Existence of dynamical low-rank approximations to parabolic problems

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

doi:10.48550/arxiv.2002.12197

Cited by 1 publication

(2 citation statements)

References 41 publications

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“…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.…”

Section: The Integrator Preserves (Skew-)symmetry If the Differential...mentioning

confidence: 99%

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An unconventional robust integrator for dynamical low-rank approximation

Ceruti¹,

Lubich²

2020

Preprint

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We propose and analyse a numerical integrator that computes a lowrank 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. Keywordsdynamical low-rank approximation • structure-preserving integrator • matrix and tensor differential equations • Tucker tensor format Mathematics Subject Classification (2000) 65L05 • 65L20 • 65L70 • 15A69

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“…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.…”

Section: The Integrator Preserves (Skew-)symmetry If the Differential...mentioning

confidence: 99%

“…Let A 1 be the solution at time t 1 = t 0 + h of the full problem (1) with initial condition A 0 . Assume that conditions 1.-3. of Theorem 2 are fulfilled.…”

Section: Lemmamentioning

confidence: 99%

An unconventional robust integrator for dynamical low-rank approximation

Ceruti¹,

Lubich²

2020

Preprint

View full text Add to dashboard Cite

show abstract

Existence of dynamical low-rank approximations to parabolic problems

Cited by 1 publication

References 41 publications

An unconventional robust integrator for dynamical low-rank approximation

An unconventional robust integrator for dynamical low-rank approximation

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