  conference
28/09/2021 29/09/2021

# Hierarchical Numerical Methods for PDEs

The hierarchical construction of complex models from basic concepts is a main principle of mathematics. Likewise, hierarchical methods are one of the main cornerstones in modern numerical analysis and scientific computing. The hierarchical approach, which underlies such different techniques as multigrid methods, H-matrices and hierarchical tensor decomposition, allows to overcome the computational barriers posed by large scale, ill conditioning or high dimensionality. This workshop focuses on recent advances on hierarchical numerical methods for PDEs and related problems.

The workshop is dedicated to Wolfgang Hackbusch who was awarded the 2020 Heinz Gumin Prize for Mathematics.

TU Hamburg

### Mateusz Michalek

University of Konstanz

EC Nantes

### Stefan Sauter

University of Zurich

KAUST

TU Munich

Penn State

### Harry Yserentant

TU Berlin
 Sep 28, 2021 09:15 09:30 Ivan Graham (University of Bath) Convergence of iterative solvers for the Helmholtz equation The Restricted Additive Schwarz method with impedance transmission conditions (often called the ORAS method) is a domain decomposition method which can be used as an iterative solver or as a preconditioner for the solution of discretized Helmholtz boundary-value problems. It is a very simple parallel one-level method, applicable in very general geometries and it has been successfully combined with coarse spaces to obtain two- or multi-level methods. Its multiplicative variants are related to 'sweeping methods' which have enjoyed considerable recent practical interest. To date there is relatively little convergence analysis for this method. In the talk we present a novel analysis of the ORAS method. The main components of the talk are: (i) ORAS is a non-conforming finite element approximation of a classical parallel Schwarz method formulated at the PDE level; (ii) The parallel Schwarz method is well-posed in suitable function spaces of Helmholtz-harmonic functions in general geometries; (iii) The parallel Schwarz method is proved to be power contractive for domain decompositions of 'strip-type' and is observed to be power contractive for general domain decompositions in 2D experiments; (iv) Working in suitable Helmholtz-harmonic finite element spaces and for fine enough meshes, the ORAS method is proved to enjoy the same convergence estimates as the parallel Schwarz method; in particular its power contractive property is independent of the polynomial order of the finite element spaces used. The proof of (iv) uses a new finite element error estimate, proving higher order convergence for certain Helmholtz problems at interior interfaces.The work on ORAS is recent joint work with Shihua Gong and Euan Spence (Bath) while the results on the parallel Schwarz method were obtained with Shihua Gong, Martin Gander (Geneva), David Lafontaine (Bath) and Euan Spence. 10:15 10:45 Sabine Le Borne (TU Hamburg) Potential (&) pitfalls in RBF-FD discretization of PDEs There exist several discretization techniques for the numerical solution of partial differential equations (PDEs). In addition to classical finite difference, finite element and finite volume techniques, a more recent approach employs radial basis functions to generate differentiation stencils on unstructured point sets. This approach, abbreviated by RBF-FD (radial basis function - finite difference), has gained in popularity since it enjoys several advantages: It is (relatively) straightforward, does not require a mesh and generalizes easily to higher spatial dimensions. However, its application is not quite as blackbox as it may appear at first sight. The computed solution might suffer severely from various sources of errors if RBF-FD parameters are not selected carefully. Through comprehensive numerical experiments, we study the influence of several of these parameters on the condition numbers of intermediate (local) weight matrices, on the condition number of the resulting (global) stiffness matrix and ultimately on the approximation error of the computed discrete solution to the partial differential equation. The parameters of investigation include the type of RBF (and its shape or other parameters if applicable), the degree of polynomial augmentation, the discretization stencil size, the underlying type of point set (structured/unstructured), and the total number of (interior and boundary) points to discretize the PDE, here chosen as a three-dimensional Poisson's problem with Dirichlet boundary conditions. The goal is to illustrate and steer away from potential pitfalls in RBF-FD discretization where a computationally more expensive set-up not always leads to a more accurate numerical solution, and to guide toward a compatible selection of RBF-FD parameters. 11:30 Anthony Nouy (EC Nantes) Approximation theory of tree tensor networks Tree Tensor networks (TTNs) are prominent model classes for the approximation of high-dimensional functions in computational and data science. After an introduction to approximation tools based on tensorization of functions and TTNs, we introduce their approximation classes and present some recent results on their properties. In particular, we show that classical smoothness (Besov) spaces are continuously embedded in TTNs approximation classes. For such spaces, TNs achieve (near to) optimal rate that is usually achieved by classical approximation tools, but without requiring to adapt the tool to the regularity of the function. The use of deep networks is shown to be essential for obtaining this property. Also, it is shown that exploiting sparsity of tensors allows to obtain optimal rates achieved by classical nonlinear approximation tools, or to better exploit structured smoothness (anisotropic or mixed) for multivariate approximation. We also show that approximation classes of tensor networks are not contained in any Besov space, unless one restricts the depth of the tensor network. That reveals again the importance of depth and the potential of tensor networks to achieve approximation or learning tasks for functions beyond standard regularity classes. 12:15 14:00 Barbara Wohlmuth (TU Munich) The challenge of non-local operators and mixed dimensions in simulation and modelling In this talk, we address some mathematical and numerical challenges of non-linear or non-local partial differential equations in applications. We use a variety of applications ranging from porous-media flow systems, fluid and structural mechanics to finance. The special challenges of variational inequalities, mixed-dimensions and non-integer differential operators are illustrated. We show the flexibility of abstract mathematical concepts and discuss limitations in theory and convergence. 14:45 Stefan Sauter (University of Zurich) Variable Order, Directional H2-Matrices for Helmholtz Problems with Complex Frequency The sparse approximation of high-frequency Helmholtz-type integral operators has many important physical applications such as problems in wave propagation and wave scattering. The discrete system matrices are huge and densely populated; hence their sparse approximation is of outstanding importance. In our talk we will generalize the directional H2-matrix techniques from the "original" Helmholtz operator (purely imaginary wave number) to general complex frequencies z with Re(z) >0. In this case, the fundamental solution decreases exponentially for large arguments. We will develop a new admissibility condition which contain Re(z) and Im(z) in an explicit way and introduce the approximation of the integral kernel function on admissible blocks in terms of frequency-dependent directional expansion functions. We present an error analysis which is explicit with respect to the expansion order and with respect to the real and imaginary part of z. This allows us to choose the variable expansion order in a quasi-optimal way depending on Re(z) but independent of, possibly large, Im(z). The complexity analysis is explicit with respect to Re(z) and Im(z) and shows how higher values of Re(z) reduce the complexity. In certain cases, it even turns out that the discrete matrix can be replaced by its nearfield part. Numerical experiments illustrate the sharpness of the derived estimates and the efficiency of our sparse approximation.This talk comprises joint work with S. Börm, Christian-Albrechts-Universität Kiel, Germany and M. Lopez-Fernandez, Sapienza Universita di Roma, Italy 15:30 16:00 Jinchao Xu (Penn State) Neural Networks and Numerical PDEs I will first give a brief introduction to neural network functions, their applications to image classifications, and their relationship with finite element and multigrid methods. I will then present some recent results on the approximation properties of neural network functions, error analysis for numerical PDEs (in view of generalization accuracy in machine learning) and optimization algorithms for the underlying non-convex problems. 18:30 Sep 29, 2021 09:15 Harry Yserentant (TU Berlin) Iterations, Tensors, and beyond This talk deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$, where $\mu$ is a positive constant. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be represented in the same way. These constructions are based on approximations of the function $1/r$ by sums of exponential functions. I will present results of similar kind for more general right-hand sides $f(x)=F(Tx)$ that are composed of a separable function on a space of a dimension $n$ greater than $m$ and a linear mapping given by a matrix $T$ of full rank. These results are based on the observation that in the high-dimensional case, for $\omega$ in most of the $\mathbb{R}^n$, the euclidean norm of the vector $T^t\omega$ in the lower dimensional space $\mathbb{R}^m$ behaves like the euclidean norm of $\omega$. This effect has much to do with the random projection theorem, which plays an important role in the data sciences, and can be seen as a concentration of measure phenomenon. 10:00 Mateusz Michalek (University of Konstanz) Matrix Product States from the Point of View of Algebraic Geometry We will present uniform Matrix Product States (uMPS) and algebraic varieties associated to them. Using tools from algebraic geometry we give answers to questions posed by Hackbusch, Morton and Critch concerning topological properties of these varieties and their defining equations. We will also present some open problems. The talk is based on a joint work with Czaplinski and Seynnaeve. 10:45 11:15 Reinhold Schneider (TU Berlin) 12:00 Gabriel Wittum (KAUST, Thuwal, Saudi Arabia)

### Peter Benner

Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg

RWTH Aachen

### André Uschmajew

Max Planck Institute for Mathematics in the Sciences, Leipzig