SIAM Journal on Scientific Computing· 2014Q1
Alternating Minimal Energy Methods for Linear Systems in Higher Dimensions
- 259citations
- Q1SCImago
- 2014year
Short summary
New algorithms solve high-dimensional, low-rank linear systems by optimizing tensor components sequentially, achieving convergence comparable to or better than DMRG.
AI-generated from the title and abstract; the full text is not read.
Key points
- Proposes algorithms for high-dimensional linear systems with low-rank solutions.
- Optimizes tensor components sequentially, similar to DMRG.
- Improves convergence using inexact gradient directions.
- Proves geometrical convergence and estimates convergence rates.
- Demonstrates efficiency for both SPD and non-SPD systems, including CME-derived models.
AI-generated from the title and abstract; the full text is not read.
Abstract
We propose algorithms for the solution of high--dimensional symmetrical positive definite (SPD) linear systems with the matrix and the right hand side given and the solution sought in a low-rank format. Similarly to density matrix renormalization group (DMRG) algorithms, our methods optimize the components of the tensor product format subsequently. To improve the convergence, we expand the search space by an inexact gradient direction. We prove the geometrical convergence and estimate the convergence rate of the proposed methods utilizing the analysis of the steepest descent algorithm. The complexity of the presented algorithms is linear in the mode size and dimension, and the demonstrated convergence is comparable to or even better than the one of the DMRG algorithm. In the numerical experiment we show that the proposed methods are also efficient for non-SPD systems, for example those arising from the chemical master equation (CME) describing the gene regulatory model at the mesoscopic scale.
The authors' abstract, as published at the source. SIAM Journal on Scientific Computing, 2014 · DOI ↗
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Field: Computational Mathematics
Computational MathematicsMathematics