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Kolmogorov widths and low-rank approximations of parametric elliptic PDEs. (English) Zbl 1358.65072

Summary: Kolmogorov \( n\)-widths and low-rank approximations are studied for families of elliptic diffusion partial differential equations (PDEs) parametrized by the diffusion coefficients. The decay of the \( n\)-widths can be controlled by that of the error achieved by best \( n\)-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the \( n\)-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or proper orthogonal decompostion methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the \( n\)-widths.

MSC:

65N99 Numerical methods for partial differential equations, boundary value problems
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy

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