Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

<jats:p>This paper analyses the following question: let <jats:bold>A</jats:bold><jats:sub><jats:italic>j</jats:italic></jats:sub>, <jats:italic>j</jats:italic> = 1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇⋅ (<jats:italic>A</jats:italic><jats:sub><jats:italic>j</jats:italic></jats:sub>∇<jats:italic>u</jats:italic><jats:sub><jats:italic>j</jats:italic></jats:sub>) + <jats:italic>k</jats:italic><jats:sup>2</jats:sup><jats:italic>n</jats:italic><jats:sub><jats:italic>j</jats:italic></jats:sub><jats:italic>u</jats:italic><jats:sub><jats:italic>j</jats:italic></jats:sub> = −<jats:italic>f</jats:italic>. How small must <jats:inline-formula><jats:alternatives><jats:tex-math>$\|A_{1} -A_{2}\|_{L^{q}}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∥</mml:mo> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$\|{n_{1}} - {n_{2}}\|_{L^{q}}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>∥</mml:mo> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>−</mml:mo> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:msub> <mml:mrow> <mml:mo>∥</mml:mo> </mml:mrow> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>q</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> be (in terms of <jats:italic>k</jats:italic>-dependence) for GMRES applied to either <jats:inline-formula><jats:alternatives><jats:tex-math>$(\mathbf {A}_1)^{-1}\mathbf {A}_2$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mrow> <mml:mi>A</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> or <jats:bold>A</jats:bold><jats:sub>2</jats:sub>(<jats:bold>A</jats:bold><jats:sub>1</jats:sub>)<jats:sup>− 1</jats:sup> to converge in a <jats:italic>k</jats:italic>-independent number of iterations for arbitrarily large <jats:italic>k</jats:italic>? (In other words, for <jats:bold>A</jats:bold><jats:sub>1</jats:sub> to be a good left or right preconditioner for <jats:bold>A</jats:bold><jats:sub>2</jats:sub>?) We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with <jats:italic>random</jats:italic> coefficients <jats:italic>A</jats:italic> and <jats:italic>n</jats:italic>. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different <jats:italic>A</jats:italic> and <jats:italic>n</jats:italic>, and the answer to the question above dictates to what extent a previously calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.</jats:p>

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