Title:

The Helmholtz equation in heterogeneous and random media : analysis and numerics

The Helmholtz equation is the simplest possible model of wave propagation, describing timeharmonic solutions of the wave equation. We study the elmholtz equation with heterogeneous and random coefficients, corresponding to wave propagation through a medium with spatially variable and random physical characteristics. In Chapter 2 we study hfiniteelement approximations of the (deterministic) heterogeneous Helmholtz equation. We prove the first sharp error bounds (explicit in the frequency) for the higherorder hfiniteelement method for the Helmholtz equation in heterogeneous media. In Chapter 3 we move on to the stochastic Helmholtz equation, i.e., the Helmholtz equation with random field coefficients. We prove the first frequencyindependent wellposedness results for the stochastic Helmholtz equation. (I.e. we prove existence and uniqueness of a solution, and we prove a frequencyexplicit bound on the solution, under a suitable assumption on the coefficients that corresponds to the problem being ‘nontrapping’ almost surely.) To prove this well posedness, we develop general, abstract arguments that allow us to prove wellposedness for three different formulations of stochastic PDEs; these arguments can be applied more widely than just to the stochastic Helmholtz equation. In Chapter 4 we study nearby preconditioning, a computational technique for speeding up solving many linear systems arising from discretisations of realisations of the stochastic Helmholtz equation. This speedup is achieved by reusing the preconditioner corresponding to one realisation for many other realisations. We prove rigorous results on when nearby preconditioning is effective, and investigate this effectiveness numerically in a range of situations. Also in Chapter 4, we combine nearby preconditioning with a QuasiMonteCarlo method (i.e., a highdimensional integration rule) to compute quantities of interest of the stochastic Helmholtz equation. We see that nearby preconditioning offers a significant computational saving, with 98% of linearsystem solves being made with a previouslycalculated preconditioner. As a byproduct of these results we also provide some preliminary computational evidence, of independent interest, that when using QMC methods for the Helmholtz equation the total number of realisations used must increase with the frequency. Finally, in Chapter 5, we study the MultiLevel MonteCarlo method for the Helmholtz equation, where one reduces the variance in an Uncertainty Quantification calculation by solving the underlying PDE on a hierarchy of meshes. We generalise the standard MultiLevel MonteCarlo convergence theory to the frequencydependent case and prove rigorous frequencyexplicit bounds on the computational cost of Monte Carlo and MultiLevel MonteCarlo methods for the Helmholtz equation. We see that in many cases the MultiLevel MonteCarlo method gives theoretical speedup over the MonteCarlo Method.
