Erik Burman

In this paper, we analyze posterior consistency of a Bayesian data assimilation problem under discretization. We prove convergence rates for the discrete posterior to the ground truth solution under both conforming discretization and finite element discretization (usually nonconforming). The analysis is based on the coupling of asymptotics between the number of samples and the dimension of discrete spaces. In the finite element discretization, tailor-made discrete priors, instead of the discretization of continuous priors, are used to generate posterior contraction.

We revisit the direct method for reconstructing coefficients in second-order elliptic partial differential equations. Two model problems are considered: first, the reconstruction of the diffusion coefficient in a scalar elliptic problem, and second, the reconstruction of the shear modulus in the elastography problem. To highlight the versatility of the framework, different notions of stability are exploited in the two situations. In the scalar case, the system is interpreted as a hyperbolic transport equation and an inf-sup condition on the discrete level is leveraged for the analysis of the numerical method. We obtain error estimates on the reconstruction coefficient that are suboptimal by half an order, which is known to be sharp on general meshes. In the vector case, the minimization of the residual in dual norm and a stability result on the continuous problem lead to error estimates that are optimal compared to the approximation. For both problems, the theoretical results are illustrated by some numerical examples.

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility methods for ill-posed problems with respect to Poisson’s equations. By carefully choosing test and trial spaces, a formulation that is stable in a certain residual norm is obtained. Numerical stability and optimal convergence are established based on the conditional stability property of the problem. Tikhonov regularization is necessary for high-order polynomial approximation, but its weak consistency may be tuned to allow for optimal convergence. For low-order elements, a simple numerical scheme with optimal convergence is obtained without stabilization. We also provide a guideline for feasible pairs of finite element spaces that satisfy suitable stability assumptions. Numerical experiments are provided to illustrate the theoretical results.

We revisit the direct method for the reconstruction of coefficients in second order elliptic partial differential equations. Two model problems are considered: first the reconstruction of the diffusion coefficient in a scalar elliptic problem and second the reconstruction of the shear modulus in the elastography problem. To highlight the versatility of the framework, different notions of stability are exploited in the two situations. In the scalar case, the system is interpreted as a hyperbolic transport equation and an inf-sup condition on the discrete level is leveraged for the analysis of the numerical method. We obtain error estimates on the reconstruction coefficient that are suboptimal by half an order, which is known to be sharp on general meshes. In the vector case, the minimization of the residual in dual norm and a stability result on the continuous problem lead to error estimates that are optimal compared to the approximation. For both problems, the theoretical results are illustrated by some numerical examples.

We consider a unique continuation problem for the wave equation given data in a volumetric subset of the space time domain. In the absence of data on the lateral boundary of the space-time cylinder we prove that the solution can be continued with Hölder stability into a certain proper subset of the space-time domain. Additionally, we show that unique continuation of the solution to the entire space-time cylinder with Lipschitz stability is possible given the knowledge of a suitable finite dimensional space in which the trace of the solution on the lateral boundary is contained. These results allow us to design a finite element method that provably converges to the exact solution at a rate that mirrors the stability properties of the continuous problem.

In this paper, we present and analyze a stabilized hybridized Nitsche method for elliptic problems with sign-changing coefficients without imposing symmetry assumptions on the mesh around the material interfaces. The use of a stabilized primal-dual formulation allows us to cope with the sign-changing nature of the problem and to prove optimal error estimates under two assumptions on the continuous problem, namely that it admits a unique solution and that the contrast at the sign-changing interface lies outside a certain critical interval. The method can be used on arbitrary shape-regular meshes (fitted to material interfaces) and yields optimal convergence rates for smooth solutions. As an illustration, the method is applied to simulate a realistic acoustic cloaking device.