• Multiscale Problems
• Workshop on Numerical Inverse and Stochastic Homogenization

• Schedule
• Participants

We consider the convergence of sparse Hermite expansions of solutions of elliptic PDEs with lognormally distributed diffusion coefficients. An important ingredient is the choice of an expansion of the Gaussian random field in terms of independent scalar Gaussian random variables, which can be interpreted as a choice of stochastic coordinates.

Here it turns out that multilevel-type expansions can lead to better results than Karhunen-Loève decompositions. We establish convergence rates and moreover consider the construction of such multilevel expansions for the important case of Matérn covariances.

This talk is based on joint works with Albert Cohen, Ronald DeVore and Giovanni Migliorati.

Hierarchical matrices are well suited for treating non-local operators with logarithmic-linear complexity. In particular, the inverse and the factors of the LU decomposition of finite element discretizations of elliptic boundary value problems can be approximated with such structures. However, a proof for this shows a strong dependence of the local rank on the ratio of the largest and smallest coefficient in the differential operator with respect to the L2-norm. Nevertheless, this kind of dependence cannot be observed in practice. The aim of this talk is to show that the above dependence can be avoided also theoretically with respect to a suitable norm. From this, a logarithmic dependence with respect to the L2-norm can be deduced.

This talk focuses on the modeling of weakly elastic materials composed of an incompressible elastic matrix and small compressible gaseous inclusions. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive. We will extend a recently proposed homogenized model to a time harmonic regime, in order to describe the solid-gas mixture from a macroscopic point of view in terms of an effective elasticity tensor. Furthermore, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters. This simplified description is used to to set up an efficient variational approach for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

Recorded Talk

In this talk, I will present our latest results on adaptive multiscale reduction. We will consider the generalized multiscale finite element method, and develop strategies to adaptively enrich the approximation space by adding multiscale basis functions. We will also present the convergence of these methods. In addition, we will apply the methods to several applications, and present the numerical results.

The research is partially supported by the Hong Kong RGC General Research Fund (Project: 14317516).

In this talk, I will discuss multiscale model reduction techniques for problems in heterogeneous media. I will discuss homogenization-based multiscale methods and their relations to multiscale finite element methods. I will describe a general multiscale framework for constructing local (space-time) reduced order models for problems with multiple scales and high contrast and their relation to multi-continuum approaches. Some issues related to the construction of multiscale basis functions, main ingredients of the method, and a number of applications will be described. A generalization of these approaches to stochastic problems will also be discussed.

We consider the homogenization of linear elliptic equations with random coefficient fields. Both in periodic and in stochastic homogenization, the homogenization error - that is, the difference between the solution of the equation with oscillating coefficients and the solution of the (homogenized) effective equation - is typically of the order of the scale of the microstructure, at least when measured in L^p type norms. In weak spatial norms, however, one expects to obtain a higher-order approximation of the solution to the equation with oscillating coefficients in terms of solutions to macroscopic equations. While this is a classical result in the setting of periodic homogenization, in stochastic homogenization no analogue has been available so far. By deriving near-optimal estimates on the second-order homogenization corrector in stochastic homogenization, we establish near-optimal estimates for the homogenization error in weak spatial norms in up to four spatial dimensions.

Joint work with Peter Bella, Benjamin Fehrman and Felix Otto.

The talk discusses a re-interpretation of existing multiscale methods by means of a discrete integral operator acting on standard finite element spaces. The exponential decay of the involved integral kernel motivates the use of a diagonal approximation and, hence, a localized piecewise constant coefficient. This local model turns out to be appropriate when the localized coefficient satisfies a certain homogenization criterion, which can be verified a posteriori. An a priori error analysis of the local model is presented and illustrated in numerical experiments.

In this talk I will present a theory of fluctuations in stochastic homogenization jointly developed with M. Duerinckx and F. Otto. In particular I shall introduce a crucial quantity, the homogenization commutator, that drives the fluctuations of all the quantities of interest in stochastic homogenization, in a path-wise way.

Recorded Talk

In this talk, I shall be attempting to give an overview of a new weak convergence type tool developed by myself, Thomas Holding (Warwick) and Jeffrey Rauch (Michigan) to handle multiple scales in advection-diffusion type models used in the turbulent diffusion theories. Loosely speaking, our strategy is to recast the advection-diffusion equation in moving coordinates dictated by the flow associated with a mean advective field. Crucial to our analysis is the introduction of a fast time variable. We introduce a notion of "convergence along mean flows" which is a weak multiple scales type convergence -- in the spirit of two-scale convergence theory. We have used ideas from the theory of "homogenization structures" developed by G. Nguetseng. We give a sufficient structural condition on the "Jacobian matrix" associated with the flow of the mean advective field which guarantees the homogenization of the original advection-diffusion problem as the microscopic lengthscale vanishes. We also show the robustness of this structural condition by giving an example where the failure of such a structural assumption leads to a degenerate limit behavior. More details on this new tool in homogenization theory can be found in the following paper:

• T. Holding, H. Hutridurga, J. Rauch. Convergence along mean flows, in press SIAM J Math. Anal., arXiv e-print: arXiv:1603.00424.

In a sequel to the above mentioned work, we are preparing a work where we address the growth in the Jacobian matrix - termed as Lagrangian stretching in Fluid dynamics literature - and its consequences on the vanishing microscopic lengthscale limit. To this effect, we introduce a new kind of multiple scales convergence in weighted Lebesgue spaces. This helps us recover some results in Freidlin-Wentzell theory. This talk aims to present both these aspects of our work in a unified manner.

Recorded Talk

In this presentation I will discuss a semi-Lagrangian discretisation of the Monge-Ampère operator on P1 finite element spaces. The wide stencil of the scheme is designed to ensure uniform stability of numerical solutions. Monge-Ampère equations arise in the Inverse Reflector Problem where the geometry of a reflecting surface is reconstructed from the illumination pattern on a target screen and the characteristics of the light source.

Monge-Ampère type equations, along with Hamilton-Jacobi-Bellman type equations are two major classes of fully nonlinear second order partial differential equations (PDEs). From the PDE point of view, Monge-Ampère type equations are well understood. On the other hand, from the numerical point of view, the situation is far from ideal. Very few numerical methods, which can reliably and efficiently approximate viscosity solutions of Monge-Ampère type PDEs on general convex domains. There are two main difficulties which contribute to the situation:

• Firstly, it is well known that the fully nonlinear structure and nonvariational concept of viscosity solutions of the PDEs prevent a direct formulation of any Galerkin-type numerical methods.
• Secondly, the Monge-Ampère operator is not an elliptic operator in generality, instead, it is only elliptic in the set of convex functions and the uniqueness of viscosity solutions only holds in that space. This convexity constraint, imposed on the admissible space, causes a daunting challenge for constructing convergent numerical methods; it indeed screens out any trivial finite difference and finite element analysis because the set of convex finite element functions is not dense in the set of convex functions.

The goal of our work is to develop a new approach for constructing convergent numerical methods for the Monge-Ampère Dirichlet problem, in particular, by focusing on overcoming the second difficulty caused by the convexity constraint. The crux of the approach is to first establish an equivalent (in the viscosity sense) Bellman formulation of the Monge-Ampère equation and then to design monotone numerical methods for the resulting Bellman equation on general triangular grids. An aim in the design of the numerical schemes was to make Howard's algorithm available, which is a globally superlinearly converging semi-smooth Newton solver as this allows us to robustly compute numerical approximations on very fine meshes of non-smooth viscosity solutions. An advantage of the rigorous convergence analysis of the numerical solutions is the comparison principle for the Bellman operator, which extends to non-convex functions. We deviate from the established Barles-Souganidis framework in the treatment of the boundary conditions to address challenges arising from consistency and comparison. The proposed approach also bridges the gap between advances on numerical methods for these two classes of second order fully nonlinear PDEs.

The contents of the presentation are based on joint work with X. Feng from the University of Tennessee.

Recorded Talk

Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. Based on an iterative counterpart relying on two-level subspace decomposition, we present and analyze a class of new methods that is very closely related to the approach of Malqvist and Peterseim [Math. Comp. 83, 2014]. As in the approach of Malqvist and Peterseim, these new methods do not make explicit or implicit use of a scale separation. Their comparatively simple analysis is based on the theory of additive Schwarz or subspace decomposition methods.

Joint work with Daniel Peterseim and Harry Yserentant.

Many natural phenomena and technological applications are modelled by deterministic or random dynamical systems. Often, these dynamical systems are characterized by the presence of processes occurring across different length and/or time scales. Examples range from biological systems and problems in atmosphere and ocean sciences to molecular dynamics, materials science, and fluid and solid mechanics, to name but a few. Of main interest for these systems is typically only the dynamics at the longest scale and multiscale methods (e.g. averaging and homogenization) provide an analytic framework for the rigorous derivation of coarse-grained dynamical systems that represent the full multiscale systems at this scale. These analytic techniques are, however, often not applicable in practice due to the complex structure of the underlying multiscale system or simply because the multiscale system is not known entirely. Instead it is desirable to infer stochastic coarse-grained models from observational data of the underlying multiscale process. Estimators such as the maximum likelihood estimator can, however, be strongly biased in this setting. In this talk we address this shortcoming and discuss a novel parametric inference methodology for stochastic coarse-grained models that does not suffer from it. Moreover, we exemplify through a real-world data set how these data-driven coarse-graining techniques can be used to study the statistical properties of a given temporal process.

The Multiscale Finite Element Method (MsFEM) is a Finite Element type approach for multiscale PDEs, where the basis functions used to generate the approximation space are precomputed and are specifically adapted to the problem at hand. The computation is performed in a two-stage procedure: (i) an offline stage, in which local basis functions are computed, and (ii) an online stage, in which the global problem is solved using an inexpensive Galerkin approximation. Several variants of the approach have been proposed and a priori error estimates have been established.

As for any numerical method, a crucial issue is to control the accuracy of the numerical solution provided by the MsFEM approach. In this work, we develop an a posteriori error estimate and the associated adaptive procedure. The estimate is based on the concept of Constitutive Relation Error (CRE), which we extend to the multiscale framework.

We introduce a guaranteed and fully computable a posteriori error estimate, both for the global error and for the error on quantities of interest, using adjoint-based techniques. We discuss the accuracy of such estimates and show how they can be used to efficiently drive an adaptive discretization. Time permitting, we also investigate the additional use of model reduction techniques (such as the Proper Generalized Decomposition (PGD)) within the MsFEM approach in order to further decrease the computational costs.

Joint work with L. Chamoin.

We will present the Local Orthogonal Decomposition technique for solving partial differential equations with multiscale data. In particular we will discuss convergence in presence of high contrast data. We will consider some different applications, including time dependent problems, where the computed basis can be reused in each time step.

Recorded Talk

Model reduction approaches for parameterized problems have seen tremendous development in recent years. A particular instance of projection based model reduction is the reduced basis (RB) method, which is based on the construction of low-dimensional approximation spaces from snapshot computations, i.e. solutions of the underlying parameterized problem for suitably chosen parameter values. In this talk we will address error control for localized model reduction methods and its usage for the construction of efficient numerical schemes for parameterized single and multiscale PDEs. Thereby we overcome the classical paradigm of offline-online splitting by allowing for arbitrary local modifications and local basis enrichment in the so called online-phase [1, 2, 3]. We also refer to [5, 6] for an alternative approach based on discontinuous global approximation spaces. Several numerical examples and applications will be shown to demonstrate the efficiency of the resulting adaptive approaches. The numerical results were obtained with the newly developed model reduction algorithms implemented in pyMOR (see pymor.org and [4]).

References:
[1] A. Buhr, and M. Ohlberger. Interactive Simulations Using Localized Reduced Basis Methods. Proceedings of MATHMOD 2015 - 8th Vienna International Conference on Mathematical Modelling, IFAC Mathematical Modelling, 48(1), pages 729–730, 2015.
[2] A. Buhr, C. Engwer, M. Ohlberger, S. Rave. ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications. Applied Mathematics Muenster, University of Muenster arXiv:1512.07840 [math.NA], Preprint (Submitted) - december 2015.
[3] A. Buhr, C. Engwer, M. Ohlberger, S. Rave. ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics. Applied Mathematics Muenster, University of Muenster arXiv:1606.06206 [math.NA], Preprint (Submitted) - june 2016.
[4] R. Milk, S. Rave, F. Schindler. pyMOR- generic algorithms and interfaces for model order reduction. SIAM J. Sci. Comput. 38(5):S194–S216, 2016.
[5] M. Ohlberger, S. Rave, F. Schindler. True Error Control for the Localized Reduced Basis Method for Parabolic Problems. Applied Mathematics Muenster, University of Muenster arXiv:1606.09216 [math.NA], Preprint (Submitted) - june 2016.
[6] M. Ohlberger, F. Schindler. Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment. SIAM J. Sci. Comput. 37(6):A2865–A2895, 2015.

The idea of using tensors in the context of multi scale problems is very simple:
1. Discretize the problem using low-order FEM on tensor-product grid with 2^d elements that resolves the finest scale.
2. Reshape the solution vector into 2x2x2x…x2 tensor.
3. Approximate this tensor in a low-rank tensor-train format.

Recent theoretical results (Schwab, Kazeev for two-dimensional elliptic problems, Schwab, Kazeev, Oseledets, Rakhuba for one-dimensional multi scale problems) demonstrate that the total number of parameters required is logarithmic both in the desired accuracy and in the multi scale parameter for important classes of coefficients. Practical computation of this approximant leads to several difficulties: since TT-parameterization is nonlinear, we are left with a non-convex optimization problem over low-rank tensor manifold. Second problem, that was unexpected to us, is the instability of low-order FEM for very small mesh sizes, which is evident from the inspection of rounding errors on the accuracy.  In order to alleviate this problem in computations, we developed a new discretization scheme for two-dimensional elliptic problem, which is robust for very small mesh sizes, yet can be efficiently implemented in the tensor format.

1. Vladimir Kazeev, Ivan Oseledets, Maxim Rakhuba, and Christoph Schwab. QTT-finite-element approximation for multiscale problems I: model problems in one dimension. Adv. Comp. Math., 2016
2. A. V. Chertkov, I. V Oseledets, and M. V. Rakhuba. Robust discretization in quantized tensor train format for elliptic problems in two dimensions. arXiv preprint 1612.01166, 2016. URL: arxiv.org/abs/1612.01166.

This talk will be based on recent joint works with C. Schwab, V. Kazeev, M. Rakhuba and A. Chertkov.

Recorded Talk

The talk will discuss elastic shape optimization in the case of stochastic loading. Complicated geometries will be treated both by a full resolution of the geometry and by a two-scale approach. Stochastic loading will be considered with different perceptions of risk aversion. Furthermore, the paradigm of stochastic dominance will be transferred from finite-dimensional stochastic programming to shape optimization. This allows for flexible risk aversion via comparison with benchmark configurations.

This is joint work with Sergio Conti, Benedikt Geihe, Rüdiger Schultz, Sascha Tölkes.

In this talk we will present a family of non-conforming "Crouzeix-Raviart" type finite elements in two and three dimensions. They consist of local polynomials of maximal degree p on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices.

We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions goes back to the seminal paper of Crouzeix and Raviart in 1973. However, the definition is implicit and the derivation of an explicit representation of the local basis functions for general p in 3D was an open problem.

We present explicit representations for these functions by developing some theoretical tools for fully symmetric and reflection symmetric orthogonal polynomials on triangles and their representation.

Finally we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space.

This talk comprises joint work with P. Ciarlet Jr., ENSTA, Paris and Charles F. Dunkl, Virginia Tech.

Recorded Talk

A very efficient algorithm has recently been introduced in [1] in order to approximate the solution of implicit solvation models for molecules. The main ingredient of this algorithm relies in the clever use of a boundary integral formulation of the problem to solve in combination with spherical harmonics. The aim of this talk is to present how such an algorithm can be adapted in order to compute efficiently effective coefficients in stochastic homogenization for random media with spherical inclusions. To this aim, the definition of new approximate corrector problems and approximate effective coefficients is needed and convergence results in the spirit of [2] are proved for this new formulation. Some numerical test cases will illustrate the behavior of this method.

References:
[1] "Domain decomposition for implicit solvation models", Eric Cancès, Yvon Maday, Benjamin Stamm, The Journal of Chemical Physics 139 (2013) 054111
[2] "Approximations of effective coefficients in stochastic homogenization", Alain Bourgeat, Andrey Piatnitski, Annales de l'institut Henri Poincaré (B) Probabilités et Statistiques 40 (2004) page 153-165

Recorded Talk

Optimal transport has become a well developed topic in analysis since it was first proposed by Monge in 1781. Due to their ability to incorporate differences in both intensity and spatial information, the related Wasserstein metrics have been adopted in a variety of applications, including seismic inversion. Quadratic Wasserstein metric (W2) has ideal properties like convexity and insensitivity to noise, while conventional L2 norm is known to suffer from local minima. We propose two ways of using W2 in seismic inversion, a trace-by-trace comparison solved by sorting, and the global comparison which requires numerical solution to Monge-Ampère equation.

Multiscale problems arise naturally from many scientific and engineering areas such as geophysics, material sciences and biology. Numerical homogenization concerns the (coarse) finite dimensional approximation of the solution space of, for example, divergence form elliptic equation with L^∞ coefficients which allows for nonseparable scales. Based on a Bayesian reformulation of numerical homogenization, we propose a class of numerical homogenization methods which allow for exponential decaying bases, localization, as well as optimal convergence rates. This can be used to construct efficient and robust fine scale fast solvers such as multi-resolution decomposition (the so-called "gamblet" decomposition) or multigrid solver with bounded condition number on each subband, and enables the resolution of boundary value problems in near-linear complexity and rigorous a priori error bounds. The method can be generalized to time dependent problems such as wave propagation in heterogeneous media, and multi-scale eigenvalue problems.

Recorded Talk