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An unconventional robust integrator for dynamical low-rank approximation
Gianluca Ceruti, Christian Lubich
TL;DR
The paper addresses robust dynamical low-rank approximation for large time-dependent matrices and Tucker tensors when small singular values make standard factor integration problematic. It proposes a different integrator with projector-splitting-level robustness, while avoiding backward substeps and supporting parallelism and symmetry preservation, subject to limits on time reversibility.
Problem
Standard factor integration can suffer severe step-size restrictions from small singular values in dynamical low-rank approximation.
Method
The paper proposes and analyses a different low-rank integrator for matrices and extends it to Tucker tensors of fixed multilinear rank.
Results
The new integrator has the same exactness property and robust error bound as the matrix and Tucker projector-splitting integrators.
Takeaways & Limitations
The method avoids backward time integration, offers enhanced parallelism, and preserves symmetry or anti-symmetry when the differential equation does.
Takeaways & Limitations
The new integrator cannot be efficiently extended to a time-reversible integrator.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper proposes a low-rank matrix and Tucker-tensor integrator that retains robustness to small singular values while avoiding backward substeps and enabling additional structural and parallel advantages.
- Standard integration of dynamical low-rank factor equations suffers severe step-size restrictions when retained singular values are small.The projector-splitting approach avoids this difficulty through tangent-space projection splitting.
- Unlike projector splitting, the method avoids backward time integration, offers more parallelism, and preserves symmetry or anti-symmetry when the differential equation does.The backward substep can be unstable for strongly dissipative problems.
- The method has a limitation relative to projector splitting: it cannot be efficiently extended to a time-reversible integrator.For time-dependent Schrödinger problems, it preserves the norm but lacks the projector-splitting integrator’s demonstrated energy conservation.
- The new integrator has the same robust error behaviour and exactness properties as the matrix and Tucker projector-splitting integrators.The paper establishes these properties for both the matrix and Tucker tensor cases.
- The paper extends the proposed approach from time-dependent matrices to tensors represented in the Tucker format with fixed multilinear rank.The paper also illustrates both matrix and Tucker tensor integrators numerically.
2 Recap: the matrix projector-splitting integrator
Dynamical low-rank approximation evolves a fixed-rank factorization through tangent-space projection, while projector splitting integrates the resulting subproblems efficiently and robustly.
- Dynamical low-rank approximation projects the differential-equation right-hand side onto the tangent space of fixed-rank matrices.The approximation is represented as Y(t) = U(t)S(t)V(t)⊤ with orthonormal factor matrices and an invertible core matrix.
- The projector-splitting method decomposes the tangent projection into three subprojections and solves the corresponding K-, S-, and L-steps consecutively.The steps update the factor matrices and the small core matrix over one time interval.
- Projector splitting exactly reproduces rank-r matrix solutions under its stated invertibility assumptions.Starting from the exact solution at t0, one step yields Y1 = A(t1).
- Small singular values are common in accurate low-rank approximations, making robustness essential because factor equations contain S(t)^−1.The smallest retained singular value cannot generally be much larger than the largest discarded singular value.
- Its robust error bound is independent of singular values of the exact or approximate solution.The bound assumes Lipschitz continuity and boundedness, a small non-tangential component, and a small initial error.
- Inexact substep solutions add an error controlled by local substep errors, with constants still independent of small singular values.
3 A new robust low-rank matrix integrator
The new matrix integrator matches the projector-splitting method’s exactness and robust error behavior while enabling parallel substeps, forward-only small-matrix integration, and symmetry preservation.
- Its matrix differential equations and QR decompositions can be solved in parallel, increasing available computational parallelism.
- The small r ×r matrix equation is integrated forward rather than backward, avoiding an unstable substep for strongly dissipative problems.
- The algorithm updates the rank-r factors through K-, L-, QR, and S-steps, with the substep differential equations solved using standard numerical integrators.
- The alternative fully parallel variant requires inverse bounds that may force step sizes to be small relative to the smallest nonzero singular value and behaved less reliably in experiments.
- The new integrator has the same exactness property and robust error bound as the matrix projector-splitting integrator.
- The method preserves symmetry or skew-symmetry when the differential equation preserves that structure, unlike the projector-splitting integrator.
4 Recap: the Tucker tensor projector-splitting integrator
Dynamical low-rank approximation represents time-dependent tensors with fixed multilinear rank in Tucker form and projects their differential equations onto the corresponding tangent space.
- The Tucker formulation approximates a time-dependent tensor by a solution of its differential equation projected onto the tangent space of fixed multilinear rank.
- Tucker tensors use orthonormal slim basis matrices and a full-multilinear-rank core tensor.
- The Tucker tangent-space projection is an alternating sum of 2d −1 subprojections, enabling a projector-splitting integrator that processes the modes sequentially.
5 A new robust low-rank Tucker tensor integrator
The new Tucker tensor integrator extends the matrix method while retaining exactness and robust error bounds independent of small singular values. Its simultaneous factor updates enable parallelism, avoid backward integration, preserve (anti-)symmetry, but do not efficiently support time reversibility.
- The matrix integrator extends naturally to Tucker tensors while retaining exactness and robust error bounds independent of small matricization singular values.
- Unlike projector-splitting Tucker integrators, it uses no backward-time equations or reduced-square matrix equations and preserves (anti-)symmetry when the differential equation does.
- The new Tucker integrator has no apparently efficient construction of a time-reversible method.
- Its Tucker step first computes parallel QR factorizations and factor updates, then advances the core tensor differential equation to obtain the updated tensor.
- The integrator updates all factor matrices simultaneously, enabling parallel differential-equation solves and QR decompositions.
Y0 Xd
The analysis establishes exactness and robust global error bounds for the Tucker integrator under regularity, tangential-error, and initial-error assumptions. These bounds, including local-error constants, are independent of singular values, and the method preserves tensor symmetry properties under permutation-compatible dynamics.
- 5.3 Robust error bound: The global error bound assumes bounded Lipschitz dynamics, an ε-small non-tangential component, and a δ-small initial error.
- 5.3 Robust error bound: The global-error constants depend on the Lipschitz and boundedness parameters, final time, and tensor order, but not on singular values of exact or approximate matricizations.
- 5.3 Robust error bound: The local-error constant is independent of singular values of the exact or approximate solution.
- 5.4 Symmetric and anti-symmetric low-rank Tucker tensors: If the right-hand side is permutation-compatible, the new integrator preserves symmetry or anti-symmetry from symmetric or anti-symmetric initial data.
6 Numerical Experiments
Numerical experiments compare the new method with explicit Runge–Kutta and projector-splitting approaches for matrix and tensor problems. They show robustness to small singular values and indicate that more accurate substep integration permits larger time steps for a prescribed error.
- 6 Numerical Experiments: The new low-rank integrator avoids the step-size restriction required by explicit Runge–Kutta in the presence of small singular values.
- 6.2 Error behaviour: Figures 2 and 3 compare singular values and final-time errors across ranks, step sizes, and substep integrators for matrix and tensor differential equations.
- 6.2 Error behaviour: Higher-accuracy substep solves allow larger step sizes while achieving a prescribed error.
- 6 Numerical Experiments: A moderately stiff Schrödinger problem compares projector-splitting and the new matrix integrator at final time T = 5 across ranks 1 through 20.