1,536 research outputs found
Fast multidimensional convolution in low-rank formats via cross approximation
We propose a new cross-conv algorithm for approximate computation of
convolution in different low-rank tensor formats (tensor train, Tucker,
Hierarchical Tucker). It has better complexity with respect to the tensor rank
than previous approaches. The new algorithm has a high potential impact in
different applications. The key idea is based on applying cross approximation
in the "frequency domain", where convolution becomes a simple elementwise
product. We illustrate efficiency of our algorithm by computing the
three-dimensional Newton potential and by presenting preliminary results for
solution of the Hartree-Fock equation on tensor-product grids.Comment: 14 pages, 2 figure
FFT-based Kronecker product approximation to micromagnetic long-range interactions
We derive a Kronecker product approximation for the micromagnetic long range
interactions in a collocation framework by means of separable sinc quadrature.
Evaluation of this operator for structured tensors (Canonical format, Tucker
format, Tensor Trains) scales below linear in the volume size. Based on
efficient usage of FFT for structured tensors, we are able to accelerate
computations to quasi linear complexity in the number of collocation points
used in one dimension. Quadratic convergence of the underlying collocation
scheme as well as exponential convergence in the separation rank of the
approximations is proved. Numerical experiments on accuracy and complexity
confirm the theoretical results.Comment: 4 figure
Vico-Greengard-Ferrando quadratures in the tensor solver for integral equations
Convolution with Green's function of a differential operator appears in a lot
of applications e.g. Lippmann-Schwinger integral equation. Algorithms for
computing such are usually non-trivial and require non-uniform mesh. However,
recently Vico, Greengard and Ferrando developed method for computing
convolution with smooth functions with compact support with spectral accuracy,
requiring nothing more than Fast Fourier Transform (FFT). Their approach is
very suitable for the low-rank tensor implementation which we develop using
Quantized Tensor Train (QTT) decomposition
Memory footprint reduction for the FFT-based volume integral equation method via tensor decompositions
We present a method of memory footprint reduction for FFT-based,
electromagnetic (EM) volume integral equation (VIE) formulations. The arising
Green's function tensors have low multilinear rank, which allows Tucker
decomposition to be employed for their compression, thereby greatly reducing
the required memory storage for numerical simulations. Consequently, the
compressed components are able to fit inside a graphical processing unit (GPU)
on which highly parallelized computations can vastly accelerate the iterative
solution of the arising linear system. In addition, the element-wise products
throughout the iterative solver's process require additional flops, thus, we
provide a variety of novel and efficient methods that maintain the linear
complexity of the classic element-wise product with an additional
multiplicative small constant. We demonstrate the utility of our approach via
its application to VIE simulations for the Magnetic Resonance Imaging (MRI) of
a human head. For these simulations we report an order of magnitude
acceleration over standard techniques.Comment: 11 pages, 10 figures, 5 tables, 2 algorithms, journa
Range-separated tensor formats for numerical modeling of many-particle interaction potentials
We introduce and analyze the new range-separated (RS) canonical/Tucker tensor
format which aims for numerical modeling of the 3D long-range interaction
potentials in multi-particle systems. The main idea of the RS tensor format is
the independent grid-based low-rank representation of the localized and global
parts in the target tensor which allows the efficient numerical approximation
of -particle interaction potentials. The single-particle reference potential
like is split into a sum of localized and long-range low-rank
canonical tensors represented on a fine 3D Cartesian grid.
The smoothed long-range contribution to the total potential sum is represented
on the 3D grid in storage via the low-rank canonical/Tucker tensor. We
prove that the Tucker rank parameters depend only logarithmically on the number
of particles and the grid-size . Agglomeration of the short range part
in the sum is reduced to an independent treatment of localized terms with
almost disjoint effective supports, calculated in operations. Thus, the
cumulated sum of short range clusters is parametrized by a single low-rank
canonical reference tensor with a local support, accomplished by a list of
particle coordinates and their charges. The RS canonical/Tucker tensor
representations reduce the cost of multi-linear algebraic operations on the 3D
potential sums arising in modeling of multi-dimensional data by radial basis
functions, say, in computation of the electrostatic potential of a protein, in
3D integration and convolution transforms, computation of gradients, forces and
the interaction energy of a many-particle systems, and in low parametric
fitting of multi-dimensional scattered data by reducing all of them to 1D
calculations.Comment: 39 pages, 27 figure
Kriging in Tensor Train data format
Combination of low-tensor rank techniques and the Fast Fourier transform
(FFT) based methods had turned out to be prominent in accelerating various
statistical operations such as Kriging, computing conditional covariance,
geostatistical optimal design, and others. However, the approximation of a full
tensor by its low-rank format can be computationally formidable. In this work,
we incorporate the robust Tensor Train (TT) approximation of covariance
matrices and the efficient TT-Cross algorithm into the FFT-based Kriging. It is
shown that here the computational complexity of Kriging is reduced to
, where is the mode size of the estimation grid,
is the number of variables (the dimension), and is the rank of the TT
approximation of the covariance matrix. For many popular covariance functions
the TT rank remains stable for increasing and . The advantages of
this approach against those using plain FFT are demonstrated in synthetic and
real data examples.Comment: 19 pages,4 figures, 1 table, UNCECOMP 2019 3rd International
Conference on Uncertainty Quantification in Computational Sciences and
Engineering 24-26 June 2019, Crete, Greece https://2019.uncecomp.org
Tensor Numerical Methods for High-dimensional PDEs: Basic Theory and Initial Applications
We present a brief survey on the modern tensor numerical methods for
multidimensional stationary and time-dependent partial differential equations
(PDEs). The guiding principle of the tensor approach is the rank-structured
separable approximation of multivariate functions and operators represented on
a grid. Recently, the traditional Tucker, canonical, and matrix product states
(tensor train) tensor models have been applied to the grid-based electronic
structure calculations, to parametric PDEs, and to dynamical equations arising
in scientific computing. The essential progress is based on the quantics tensor
approximation method proved to be capable to represent (approximate) function
related -dimensional data arrays of size with log-volume complexity,
. Combined with the traditional numerical schemes, these novel
tools establish a new promising approach for solving multidimensional integral
and differential equations using low-parametric rank-structured tensor formats.
As the main example, we describe the grid-based tensor numerical approach for
solving the 3D nonlinear Hartree-Fock eigenvalue problem, that was the starting
point for the developments of tensor-structured numerical methods for
large-scale computations in solving real-life multidimensional problems. We
also address new results on tensor approximation of the dynamical Fokker-Planck
and master equations in many dimensions up to . Numerical tests
demonstrate the benefits of the rank-structured tensor approximation on the
aforementioned examples of multidimensional PDEs. In particular, the use of
grid-based tensor representations in the reduced basis of atomics orbitals
yields an accurate solution of the Hartree-Fock equation on large grids with a grid size of up to
Tucker Tensor analysis of Matern functions in spatial statistics
In this work, we describe advanced numerical tools for working with
multivariate functions and for the analysis of large data sets. These tools
will drastically reduce the required computing time and the storage cost, and,
therefore, will allow us to consider much larger data sets or finer meshes.
Covariance matrices are crucial in spatio-temporal statistical tasks, but are
often very expensive to compute and store, especially in 3D. Therefore, we
approximate covariance functions by cheap surrogates in a low-rank tensor
format. We apply the Tucker and canonical tensor decompositions to a family of
Matern- and Slater-type functions with varying parameters and demonstrate
numerically that their approximations exhibit exponentially fast convergence.
We prove the exponential convergence of the Tucker and canonical approximations
in tensor rank parameters. Several statistical operations are performed in this
low-rank tensor format, including evaluating the conditional covariance matrix,
spatially averaged estimation variance, computing a quadratic form,
determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a
large covariance matrix. Low-rank tensor approximations reduce the computing
and storage costs essentially. For example, the storage cost is reduced from an
exponential to a linear scaling , where
is the spatial dimension, is the number of mesh points in one
direction, and is the tensor rank. Prerequisites for applicability of the
proposed techniques are the assumptions that the data, locations, and
measurements lie on a tensor (axes-parallel) grid and that the covariance
function depends on a distance, .Comment: 23 pages, 2 diagrams, 2 tables, 9 figure
Low-Rank Tucker Approximation of a Tensor From Streaming Data
This paper describes a new algorithm for computing a low-Tucker-rank
approximation of a tensor. The method applies a randomized linear map to the
tensor to obtain a sketch that captures the important directions within each
mode, as well as the interactions among the modes. The sketch can be extracted
from streaming or distributed data or with a single pass over the tensor, and
it uses storage proportional to the degrees of freedom in the output Tucker
approximation. The algorithm does not require a second pass over the tensor,
although it can exploit another view to compute a superior approximation. The
paper provides a rigorous theoretical guarantee on the approximation error.
Extensive numerical experiments show that that the algorithm produces useful
results that improve on the state of the art for streaming Tucker
decomposition.Comment: 34 pages, 14 figure
Regularized Computation of Approximate Pseudoinverse of Large Matrices Using Low-Rank Tensor Train Decompositions
We propose a new method for low-rank approximation of Moore-Penrose
pseudoinverses (MPPs) of large-scale matrices using tensor networks. The
computed pseudoinverses can be useful for solving or preconditioning of
large-scale overdetermined or underdetermined systems of linear equations. The
computation is performed efficiently and stably based on the modified
alternating least squares (MALS) scheme using low-rank tensor train (TT)
decompositions and tensor network contractions. The formulated large-scale
optimization problem is reduced to sequential smaller-scale problems for which
any standard and stable algorithms can be applied. Regularization technique is
incorporated in order to alleviate ill-posedness and obtain robust low-rank
approximations. Numerical simulation results illustrate that the regularized
pseudoinverses of a wide class of non-square or nonsymmetric matrices admit
good approximate low-rank TT representations. Moreover, we demonstrated that
the computational cost of the proposed method is only logarithmic in the matrix
size given that the TT-ranks of a data matrix and its approximate pseudoinverse
are bounded. It is illustrated that a strongly nonsymmetric
convection-diffusion problem can be efficiently solved by using the
preconditioners computed by the proposed method.Comment: 28 page
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