A projector-splitting integrator for dynamical low-rank approximation

By Christian Lubich, Ivan V. Oseledets. 2013

240 citations recorded by the source corpus.

The dynamical low-rank approximation of time-dependent matrices is a low-rank factorization updating technique. It leads to differential equations for factors of the matrices, which need to be solved numerically. We propose and analyze a fully explicit, computationally inexpensive integrator that is based on splitting the orthogonal projector onto the tangent space of the low-rank manifold. As is shown by theory and illustrated by numerical experiments, the integrator enjoys robustness properties that are not shared by any standard numerical integrator. This robustness can be exploited to change the rank adaptively. Another application is in optimization algorithms for low-rank matrices where truncation back to the given low rank can be done efficiently by applying a step of the integrator proposed here.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

1. Andrzej S. Cichocki
2. Ivan V. Oseledets
3. Dmitry V. Savostyanov
4. V. Dolgov
5. Qibin Zhao
6. Anh Huy Phan
7. Danilo P. Mandic
8. Boris N. Khoromskij

In Computational Theory and Mathematics

1. Dmitry V. Savostyanov
2. V. Dolgov
3. Andrzej S. Cichocki
4. Ivan V. Oseledets
5. Qibin Zhao
6. Anh Huy Phan
7. Danilo P. Mandic
8. Boris N. Khoromskij

In Statistical and Nonlinear Physics

1. Dmitry V. Savostyanov
2. V. Dolgov
3. Andrzej S. Cichocki
4. Qibin Zhao
5. Namgil Lee
6. Danilo P. Mandic
7. Anh Huy Phan
8. Ivan V. Oseledets

Citations in the corpus

• A literature survey of low‐rank tensor approximation techniques (2013, neutral): “The efficient and robust numerical integration of (12) is crucial to the success of dynamical low-rank methods; apart from the references above, this aspect has been discussed in [184,185].”
• ALTERNATING MINIMAL ENERGY METHODS FOR LINEAR SYSTEMS IN HIGHER DIMENSIONS∗ (2014, methodology): “the TT cores x was proposed in [42] as the KSL scheme.”
• Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, unknown): “a† = ( 0 0 1 0 ) , I = ( 1 0 0 1 ) , s = ( 1 0 0 −1 ) , (142)”
• Tensor Networks for Dimensionality Reduction and Large-scale Optimization: Part 2 Applications and Future Perspectives (2017, result): “This leads to a much better convergence, and moreover an exactness result [Lubich and Oseledets, 2014] can be proven if A1 and A0 are on the manifold, and Y0 = A0, then this scheme is exact.”
• Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions (2013, unknown): “the TT cores x was proposed in [42] as the KSL scheme.”