Vesa Kaarnioja’s research while affiliated with Freie Universität Berlin and other places

The theoretical development of quasi-Monte Carlo (QMC) methods for uncertainty quantification of partial differential equations (PDEs) is typically centered around simplified model problems such as elliptic PDEs subject to homogeneous zero Dirichlet boundary conditions. In this paper, we present a theoretical treatment of the application of randomly shifted rank-1 lattice rules to electrical impedance tomography (EIT). EIT is an imaging modality, where the goal is to reconstruct the interior conductivity of an object based on electrode measurements of current and voltage taken at the boundary of the object. This is an inverse problem, which we tackle using the Bayesian statistical inversion paradigm. As the reconstruction, we consider QMC integration to approximate the unknown conductivity given current and voltage measurements. We prove under moderate assumptions placed on the parameterization of the unknown conductivity that the QMC approximation of the reconstructed estimate has a dimension-independent, faster-than-Monte Carlo cubature convergence rate. Finally, we present numerical results for examples computed using simulated measurement data.

There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Recent studies by Chernov and Le (Comput. Math. Appl., 2024, and SIAM J. Numer. Anal., 2024) as well as Harbrecht, Schmidlin, and Schwab (Math. Models Methods Appl. Sci., 2024) analyze the setting wherein the input random field is assumed to be uniformly bounded with respect to the uncertain parameters. In this paper, we relax this assumption and allow for parameter-dependent bounds. The parametric inputs are modeled as generalized Gaussian random variables, and we analyze the application of quasi-Monte Carlo (QMC) integration to assess the PDE response statistics using randomly shifted rank-1 lattice rules. In addition to the QMC error analysis, we also consider the dimension truncation and finite element errors in this setting.

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.

Convergence rates for $L_2$ approximation in a Hilbert space H are a central theme in numerical analysis. The present work is inspired by Schaback (Math. Comp., 1999), who showed, in the context of best pointwise approximation for radial basis function interpolation, that the convergence rate for sufficiently smooth functions can be doubled, compared to the general rate for functions in the "native space" H. Motivated by this, we obtain a general result for H-orthogonal projection onto a finite dimensional subspace of H: namely, that any known $L_2$ convergence rate for all functions in H translates into a doubled $L_2$ convergence rate for functions in a smoother normed space B, along with a similarly improved error bound in the H-norm, provided that $L_2$, H and B are suitably related. As a special case we improve the known $L_2$ and H-norm convergence rates for kernel interpolation in reproducing kernel Hilbert spaces, with particular attention to a recent study (Kaarnioja, Kazashi, Kuo, Nobile, Sloan, Numer. Math., 2022) of periodic kernel-based interpolation at lattice points applied to parametric partial differential equations. A second application is to radial basis function interpolation for general conditionally positive definite basis functions, where again the $L_2$ convergence rate is doubled, and the convergence rate in the native space norm is similarly improved, for all functions in a smoother normed space B.

We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model proposed by Kaarnioja, Kuo, and Sloan (SIAM J.~Numer.~Anal.~2020), in which a countable number of independent random variables enter the random field as periodic functions, it was shown by Kaarnioja, Kazashi, Kuo, Nobile, and Sloan (Numer.~Math.~2022) that the lattice-based kernel interpolant can be constructed for the PDE solution as a function of the stochastic variables in a highly efficient manner using fast Fourier transform (FFT). In this work, we discuss the connection between our model and the popular ``affine and uniform model'' studied widely in the literature of uncertainty quantification for PDEs with uncertain coefficients. We also propose a new class of weights entering the construction of the kernel interpolant -- \emph{serendipitous weights} -- which dramatically improve the computational performance of the kernel interpolant for PDE problems with uncertain coefficients, and allow us to tackle function approximation problems up to very high dimensionalities. Numerical experiments are presented to showcase the performance of the serendipitous weights.

Parametric mathematical models such as partial differential equations with random coefficients have received a lot of attention within the field of uncertainty quantification. The model uncertainties are often represented via a series expansion in terms of the parametric variables. In practice, this series expansion needs to be truncated to a finite number of terms, introducing a dimension truncation error to the numerical simulation of a parametric mathematical model. There have been several studies of the dimension truncation error corresponding to different models of the input random field in recent years, but many of these analyses have been carried out within the context of numerical integration. In this paper, we study the $L^2$ dimension truncation error of the parametric model problem. Estimates of this kind arise in the assessment of the dimension truncation error for function approximation in high dimensions. In addition, we show that the dimension truncation error rate is invariant with respect to certain transformations of the parametric variables. Numerical results are presented which showcase the sharpness of the theoretical results.