Solution of the Fokker–Planck Equation by Cross Approximation Method in the Tensor Train Format

Chertkov, Andrei; Oseledets, Ivan

doi:10.3389/frai.2021.668215

Cited by 12 publications

(7 citation statements)

References 25 publications

(57 reference statements)

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“…Proof The equivalence between (47) and ( 48) is an immediate consequence of (7). We now prove that ( 46) is equivalent to (47).…”

Section: Proposition 3 (Geometric Interpretation Of Rank Addition) Letmentioning

confidence: 76%

“…In order to reduce the number of degrees of freedom in the solution tensor f (t), we seek a representation of the solution on a low-rank tensor format [7,13,22] for all t ≥ 0. To complement the low-rank structure of f (t), we also represent the operator N in a compatible low-rank format, allowing for an efficient computation of N( f (t)) at each time t in (2).…”

Section: Introductionmentioning

confidence: 99%

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Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

2022

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We develop new adaptive algorithms for temporal integration of nonlinear evolution equations on tensor manifolds. These algorithms, which we call step-truncation methods, are based on performing one time step with a conventional time-stepping scheme, followed by a truncation operation onto a tensor manifold. By selecting the rank of the tensor manifold adaptively to satisfy stability and accuracy requirements, we prove convergence of a wide range of step-truncation methods, including explicit one-step and multi-step methods. These methods are very easy to implement as they rely only on arithmetic operations between tensors, which can be performed by efficient and scalable parallel algorithms. Adaptive step-truncation methods can be used to compute numerical solutions of high-dimensional PDEs, which, have become central to many new areas of application such optimal mass transport, random dynamical systems, and mean field optimal control. Numerical applications are presented and discussed for a Fokker-Planck equation with spatially dependent drift on a flat torus of dimension two and four.

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“…Proof The equivalence between (47) and ( 48) is an immediate consequence of (7). We now prove that ( 46) is equivalent to (47).…”

Section: Proposition 3 (Geometric Interpretation Of Rank Addition) Letmentioning

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

2022

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“…In References 15‐17, it was observed that the straightforward application of TT optimization‐based solvers to linear systems arising from PDEs with matrices in the QTT format, leads to severe numerical instabilities. This problem was formalized in Reference 18 and originates from both ill conditioning of discretized differential operators and the ill conditioning of the tensor representations themselves.…”

Section: Introductionmentioning

confidence: 99%

Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

Vysotsky

Rakhuba

2022

Numerical Linear Algebra App

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In this paper, we are concerned with the inversion of circulant matrices and their quantized tensor‐train (QTT) structure. In particular, we show that the inverse of a complex circulant matrix A$$ A $$, generated by the first column of the form false(a0,…,amprefix−1,0,…,0,aprefix−n,…,aprefix−1false)⊤$$ {\left({a}_0,\dots, {a}_{m-1},0,\dots, 0,{a}_{-n},\dots, {a}_{-1}\right)}^{\top } $$ admits a QTT representation with the QTT ranks bounded by false(m+nfalse)$$ \left(m+n\right) $$. Under certain assumptions on the entries of A$$ A $$, we also derive an explicit QTT representation of Aprefix−1$$ {A}^{-1} $$. The latter can be used, for instance, to overcome stability issues arising when numerically solving differential equations with periodic boundary conditions in the QTT format.

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“…Tensor networks have been successfully applied to a wide range of studies, including applications to solving partial differential equations, − quantum dynamics methods, − machine learning, electronic structure calculations, − and calculations of vibrational states. − …”

Section: Introductionmentioning

confidence: 99%

Functional Tensor-Train Chebyshev Method for Multidimensional Quantum Dynamics Simulations

Soley

Bergold

Gorodetsky

et al. 2021

J. Chem. Theory Comput.

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Methods for efficient simulations of multidimensional quantum dynamics are essential for theoretical studies of chemical systems where quantum effects are important, such as those involving rearrangements of protons or electronic configurations. Here, we introduce the functional tensor-train Chebyshev (FTTC) method for rigorous nuclear quantum dynamics simulations. FTTC is essentially the Chebyshev propagation scheme applied to the initial state represented in a continuous analogue tensor-train format. We demonstrate the capabilities of FTTC as applied to simulations of proton quantum dynamics in a 50-dimensional model of hydrogen-bonded DNA base pairs.

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Solution of the Fokker–Planck Equation by Cross Approximation Method in the Tensor Train Format

Cited by 12 publications

References 25 publications

Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

Adaptive Integration of Nonlinear Evolution Equations on Tensor Manifolds

Tensor rank bounds and explicit QTT representations for the inverses of circulant matrices

Functional Tensor-Train Chebyshev Method for Multidimensional Quantum Dynamics Simulations

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