Existence of dynamical low-rank approximations to parabolic problems

Bachmayr, Markus; Eisenmann, Henrik; Kieri, Emil; Uschmajew, André

doi:10.1090/mcom/3626

Cited by 12 publications

(14 citation statements)

References 59 publications

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“…Theorem 2 (Robust error bound, [11, Theorem 2.1]) Let A(t) denote the solution of the matrix differential equation (1). Assume that the following conditions hold in the Frobenius norm • = • F : 1.…”

Section: L-stepmentioning

confidence: 99%

“…This does not appear in the new algorithm. We mention that in [1], the problem of the backward substep for parabolic problems has recently been addressed in a different way. In the alternative variant of the projector-splitting integrator proposed in [4], which is based on a rearrangement of the terms in (4), the S-step is integrated forward in time.…”

Section: The Integrator Preserves (Skew-)symmetry If the Differential Equation Doesmentioning

confidence: 99%

“…We denote the solution value at t 1 by A 1 . The basis matrix computed in the first part of the integrator is denoted by U 1 i for each i = 1, . .…”

Section: The Error In the Initial Value Is δ-Smallmentioning

confidence: 99%

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An unconventional robust integrator for dynamical low-rank approximation

Ceruti

Lubich

2021

Bit Numer Math

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We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.

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Section: L-stepmentioning

confidence: 99%

Section: The Integrator Preserves (Skew-)symmetry If the Differential Equation Doesmentioning

confidence: 99%

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An unconventional robust integrator for dynamical low-rank approximation

Ceruti

Lubich

2021

Bit Numer Math

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“…Recently, the existence and uniqueness of the dynamical low rank approximation for a class of random semi-linear evolutionary equations was established in [19] and for linear parabolic equations in two space dimensions with a symmetric operator L in [3].…”

Section: Lemmamentioning

confidence: 99%

“…where C L > 0 is the coercivity constant defined in (4) and C P is the continuous embedding constant defined in (3).…”

Section: Stability Of the Continuous Problemmentioning

confidence: 99%

Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

2021

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We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a “parabolic” type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.

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Dynamical low‐rank approximations of solutions to the Hamilton–Jacobi–Bellman equation

Eigel

Schneider

Sommer

2022

Numerical Linear Algebra App

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We present a novel method to approximate optimal feedback laws for nonlinear optimal control based on low‐rank tensor train (TT) decompositions. The approach is based on the Dirac–Frenkel variational principle with the modification that the optimization uses an empirical risk. Compared to current state‐of‐the‐art TT methods, our approach exhibits a greatly reduced computational burden while achieving comparable results. A rigorous description of the numerical scheme and demonstrations of its performance are provided.

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Existence of dynamical low-rank approximations to parabolic problems

Cited by 12 publications

References 59 publications

An unconventional robust integrator for dynamical low-rank approximation

An unconventional robust integrator for dynamical low-rank approximation

Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

Dynamical low‐rank approximations of solutions to the Hamilton–Jacobi–Bellman equation

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