Distributed Hierarchical SVD in the Hierarchical Tucker Format

We consider tensors in the Hierarchical Tucker format and suppose the tensor data to be distributed among several compute nodes. We assume the compute nodes to be in a one-to-one correspondence with the nodes of the Hierarchical Tucker format such that connected nodes can communicate with each other. An appropriate tree structure in the Hierarchical Tucker format then allows for the parallelization of basic arithmetic operations between tensors with a parallel runtime which grows like $\log(d)$, where $d$ is the tensor dimension. We introduce parallel algorithms for several tensor operations, some of which can be applied to solve linear equations $\mathcal{A}X=B$ directly in the Hierarchical Tucker format using iterative methods like conjugate gradients or multigrid. We present weak scaling studies, which provide evidence that the runtime of our algorithms indeed grows like $\log(d)$. Furthermore, we present numerical experiments in which we apply our algorithms to solve a parameter-dependent diffusion equation in the Hierarchical Tucker format by means of a multigrid algorithm

A literature survey of low-rank tensor approximation techniques

During the last years, low-rank tensor approximation has been established as a new tool in scientific computing to address large-scale linear and multilinear algebra problems, which would be intractable by classical techniques. This survey attempts to give a literature overview of current developments in this area, with an emphasis on function-related tensors

Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations

A recurring theme in attempts to break the curse of dimensionality in the numerical approximations of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class $\Sigma_n$ of functions, which can be written as a sum of rank-one tensors using a total of at most $n$ parameters and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, among other results, that whenever the right-hand side $f$ of the elliptic PDE can be approximated with a certain rate $\mathcal{O}(n^{-r})$ in the norm of ${\mathrm H}^{-1}$ by elements of $\Sigma_n$, then the solution $u$ can be approximated in ${\mathrm H}^1$ from $\Sigma_n$ to accuracy $\mathcal{O}(n^{-r'})$ for any $r'\in (0,r)$. Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second "basis-free" model of tensor sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model high-dimensional elliptic PDEs with tensor-sparse data.Comment: 41 pages, 1 figur

Theorie und Anwendungen Hierarchischer Matrizen

The modeling of physical properties often leads to the task of solving partial differential equations or integral equations. The results of some discretisation and linearisation process are matrix equations or linear systems of equations with special features. In the case of partial differential equations one exploits the local character of the differentiation by using some finite element method or finite difference scheme and gains a sparse system matrix. In the case of (nonlocal) integral operators low rank approximations seem to be the method of choice. These are either given explicitly by some multipole method or panel clustering technique or implicitly by rank revealing decompositions. Both types of matrices can be represented as so-called H-matrices. In this thesis we investigate algorithms that perform the addition, multiplication and inversion of H-matrices approximately. Under moderate assumptions the complexity of these new arithmetics is almost linear (linear up to logarithmic terms of order 1 to 3). The arithmetic operations can be performed adaptively, that is up to some given accuracy epsilon the relative error of the operations is zero. The question arises under which circumstances the inverse of an H-matrix can be approximated by an H-matrix. For the techniques used in this thesis we need very restrictive assumptions, but the numerical examples in the last part indicate that the approximability does not depend on these assumptions

Numerical upscaling of parametric microstructures in a possibilistic uncertainty framework with tensor trains

A fuzzy arithmetic framework for the efficient possibilistic propagation of shape uncertainties based on a novel fuzzy edge detection method is introduced. The shape uncertainties stem from a blurred image that encodes the distribution of two phases in a composite material. The proposed framework employs computational homogenisation to upscale the shape uncertainty to a effective material with fuzzy material properties. For this, many samples of a linear elasticity problem have to be computed, which is significantly sped up by a highly accurate low-rank tensor surrogate. To ensure the continuity of the underlying mapping from shape parametrisation to the upscaled material behaviour, a diffeomorphism is constructed by generating an appropriate family of meshes via transformation of a reference mesh. The shape uncertainty is then propagated to measure the distance of the upscaled material to the isotropic and orthotropic material class. Finally, the fuzzy effective material is used to compute bounds for the average displacement of a non-homogenized material with uncertain star-shaped inclusion shapes

Solving an Elliptic PDE Eigenvalue Problem via Automated Multi-Level Substructuring and Hierarchical Matrices

We propose a new method for the solution of discretised elliptic PDE eigenvalue problems. The new method combines ideas of domain decomposition, as in the automated multi-level substructuring (short AMLS), with the concept of hierarchical matrices (short H-matrices) in order to obtain a solver that scales almost optimal in the size of the discrete space. Whereas the AMLS method is very effective for PDEs posed in two dimensions, it is getting very expensive in the three-dimensional case, due to the fact that the interface coupling in the domain decomposition requires dense matrix operations. We resolve this problem by use of data-sparse hierarchical matrices. In addition to the discretisation error our new approach involves a projection error due to AMLS and an arithmetic error due to H-matrix approximation. A suitable choice of parameters to balance these errors is investigated in examples. Mathematics Subject Classification (2000) 65F15, 65F30, 65F50, 65H17, 65N25, 65N5

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length

Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length