multiple shooting formulations of more general optimal control problems constrained by time-dependent PDEs. The computation of a Hessian vector product requires the solution of the linearized state equation with initial value given by the vector to which the Hessian is applied to, followed by the solution of the second order adjoint equation. Projection based ROMs of these two linear differential equations are used to generate the Hessian approximation. The challenge is that in general no fixed ROM well-approximates neither the application of the Hessian to all possible vectors of initial data nor the solution of the corresponding linear equation for all possible right hand sides. The new approach, after having selected a basic ROM, augments this basic ROM by one vector. This vector is either the right hand side or the vector of initial data to which the Hessian is applied to. Although the size of the ROM increases only by one, this new augmented ROM produces substantially better approximations than the basic ROM. I will also analyze the use of these ROM Hessians in a conjugate gradient (CG) method and the sequential application of this method. This is joint work with Doerte Jando (Uni Heidelberg)

In this talk we will introduce some recent developments on reduced order modelling (ROM) of parametrized systems in incompressible computational fluid dynamics (CFD) problems. A challenge is related to extension of ROMs to problems characterized by large values of Reynolds numbers, i.e. problems with dominant convection fields. To this end, two complementary approaches are pursued in this talk. As a proof of concept, for low to moderate Reynolds numbers, we recast classical stabilization techniques (e.g. Brezzi-Pitkaranta, streamline upwind Petrov-Galerkin, Galerkin least squares) into a reduced order setting. The interplay between reduced order stabilization and supremizer enrichment of the velocity space is analyzed. For higher Reynolds number, ROM of a Smagorinsky turbulence model is detailed, and tested through some numerical examples. Time permitting, some representative applications in a broader CFD context may also be presented, such as recent work on environmental flows and marine sciences. Methodology and results presented in this talk are jointly developed by researchers in the AROMA-CFD group of Prof. Gianluigi Rozza, SISSA mathLab, International School for Advanced Studies, Trieste, Italy.

Partial differential equations with stochastic or parameter-dependent coefficients constitute an important uncertainty quantification task and a challenging high-dimensional problem. It has been approached with many techniques, such as Monte Carlo, Sparse Grids, Reduced Basis and tensor decompositions. The latter two offer potentially a lower complexity by finding a tailored low-rank solution approximation. However, generic low-rank methods, such as Alternating Least Squares (ALS), can disturb the sparsity of the original system. As a result, realistic scenarios with fine unstructured grids and large ranks resisted immediate treatment with these algorithms. We propose to combine the ALS steps for the spatial variables and the cross approximation steps for the other parameters. This scheme respects the sparsity (in fact, block-diagonality) of the system matrix and allows to reuse dedicated solvers for the deterministic problem. We show that the new algorithm can be significantly faster than the Sparse Grids and Quasi Monte Carlo methods for smooth random coefficients.

In this talk we present several successful applications of the space mapping optimization technique, which allows to fasten the design time significantly by exploiting underlying model hierarchies. This shall be exemplified by various problems, ranging from flow control over filter design to semiconductor design.

Approximate schedule:

9:00 bus leaves Grey College
10:00 arrive at Steel Rigg
10:45 walk to Housesteads (optional, ca 6km, can take bus instead)
12:45 bus leaves Housesteads for Hexham
13:15 arrive at Hexham
15:00 bus leaves Hexham
16:00 arrive at Grey College

We discuss the model order reduction problem for parameter-dependent dynamical systems. We survey the many approaches suggested in the previous decade, and will emphasize their differences as well as their commonalities. We will also illustrate the performance of various methods for real-world problems from different application areas.

We consider stochastic linear systems and their relation to linear and bilinear deterministic systems. There are some common features which can be exploited to design schemes for model order reduction of stochastic systems. In particular we derive a generalization of the method of balanced truncation. However, both analytically and computationally the situation is more involved in the stochastic setup. Therefore we elaborate on the preservation of stability, error bounds and computational schemes. Roughly, the plan is the following: In the first talk we introduce stochastic systems and basic ideas of model order reduction. The second talk is devoted to the analysis of the reduced systems, while in the third talk we consider computational issues and numerical examples.

When solving linear stochastic partial differential equations umerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As so-called type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Levy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this talk, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Levy driven systems.

The generalised singular perturbation approximation (GSPA) is a model reduction scheme for linear control systems. It is a state-space approach to truncation with the defining property that the transfer function of the approximation interpolates the original transfer function at a single prescribed point in the closed right half complex plane. Thus the GSPA also provides another procedure for rational approximation of rational matrix-valued functions of a complex variable. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at infinity and at zero, respectively. Suitably modified, the GSPA is shown to preserve dissipativity of the truncations and existing a priori error bounds for dissipative balanced truncation schemes are satisfied as well.

The model reduction problem for networks of interconnected dynamical systems is studied in this talk. For such networked systems, reduction is performed by clustering subsystems that show similar behavior and subsequently aggregating their states, leading to a reduced-order networked system that allows for an insightful physical interpretation. The clusters are chosen on the basis of the analysis of controllability and observability properties of associated edge systems, representing the importance of the couplings and providing a measure of the similarity of the behavior of neighboring subsystems. This reduction procedure is shown to preserve synchronization properties (i.e., the convergence of the subsystem trajectories to each other) and allows for the a priori computation of a bound on the reduction error with respect to external inputs and outputs.

We propose and analyze a multilevel weighted least squares polynomial approximation method. Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method, which employs samples with different accuracies and is able to match the accuracy of single level approximations at reduced computational work. We prove complexity bounds under certain assumptions on polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Numerical experiments underline the practical applicability of our method.

We propose an iterative algorithm called RKFIT for the approximate solution of nonlinear rational least squares problems. At each iteration RKFIT constructs a rational Krylov space and manipulates an associated Arnoldi decomposition to find improved poles of a rational approximant. In a diagonal special case, RKFIT is closely related to the popular vector fitting (VFIT) algorithm by Gustavsen and Semlyen. Both approaches differ in the representation of the rational approximants and their pole relocation steps. We demonstrate that RKFIT is often more robust and faster convergent than VFIT.

We consider stochastic linear systems and their relation to linear and bilinear deterministic systems. There are some common features which can be exploited to design schemes for model order reduction of stochastic systems. In particular we derive a generalization of the method of balanced truncation. However, both analytically and computationally the situation is more involved in the stochastic setup. Therefore we elaborate on the preservation of stability, error bounds and computational schemes. Roughly, the plan is the following: In the first talk we introduce stochastic systems and basic ideas of model order reduction. The second talk is devoted to the analysis of the reduced systems, while in the third talk we consider computational issues and numerical examples.

We discuss the model order reduction problem for parameter-dependent dynamical systems. We survey the many approaches suggested in the previous decade, and will emphasize their differences as well as their commonalities. We will also illustrate the performance of various methods for real-world problems from different application areas.

We discuss several model reduction techniques that are used to determine reduced-order models of fluid flows. In particular, we will discuss two techniques: balanced truncation and dynamic mode decomposition. Balanced truncation is a technique for model reduction of linear input-output systems that can significantly outperform more conventional methods such as Proper Orthogonal Decomposition. Its advantages are especially apparent for highly non-normal linear systems, as arise in many transitional shear flows. Dynamic mode decomposition is a method for identifying characteristic frequencies and growth rates, and their associated spatial structures, directly from data. It may also be used to determine predictive models of dynamics. We show examples of both of these methods as applied to various fluid flows.

Proper Orthogonal Decomposition (POD), or Principal Component Analysis (PCA), is an important technique for the construction of low-dimensional approximation spaces from high-dimensional input data. For large-scale applications and an increasing amount of input data vectors, computing the POD often becomes prohibitively expensive, however. In this talk we discuss a generic and easy to implement approach to compute an approximate POD based on tree hierarchies of small, fast to compute sub-PODs. The tree can be freely adapted to optimally suit the available computational resources. In particular, this hierarchical approximate POD (HAPOD) allows for both, simple parallelization with low communication overhead, as well as live incremental POD computation under restricted memory capacities. Rigorous error bounds ensure the quality of the computed approximation spaces as well as the runtime performance of our algorithm.

Computing the solutions $u$ of an equation $f(u, \lambda) = 0$ as the parameter $\lambda$ is varied is a central task in applied mathematics and engineering. In this talk I will present a new algorithm, deflated continuation, for this task. Deflated continuation has three main advantages. First, it is capable of computing disconnected bifurcation diagrams; previous algorithms only aimed to compute that part of the bifurcation diagram continuously connected to the initial data. Second, its implementation is extremely simple: it only requires a minor modification to any existing Newton-based solver. Third, it can scale to very large discretisations if a good preconditioner is available. Among other problems, we will apply this to a famous singularly perturbed ODE, Carrier's problem. The computations reveal a striking and beautiful bifurcation diagram, with an infinite sequence of alternating pitchfork and fold bifurcations as the singular perturbation parameter tends to zero. The analysis yields a novel and complete taxonomy of the solutions to the problem, and demonstrates that a claim of Bender & Orszag (1999) is incorrect. We will also use the algorithm to calculate distinct local minimisers of a topology optimisation problem via the combination of deflated continuation and a semismooth Newton method.

In this talk we explore the methodology of model order reduction for singularly perturbed (slow-fast) systems at a non-hyperbolic point. The presence of such non-hyperbolic point of the folded type complicates the analysis and controller design, i.e., the classical singular perturbation technique cannot be used. We use geometric desingularization, a kind of "blow up" technique, to be able to handle such points, to make some kind of separation between the slow and fast part of the system, and to design stabilizing controllers for such system based on this separation.

This talk will introduce some of the numerical methods and functionalities available within the new software package IRtools, i.e., a MATLAB toolbox for iterative regularization. Most of the solvers in IRtools are Krylov subspace methods, which are particularly suitable for solving linear ill-conditioned systems. This is joint work with Per Christian Hansen (DTU Compute, Denmark) and James Nagy (Emory University, USA).

We consider stochastic linear systems and their relation to linear and bilinear deterministic systems. There are some common features which can be exploited to design schemes for model order reduction of stochastic systems. In particular we derive a generalization of the method of balanced truncation. However, both analytically and computationally the situation is more involved in the stochastic setup. Therefore we elaborate on the preservation of stability, error bounds and computational schemes. Roughly, the plan is the following: In the first talk we introduce stochastic systems and basic ideas of model order reduction. The second talk is devoted to the analysis of the reduced systems, while in the third talk we consider computational issues and numerical examples.

We discuss the model order reduction problem for parameter-dependent dynamical systems. We survey the many approaches suggested in the previous decade, and will emphasize their differences as well as their commonalities. We will also illustrate the performance of various methods for real-world problems from different application areas.

Reduced order models (ROMs) have become prevalent in many ﬁelds of physics as they oﬀer the potential to simulate dynamical systems with substantially increased computation eﬃciency in comparison to standard techniques. Among the model reduction techniques, the proper orthogonal decomposition (POD) method has proven to be an eﬃcient means of deriving a reduced basis for high-dimensional ﬂow systems. The intrusive ROM (IROM) is normally derived by the POD and Galerkin projection methods. The IROM is appealing for non-linear and linear model reductions and has been successfully applied to numerous research ﬁelds. However, IROMs suﬀer from instability and non-linearity eﬃciency issues. In addition, they can be complex to code because they are intrusive. In most cases the source code describing the physical system has to be modiﬁed in order to generate the reduced order model. These modiﬁcations can be complex, especially in legacy codes, or may not be possible if the source code is not available (e.g. in some commercial software). To circumvent these shortcomings, non-intrusive approaches have been introduced into ROMs. The Non-Intrusive ROM (NIROM) is independent of the original physical system. In this talk, I will present our NIROM methods and their fluids, fluids-solids coupling, and multiphase applications. The basis idea of NIROM is to construct a set of hypersurfaces to replace the reduced system. The results show that the NIROM is promising and very fast.

Hilbert's 6th problem concerns the axiomatization of those parts of physics which are ready for a rigorous mathematical approach. D. Hilbert attracted special attention to the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua". We formalise this question as a problem of slow invariant manifolds for kinetic equations. We review a few instances where such hydrodynamic manifolds were found by the direct solution of the invariance equation. The dynamic equations on these manifolds give us a clue about the proper asymptotic of the continuum mechanic equations for rarefied non-equilibrium gases.

In this study we propose a-posteriori error estimation results to approximate the precision loss in quantities of interests computed using reduced order models. To generate the surrogate models we employ Proper Orthogonal Decomposition and Discrete Empirical Interpolation Method. First order expansions of the components of the quantity of interest obtained as the product between the components gradient and model residuals are summed up to generate the error estimation result. Efficient versions are derived for explicit and implicit Euler schemes and require only one reduced forward and adjoint models and high-fidelity model residuals estimation. Then we derive an adaptive DEIM algorithm to enhance the accuracy of these quantities of interests. The adaptive DEIM algorithm uses dual weighted residuals singular vectors in combination with the non-linear term basis. Both the a-posteriori error estimation result and the adaptive DEIM algorithm were assessed using the 1D-Burgers and Shallow Water Equation models and the numerical experiments shows very good agreement with the theoretical results.

We will present an approach to the decomposition of large matrix inequalities into several smaller ones with the goal to efficiently use existing semidefinite programming (SDP) solvers. The approaches will be demonstrated on an SDP problem arising in topology optimization of mechanical structures with vibration constraints. The technique is based on the decomposition of chordal graphs. We will show that it leads to a significant reduction of the CPU time.