6,814 research outputs found
Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations
We consider a framework for the construction of iterative schemes for
operator equations that combine low-rank approximation in tensor formats and
adaptive approximation in a basis. Under fairly general assumptions, we obtain
a rigorous convergence analysis, where all parameters required for the
execution of the methods depend only on the underlying infinite-dimensional
problem, but not on a concrete discretization. Under certain assumptions on the
rates for the involved low-rank approximations and basis expansions, we can
also give bounds on the computational complexity of the iteration as a function
of the prescribed target error. Our theoretical findings are illustrated and
supported by computational experiments. These demonstrate that problems in very
high dimensions can be treated with controlled solution accuracy.Comment: 51 page
Adaptive Low-Rank Methods for Problems on Sobolev Spaces with Error Control in
Low-rank tensor methods for the approximate solution of second-order elliptic
partial differential equations in high dimensions have recently attracted
significant attention. A critical issue is to rigorously bound the error of
such approximations, not with respect to a fixed finite dimensional discrete
background problem, but with respect to the exact solution of the continuous
problem. While the energy norm offers a natural error measure corresponding to
the underlying operator considered as an isomorphism from the energy space onto
its dual, this norm requires a careful treatment in its interplay with the
tensor structure of the problem. In this paper we build on our previous work on
energy norm-convergent subspace-based tensor schemes contriving, however, a
modified formulation which now enforces convergence only in . In order to
still be able to exploit the mapping properties of elliptic operators, a
crucial ingredient of our approach is the development and analysis of a
suitable asymmetric preconditioning scheme. We provide estimates for the
computational complexity of the resulting method in terms of the solution error
and study the practical performance of the scheme in numerical experiments. In
both regards, we find that controlling solution errors in this weaker norm
leads to substantial simplifications and to a reduction of the actual numerical
work required for a certain error tolerance.Comment: 26 pages, 7 figure
Efficient Resolution of Anisotropic Structures
We highlight some recent new delevelopments concerning the sparse
representation of possibly high-dimensional functions exhibiting strong
anisotropic features and low regularity in isotropic Sobolev or Besov scales.
Specifically, we focus on the solution of transport equations which exhibit
propagation of singularities where, additionally, high-dimensionality enters
when the convection field, and hence the solutions, depend on parameters
varying over some compact set. Important constituents of our approach are
directionally adaptive discretization concepts motivated by compactly supported
shearlet systems, and well-conditioned stable variational formulations that
support trial spaces with anisotropic refinements with arbitrary
directionalities. We prove that they provide tight error-residual relations
which are used to contrive rigorously founded adaptive refinement schemes which
converge in . Moreover, in the context of parameter dependent problems we
discuss two approaches serving different purposes and working under different
regularity assumptions. For frequent query problems, making essential use of
the novel well-conditioned variational formulations, a new Reduced Basis Method
is outlined which exhibits a certain rate-optimal performance for indefinite,
unsymmetric or singularly perturbed problems. For the radiative transfer
problem with scattering a sparse tensor method is presented which mitigates or
even overcomes the curse of dimensionality under suitable (so far still
isotropic) regularity assumptions. Numerical examples for both methods
illustrate the theoretical findings
A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems
In this paper, we propose a method for the approximation of the solution of
high-dimensional weakly coercive problems formulated in tensor spaces using
low-rank approximation formats. The method can be seen as a perturbation of a
minimal residual method with residual norm corresponding to the error in a
specified solution norm. We introduce and analyze an iterative algorithm that
is able to provide a controlled approximation of the optimal approximation of
the solution in a given low-rank subset, without any a priori information on
this solution. We also introduce a weak greedy algorithm which uses this
perturbed minimal residual method for the computation of successive greedy
corrections in small tensor subsets. We prove its convergence under some
conditions on the parameters of the algorithm. The residual norm can be
designed such that the resulting low-rank approximations are quasi-optimal with
respect to particular norms of interest, thus yielding to goal-oriented order
reduction strategies for the approximation of high-dimensional problems. The
proposed numerical method is applied to the solution of a stochastic partial
differential equation which is discretized using standard Galerkin methods in
tensor product spaces
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
Fast Isogeometric Boundary Element Method based on Independent Field Approximation
An isogeometric boundary element method for problems in elasticity is
presented, which is based on an independent approximation for the geometry,
traction and displacement field. This enables a flexible choice of refinement
strategies, permits an efficient evaluation of geometry related information, a
mixed collocation scheme which deals with discontinuous tractions along
non-smooth boundaries and a significant reduction of the right hand side of the
system of equations for common boundary conditions. All these benefits are
achieved without any loss of accuracy compared to conventional isogeometric
formulations. The system matrices are approximated by means of hierarchical
matrices to reduce the computational complexity for large scale analysis. For
the required geometrical bisection of the domain, a strategy for the evaluation
of bounding boxes containing the supports of NURBS basis functions is
presented. The versatility and accuracy of the proposed methodology is
demonstrated by convergence studies showing optimal rates and real world
examples in two and three dimensions.Comment: 32 pages, 27 figure
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