arXiv subjects

Oliver G. Ernst

Publications and source records attributed to Oliver G. Ernst.

8 recordsLinked to original sources

arXiv2026

Learning to Integrate

We address uncertainty quantification for a generic input distribution to some resource-intensive simulation, e.g., requiring the solution of a partial differential equation. While efficient numerical methods exist to compute integrals for high-dimensional Gaussian and other separable distributions based on sparse grids (SG), input data arising in practice often does not fall into this class. We therefore employ transport maps to transform complex distributions to multivariate standard normals. In generative learning, a number of neural network architectures have been introduced that accomplish this task approximately. Examples are affine coupling flows (ACF) and ordinary differential equation-based networks such as conditional flow matching (CFM). To compute the expectation of a quantity of interest, we numerically integrate the composition of the inverse of the learned transport map with the simulation code output. As this map is integrated over a multivariate Gaussian distribution, SG techniques can be applied. Viewing the images of the SG quadrature nodes as learned quadrature nodes for a given complex distribution motivates our title. We demonstrate our method for monomials of total degrees for which the unmapped SG rules are exact. We also apply our approach to the stationary diffusion equation with coefficients modeled by exponentiated Lévy random fields, using modal expansions of Karhunen--Loève type with 9 and 25 modes. In a series of numerical experiments, we investigate errors due to learning accuracy, quadrature, statistical estimation, truncation of the modal series of the input random field, and training data size for three normalizing flows (ACF, conditional flow matching and optimal transport flow matching). We discuss the mathematical assumptions on which our approach is based and demonstrate its shortcomings when these are violated.

math.NA↗

arXiv2023

Efficient solution of parameter identification problems with $H^1$ regularization

We consider the identification of spatially distributed parameters under $H^1$ regularization. Solving the associated minimization problem by Gauss-Newton iteration results in linearized problems to be solved in each step that can be cast as boundary value problems involving a low-rank modification of the Laplacian. Using algebraic multigrid as a fast Laplace solver, the Sherman-Morrison-Woodbury formula can be employed to construct a preconditioner for these linear problems which exhibits excellent scaling w.r.t. the relevant problem parameters. We first develop this approach in the functional setting, thus obtaining a consistent methodology for selecting boundary conditions that arise from the $H^1$ regularization. We then construct a method for solving the discrete linear systems based on combining any fast Poisson solver with the Woodbury formula. The efficacy of this method is then demonstrated with scaling experiments. These are carried out for a common nonlinear parameter identification problem arising in electrical resistivity tomography.

math.NA↗

arXiv2022

Wasserstein sensitivity of Risk and Uncertainty Propagation

When propagating uncertainty in the data of differential equations, the probability laws describing the uncertainty are typically themselves subject to uncertainty. We present a sensitivity analysis of uncertainty propagation for differential equations with random inputs to perturbations of the input measures. We focus on the elliptic diffusion equation with random coefficient and source term, for which the probability measure of the solution random field is shown to be Lipschitz-continuous in both total variation and Wasserstein distance. The result generalizes to the solution map of any differential equation with locally Hölder dependence on input parameters. In addition, these results extend to Lipschitz continuous quantities of interest of the solution as well as to coherent risk functionals of these applied to evaluate the impact of their uncertainty. Our analysis is based on the sensitivity of risk functionals and pushforward measures for locally Hölder mappings with respect to the Wasserstein distance of perturbed input distributions. The established results are applied, in particular, to the case of lognormal diffusion and the truncation of series representations of input random fields.

math.PR↗

arXiv2021

Integrability and Approximability of Solutions to the Stationary Diffusion Equation with Lévy Coefficient

We investigate the stationary diffusion equation with a coefficient given by a (transformed) Lévy random field. Lévy random fields are constructed by smoothing Lévy noise fields with kernels from the Matérn class. We show that Lévy noise naturally extends Gaussian white noise within Minlos' theory of generalized random fields. Results on the distributional path spaces of Lévy noise are derived as well as the amount of smoothing to ensure such distributions become continuous paths. Given this, we derive results on the pathwise existence and measurability of solutions to the random boundary value problem (BVP). For the solutions of the BVP we prove existence of moments (in the $H^1$-norm) under adequate growth conditions on the Lévy measure of the noise field. Finally, a kernel expansion of the smoothed Lévy noise fields is introduced and convergence in $L^n$ ($n\geq 1$) of the solutions associated with the approximate random coefficients is proven with an explicit rate.

math.AP↗

arXiv2021

Three Ways to Solve Partial Differential Equations with Neural Networks -- A Review

Neural networks are increasingly used to construct numerical solution methods for partial differential equations. In this expository review, we introduce and contrast three important recent approaches attractive in their simplicity and their suitability for high-dimensional problems: physics-informed neural networks, methods based on the Feynman-Kac formula and methods based on the solution of backward stochastic differential equations. The article is accompanied by a suite of expository software in the form of Jupyter notebooks in which each basic methodology is explained step by step, allowing for a quick assimilation and experimentation. An extensive bibliography summarizes the state of the art.

math.NA↗

arXiv2020

On Expansions and Nodes for Sparse Grid Collocation of Lognormal Elliptic PDEs

This work is a follow-up to our previous contribution ("Convergence of sparse collocation for functions of countably many Gaussian random variables (with application to elliptic PDEs)", SIAM J. Numer. Anal., 2018), and contains further insights on some aspects of the solution of elliptic PDEs with lognormal diffusion coefficients using sparse grids. Specifically, we first focus on the choice of univariate interpolation rules, advocating the use of Gaussian Leja points as introduced by Narayan and Jakeman ("Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation", SIAM J. Sci. Comput., 2014) and then discuss the possible computational advantages of replacing the standard Karhunen-Loève expansion of the diffusion coefficient with the Lévy-Ciesielski expansion, motivated by theoretical work of Bachmayr, Cohen, DeVore, and Migliorati ("Sparse polynomial approximation of parametric elliptic PDEs. part II: lognormal coefficients", ESAIM: M2AN, 2016). Our numerical results indicate that, for the problem under consideration, Gaussian Leja collocation points outperform Gauss-Hermite and Genz-Keister nodes for the sparse grid approximation and that the Karhunen-Loève expansion of the log diffusion coefficient is more appropriate than its Lévy-Ciesielski expansion for purpose of sparse grid collocation.

math.NA↗

arXiv2017

Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general $L^2$-convergence theory based on previous work by Bachmayr et al. (2016) and Chen (2016) and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We verify specifically for Gauss-Hermite nodes that this assumption holds and also show algebraic convergence w.r.t. the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.

math.NA↗

arXiv2015

Analysis of the Ensemble and Polynomial Chaos Kalman Filters in Bayesian Inverse Problems

We analyze the Ensemble and Polynomial Chaos Kalman filters applied to nonlinear stationary Bayesian inverse problems. In a sequential data assimilation setting such stationary problems arise in each step of either filter. We give a new interpretation of the approximations produced by these two popular filters in the Bayesian context and prove that, in the limit of large ensemble or high polynomial degree, both methods yield approximations which converge to a well-defined random variable termed the analysis random variable. We then show that this analysis variable is more closely related to a specific linear Bayes estimator than to the solution of the associated Bayesian inverse problem given by the posterior measure. This suggests limited or at least guarded use of these generalized Kalman filter methods for the purpose of uncertainty quantification.

math.NA↗