31 research outputs found
Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States
We resume the recent successes of the grid-based tensor numerical methods and
discuss their prospects in real-space electronic structure calculations. These
methods, based on the low-rank representation of the multidimensional functions
and integral operators, led to entirely grid-based tensor-structured 3D
Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core
Hamiltonian and two-electron integrals (TEI) in complexity using
the rank-structured approximation of basis functions, electron densities and
convolution integral operators all represented on 3D
Cartesian grids. The algorithm for calculating TEI tensor in a form of the
Cholesky decomposition is based on multiple factorizations using algebraic 1D
``density fitting`` scheme. The basis functions are not restricted to separable
Gaussians, since the analytical integration is substituted by high-precision
tensor-structured numerical quadratures. The tensor approaches to
post-Hartree-Fock calculations for the MP2 energy correction and for the
Bethe-Salpeter excited states, based on using low-rank factorizations and the
reduced basis method, were recently introduced. Another direction is related to
the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for
finite lattice-structured systems, where one of the numerical challenges is the
summation of electrostatic potentials of a large number of nuclei. The 3D
grid-based tensor method for calculation of a potential sum on a lattice manifests the linear in computational work, ,
instead of the usual scaling by the Ewald-type approaches
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
Efficient computation of highly oscillatory integrals by using QTT tensor approximation
We propose a new method for the efficient approximation of a class of highly
oscillatory weighted integrals where the oscillatory function depends on the
frequency parameter , typically varying in a large interval. Our
approach is based, for fixed but arbitrary oscillator, on the pre-computation
and low-parametric approximation of certain -dependent prototype
functions whose evaluation leads in a straightforward way to recover the target
integral. The difficulty that arises is that these prototype functions consist
of oscillatory integrals and are itself oscillatory which makes them both
difficult to evaluate and to approximate. Here we use the quantized-tensor
train (QTT) approximation method for functional -vectors of logarithmic
complexity in in combination with a cross-approximation scheme for TT
tensors. This allows the accurate approximation and efficient storage of these
functions in the wide range of grid and frequency parameters. Numerical
examples illustrate the efficiency of the QTT-based numerical integration
scheme on various examples in one and several spatial dimensions.Comment: 20 page
On identification of self-similar characteristics using the Tensor Train decomposition method with application to channel turbulence flow
A study on the application of the Tensor Train decomposition method to 3D direct numerical simulation data of channel turbulence flow is presented. The approach is validated with respect to compression rate and storage requirement. In tests with synthetic data, it is found that grid-aligned self-similar patterns are well captured, and also the application to non grid-aligned self-similarity yields satisfying results. It is observed that the shape of the input Tensor significantly affects the compression rate. Applied to data of channel turbulent flow, the Tensor Train format allows for surprisingly high compression rates whilst ensuring low relative errors
TT-NF: Tensor Train Neural Fields
Learning neural fields has been an active topic in deep learning research,
focusing, among other issues, on finding more compact and easy-to-fit
representations. In this paper, we introduce a novel low-rank representation
termed Tensor Train Neural Fields (TT-NF) for learning neural fields on dense
regular grids and efficient methods for sampling from them. Our representation
is a TT parameterization of the neural field, trained with backpropagation to
minimize a non-convex objective. We analyze the effect of low-rank compression
on the downstream task quality metrics in two settings. First, we demonstrate
the efficiency of our method in a sandbox task of tensor denoising, which
admits comparison with SVD-based schemes designed to minimize reconstruction
error. Furthermore, we apply the proposed approach to Neural Radiance Fields,
where the low-rank structure of the field corresponding to the best quality can
be discovered only through learning.Comment: Preprint, under revie