Chebyshev HOPGD with sparse grid sampling for parameterized linear systems
arxiv(2023)
摘要
We consider approximating solutions to parameterized linear systems of the
form A(μ_1,μ_2) x(μ_1,μ_2) = b, where (μ_1, μ_2) ∈ℝ^2. Here the matrix A(μ_1,μ_2) ∈ℝ^n × n is
nonsingular, large, and sparse and depends nonlinearly on the parameters
μ_1 and μ_2. Specifically, the system arises from a discretization of a
partial differential equation and x(μ_1,μ_2) ∈ℝ^n, b ∈ℝ^n. This work combines companion linearization with the Krylov
subspace method preconditioned bi-conjugate gradient (BiCG) and a decomposition
of a tensor matrix of precomputed solutions, called snapshots. As a result, a
reduced order model of x(μ_1,μ_2) is constructed, and this model can be
evaluated in a cheap way for many values of the parameters. Tensor
decompositions performed on a set of snapshots can fail to reach a certain
level of accuracy, and it is not known a priori if a decomposition will be
successful. Moreover, the selection of snapshots can affect both the quality of
the produced model and the computation time required for its construction. This
new method offers a way to generate a new set of solutions on the same
parameter space at little additional cost. An interpolation of the model is
used to produce approximations on the entire parameter space, and this method
can be used to solve a parameter estimation problem. Numerical examples of a
parameterized Helmholtz equation show the competitiveness of our approach. The
simulations are reproducible, and the software is available online.
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关键词
chebyshev hopgd,grid
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