7,437 research outputs found
A semi-Lagrangian Vlasov solver in tensor train format
In this article, we derive a semi-Lagrangian scheme for the solution of the
Vlasov equation represented as a low-parametric tensor. Grid-based methods for
the Vlasov equation have been shown to give accurate results but their use has
mostly been limited to simulations in two dimensional phase space due to
extensive memory requirements in higher dimensions. Compression of the solution
via high-order singular value decomposition can help in reducing the storage
requirements and the tensor train (TT) format provides efficient basic linear
algebra routines for low-rank representations of tensors. In this paper, we
develop interpolation formulas for a semi-Lagrangian solver in TT format. In
order to efficiently implement the method, we propose a compression of the
matrix representing the interpolation step and an efficient implementation of
the Hadamard product. We show numerical simulations for standard test cases in
two, four and six dimensional phase space. Depending on the test case, the
memory requirements reduce by a factor in four and a factor
in six dimensions compared to the full-grid method
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
Low-rank approximate inverse for preconditioning tensor-structured linear systems
In this paper, we propose an algorithm for the construction of low-rank
approximations of the inverse of an operator given in low-rank tensor format.
The construction relies on an updated greedy algorithm for the minimization of
a suitable distance to the inverse operator. It provides a sequence of
approximations that are defined as the projections of the inverse operator in
an increasing sequence of linear subspaces of operators. These subspaces are
obtained by the tensorization of bases of operators that are constructed from
successive rank-one corrections. In order to handle high-order tensors,
approximate projections are computed in low-rank Hierarchical Tucker subsets of
the successive subspaces of operators. Some desired properties such as symmetry
or sparsity can be imposed on the approximate inverse operator during the
correction step, where an optimal rank-one correction is searched as the tensor
product of operators with the desired properties. Numerical examples illustrate
the ability of this algorithm to provide efficient preconditioners for linear
systems in tensor format that improve the convergence of iterative solvers and
also the quality of the resulting low-rank approximations of the solution
Towards tensor-based methods for the numerical approximation of the Perron-Frobenius and Koopman operator
The global behavior of dynamical systems can be studied by analyzing the
eigenvalues and corresponding eigenfunctions of linear operators associated
with the system. Two important operators which are frequently used to gain
insight into the system's behavior are the Perron-Frobenius operator and the
Koopman operator. Due to the curse of dimensionality, computing the
eigenfunctions of high-dimensional systems is in general infeasible. We will
propose a tensor-based reformulation of two numerical methods for computing
finite-dimensional approximations of the aforementioned infinite-dimensional
operators, namely Ulam's method and Extended Dynamic Mode Decomposition (EDMD).
The aim of the tensor formulation is to approximate the eigenfunctions by
low-rank tensors, potentially resulting in a significant reduction of the time
and memory required to solve the resulting eigenvalue problems, provided that
such a low-rank tensor decomposition exists. Typically, not all variables of a
high-dimensional dynamical system contribute equally to the system's behavior,
often the dynamics can be decomposed into slow and fast processes, which is
also reflected in the eigenfunctions. Thus, the weak coupling between different
variables might be approximated by low-rank tensor cores. We will illustrate
the efficiency of the tensor-based formulation of Ulam's method and EDMD using
simple stochastic differential equations
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