Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems

In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. 10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties.

Eulerian time-stepping schemes for the non-stationary Stokes equations on time-dependent domains

This article is concerned with the discretisation of the Stokes equations on time-dependent domains in an Eulerian coordinate framework. Our work can be seen as an extension of a recent paper by Lehrenfeld and Olshanskii (ESAIM: M2AN 53(2):585–614, 2019), where BDF-type time-stepping schemes are studied for a parabolic equation on moving domains. For space discretisation, a geometrically unfitted finite element discretisation is applied in combination with Nitsche’s method to impose boundary conditions. Physically undefined values of the solution at previous time-steps are extended implicitly by means of so-called ghost penalty stabilisations. We derive a complete a priori error analysis of the discretisation error in space and time, including optimal $$L^2(L^2)$$ L 2 ( L 2 ) -norm error bounds for the velocities. Finally, the theoretical results are substantiated with numerical examples.

Provably non-stiff implementation of weak coupling conditions for hyperbolic problems

In the context of coupling hyperbolic problems, the maximum stable time step of an explicit numerical scheme may depend on the design of the coupling procedure. If this is the case, the coupling procedure is sensitive to changes in model parameters independent of the Courant–Friedrichs–Levy condition. This sensitivity can cause artificial stiffness that degrades the performance of a numerical scheme. To overcome this problem, we present a systematic and general procedure for weakly imposing coupling conditions via penalty terms in a provably non-stiff manner. The procedure can be used to construct both energy conservative and dissipative couplings, and the user is given control over the amount of dissipation desired. The resulting formulation is simple to implement and dual consistent. The penalty coefficients take the form of projection matrices based on the coupling conditions. Numerical experiments demonstrate that this procedure results in both optimal spectral radii and superconvergent linear functionals.

Boundary element methods for the wave equation based on hierarchical matrices and adaptive cross approximation

Time-domain Boundary Element Methods (BEM) have been successfully used in acoustics, optics and elastodynamics to solve transient problems numerically. However, the storage requirements are immense, since the fully populated system matrices have to be computed for a large number of time steps or frequencies. In this article, we propose a new approximation scheme for the Convolution Quadrature Method powered BEM, which we apply to scattering problems governed by the wave equation. We use $${\mathscr {H}}^2$$ H 2 -matrix compression in the spatial domain and employ an adaptive cross approximation algorithm in the frequency domain. In this way, the storage and computational costs are reduced significantly, while the accuracy of the method is preserved.

Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm $$<1/2$$ < 1 / 2 as an operator on the natural trace space $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) whenever $$\Gamma $$ Γ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) for any sequence of finite-dimensional subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$H^{1/2}(\Gamma )$$ H 1 / 2 ( Γ ) . Long-standing open questions are whether the essential norm is also $$<1/2$$ < 1 / 2 for D as an operator on $$L^2(\Gamma )$$ L 2 ( Γ ) for all Lipschitz $$\Gamma $$ Γ in 2-d; or whether, for all Lipschitz $$\Gamma $$ Γ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators $$\pm \frac{1}{2}I+D$$ ± 1 2 I + D are compact perturbations of coercive operators—this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces $$({{\mathcal {H}}}_N)_{N=1}^\infty $$ ( H N ) N = 1 ∞ that is asymptotically dense in $$L^2(\Gamma )$$ L 2 ( Γ ) . We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is $$\ge 1/2$$ ≥ 1 / 2 , and examples with Lipschitz constant two for which the operators $$\pm \frac{1}{2}I +D$$ ± 1 2 I + D are not coercive plus compact. We also give, for every $$C>0$$ C > 0 , examples of Lipschitz polyhedra for which the essential norm is $$\ge C$$ ≥ C and for which $$\lambda I+D$$ λ I + D is not a compact perturbation of a coercive operator for any real or complex $$\lambda $$ λ with $$|\lambda |\le C$$ | λ | ≤ C . We then, via a new result on the Galerkin method in Hilbert spaces, explore the implications of these results for the convergence of Galerkin boundary element methods in the $$L^2(\Gamma )$$ L 2 ( Γ ) setting. Finally, we resolve negatively a related open question in the convergence theory for collocation methods, showing that, for our polyhedral examples, there is no weighted norm on $$C(\Gamma )$$ C ( Γ ) , equivalent to the standard supremum norm, for which the essential norm of D on $$C(\Gamma )$$ C ( Γ ) is $$<1/2$$ < 1 / 2 .

Fortin operator for the Taylor–Hood element

We design a local Fortin operator for the lowest-order Taylor–Hood element in any dimension, which was previously constructed only in 2D. In the construction we use tangential edge bubble functions for the divergence correcting operator. This naturally leads to an alternative inf-sup stable reduced finite element pair. Furthermore, we provide a counterexample to the inf-sup stability and hence to existence of a Fortin operator for the $$P_2$$ P 2 –$$P_0$$ P 0 and the augmented Taylor–Hood element in 3D.

Kernel-independent adaptive construction of $$\mathcal {H}^2$$-matrix approximations

A method for the kernel-independent construction of $$\mathcal {H}^2$$ H 2 -matrix approximations to non-local operators is proposed. Special attention is paid to the adaptive construction of nested bases. As a side result, new error estimates for adaptive cross approximation (ACA) are presented which have implications on the pivoting strategy of ACA.

Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification

This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice—a setting that, as pointed out by Zeng et al. (Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, New York, 2006) and Zeng et al. (Constr. Approx. 30: 529–555, 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja et al. (SIAM J Numer Anal 58(2): 1068–1091, 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.