Papers › Preconditioned Chebyshev BiCG for parameterized linear systems
Preconditioned Chebyshev BiCG for parameterized linear systems
Siobhán Correnty, Elias Jarlebring, Daniel B. Szyld
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We consider the problem of approximating the solution to A(μ) x(μ) = b for many different values of the parameter μ. Here we assume A(μ) is large, sparse, and nonsingular with a nonlinear dependence on μ. Our method is based on a companion linearization derived from an accurate Chebyshev interpolation of A(μ) on the interval [-a,a], a ∈ℝ. The solution to the linearization is approximated in a preconditioned BiCG setting for shifted systems, where the Krylov basis matrix is formed once. This process leads to a short-term recurrence method, where one execution of the algorithm produces the approximation to x(μ) for many different values of the parameter μ∈[-a,a] simultaneously. In particular, this work proposes one algorithm which applies a shift-and-invert preconditioner exactly as well as an algorithm which applies the preconditioner inexactly. The competitiveness of the algorithms are illustrated with large-scale problems arising from a finite element discretization of a Helmholtz equation with parameterized material coefficient. The software used in the simulations is publicly available online, and thus all our experiments are reproducible.
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