SUMMARYIn this paper, we investigate strategies to improve the accuracy and efficiency of the ultra weak variational formulation (UWVF) of the Helmholtz equation. The UWVF is a Trefftz type, nonpolynomial method using basis functions derived from solutions of the adjoint Helmholtz equation. We shall consider three choices of basis function: propagating plane waves (original choice), Bessel basis functions, and evanescent wave basis functions. Traditionally, two‐dimensional triangular elements are used to discretize the computational domain. However, the element shapes affect the conditioning of the UWVF. Hence, we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom. Our results include the first tests of a plane wave method on meshes of mixed element types. In many modeling problems, evanescent waves occur naturally and are challenging to model. Therefore, we introduce evanescent wave basis functions for the first time in the UWVF to tackle rapidly decaying wave modes. The advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces.Copyright © 2013 John Wiley & Sons, Ltd.

We investigate the ultra weak variational formulation (UWVF) of the 2-D Helmholtz equation using a new choice of basis functions. Traditionally the UWVF basis functions are chosen to be plane waves. Here, we instead use first kind Bessel functions. We compare the performance of the two bases. Moreover, we show that it is possible to use coupled plane wave and Bessel bases in the same mesh. As test cases we shall consider propagating plane and evanescent waves in a rectangular domain and a singular 2-D Helmholtz problem in an L-shaped domain.

Time harmonic inverse scattering using accurate forward models is often computationally expensive. On the other hand, the use of computationally efficient solvers, such as the Born approximation, may fail if the targets do not satisfy the assumptions of the simplified model. In the Bayesian framework for inverse problems, one can construct a statistical model for the errors that are induced when approximate solvers are used, and hence increase the domain of applicability of the approximate model. In this paper, we investigate the error structure that is induced by the Born approximation and show that the Bayesian approximation error approach can be used to partially recover from these errors. In particular, we study the model problem of reconstruction of the index of refraction of a penetrable medium from measurements of the far field pattern of the scattered wave.

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