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PLSS: A PROJECTED LINEAR SYSTEMS SOLVER
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DOI：10.1137/22M1509783.png)
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Abstract
En 中文
We propose iterative projection methods for solving square or rectangular consistent linear systems Ax = b. Existing projection methods use sketching matrices (possibly randomized) to generate a sequence of small projected subproblems, but even the smaller systems can be costly. We develop a process that appends one column to the sketching matrix each iteration and converges in a finite number of iterations whether the sketch is random or deterministic. In general, our process generates orthogonal updates to the approximate solution x(k). By choosing the sketch to be the set of all previous residuals, we obtain a simple recursive update and convergence in at most rank(A) iterations (in exact arithmetic). By choosing a sequence of identity columns for the sketch, we develop a generalization of the Kaczmarz method. In experiments on large sparse systems, our method (PLSS) with residual sketches is competitive with LSQR and LSMR and with residual and identity sketches compares favorably with state-of-the-art randomized methods.
Keywords:
linear systems
iterative solver
randomized numerical linear algebra
projection method
LSQR
Kaczmarz method
LSMR
Craig's method
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