# Hierarchical Numerical Methods for PDEs

## Abstracts for the talks

**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 preconðitioner

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.

**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.

**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.

**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.

**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

**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.

**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.

**Harry Yserentant ***TU Berlin***Iterations, Tensors, and beyond**

This talk deals with the equation −Δu + μu = f on high-dimensional spaces ℝ^{m}, where μ 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 ω in most of the ℝ^{n}, the euclidean norm of the vector T^{t}ω in the lower dimensional space ℝ^{m} behaves like the euclidean norm of ω. 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.

## Date and Location

**September 28 - 29, 2021**

Max Planck Institute for Mathematics in the Sciences

Inselstr. 22

04103 Leipzig

## Scientific Organizers

**Peter Benner**

MPI for Dynamics of Complex Technical Systems, Magdeburg**Lars Grasedyck**

RWTH Aachen**André Uschmajew**

MPI for Mathematics in the Sciences, Leipzig

## Administrative Contact

**Katja Heid**

MPI for Mathematics in the Sciences

Leipzig

Contact by Email