Claudia Schillings

Claudia Schillings, Claudia Totzeck, Philipp Wacker

We propose an approach based on function evaluations and Bayesian inference to extract higher-order differential information of objective functions {from a given ensemble of particles}. Pointwise evaluation $\{V(x^i)\}_i$ of some potential $V$ in an ensemble $\{x^i\}_i$ contains implicit information about first or higher order derivatives, which can be made explicit with little computational effort (ensemble-based gradient inference -- EGI). We suggest to use this information for the improvement of established ensemble-based numerical methods for optimization and sampling such as Consensus-based optimization and Langevin-based samplers. Numerical studies indicate that the augmented algorithms are often superior to their gradient-free variants, in particular the augmented methods help the ensembles to escape their initial domain, to explore multimodal, non-Gaussian settings and to speed up the collapse at the end of optimization dynamics.} The code for the numerical examples in this manuscript can be found in the paper's Github repository (https://github.com/MercuryBench/ensemble-based-gradient.git).

Dishi Liu, Alexander Litvinenko, Claudia Schillings et al.

Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods since it is computationally expensive, especially for the uncertainties caused by random geometry variations which involve a large number of variables. This paper compares five methods, including quasi-Monte Carlo quadrature, polynomial chaos with coefficients determined by sparse quadrature and gradient-enhanced version of Kriging, radial basis functions and point collocation polynomial chaos, in their efficiency in estimating statistics of aerodynamic performance upon random perturbation to the airfoil geometry which is parameterized by 9 independent Gaussian variables. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods with the same computational cost.

Ilja Klebanov, Claudia Schillings, Dana Wrischnig

The Bootstrap Particle Filter (BPF) and the Ensemble Kalman Filter (EnKF) are two widely used methods for sequential Bayesian filtering: the BPF is asymptotically exact but can suffer from weight degeneracy, while the EnKF scales well in high dimension (typically with localization) yet is exact only in the linear-Gaussian case. We combine these approaches by retaining the EnKF transport step and adding a principled importance-sampling correction. Our first contribution is a general importance-sampling theory for mixture targets and proposals, including variance comparisons between individual- and mixture-based estimators. We then interpret the stochastic EnKF analysis as sampling from explicit Gaussian-mixture proposals obtained by conditioning on the current or previous ensemble, which leads to six self-normalized IS-EnKF schemes. We embed these updates into a broader class of ensemble-based filters and prove consistency and error bounds, including weight-variance comparisons and sufficient conditions ensuring finite-variance importance weights. As a second contribution, we construct transported quasi-Monte Carlo (TQMC) point sets for the Gaussian-mixture laws arising in prediction and analysis, yielding TQMC-enhanced variants that can substantially reduce sampling error without changing the filtering pipeline. Numerical experiments on benchmark models compare the proposed mixture-weighted and TQMC-enhanced filters, showing improved filtering accuracy relative to BPF, EnKF, and the standard weighted EnKF, and that the weighted schemes eliminate the EnKF error plateau often caused by analysis-target mismatch.

Vesa Kaarnioja, Claudia Schillings

This paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration problems over high-dimensional parameter and data domains arising in Bayesian optimal design problems. We provide a detailed analysis for these double integration problems using two approaches: a full tensor product and a sparse tensor product combination of quasi-Monte Carlo (QMC) cubature rules over the parameter and data domains. Specifically, we show that the latter approach significantly improves the convergence rate, exhibiting performance comparable to that of QMC integration of a single high-dimensional integral. Furthermore, we numerically verify the predicted convergence rates for an elliptic PDE problem with an unknown diffusion coefficient in two spatial dimensions, offering empirical evidence supporting the theoretical results and highlighting practical applicability.

Ana Djurdjevac, Vesa Kaarnioja, Claudia Schillings et al.

We study uncertainty quantification for partial differential equations subject to domain uncertainty. We parameterize the random domain using the model recently considered by Chernov and Le (2024) as well as Harbrecht, Schmidlin, and Schwab (2024) in which the input random field is assumed to belong to a Gevrey smoothness class. This approach has the advantage of being substantially more general than models which assume a particular parametric representation of the input random field such as a Karhunen-Loeve series expansion. We consider both the Poisson equation as well as the heat equation and design randomly shifted lattice quasi-Monte Carlo (QMC) cubature rules for the computation of the expected solution under domain uncertainty. We show that these QMC rules exhibit dimension-independent, essentially linear cubature convergence rates in this framework. In addition, we complete the error analysis by taking into account the approximation errors incurred by dimension truncation of the random input field and finite element discretization. Numerical experiments are presented to confirm the theoretical rates.

Steffen Jung, Jan Christian Schwedhelm, Claudia Schillings et al.

In recent years, optimization in the learned latent space of deep generative models has been successfully applied to black-box optimization problems such as drug design, image generation or neural architecture search. Existing models thereby leverage the ability of neural models to learn the data distribution from a limited amount of samples such that new samples from the distribution can be drawn. In this work, we propose a novel image generative approach that optimizes the generated sample with respect to a continuously quantifiable property. While we anticipate absolutely no practically meaningful application for the proposed framework, it is theoretically principled and allows to quickly propose samples at the mere boundary of the training data distribution. Specifically, we propose to use tree-based ensemble models as mathematical programs over the discrete latent space of vector quantized VAEs, which can be globally solved. Subsequent weighted retraining on these queries allows to induce a distribution shift. In lack of a practically relevant problem, we consider a visually appealing application: the generation of happily smiling faces (where the training distribution only contains less happy people) - and show the principled behavior of our approach in terms of improved FID and higher smile degree over baseline approaches.

David Sommer, Robert Gruhlke, Max Kirstein et al.

Sampling from probability densities is a common challenge in fields such as Uncertainty Quantification (UQ) and Generative Modelling (GM). In GM in particular, the use of reverse-time diffusion processes depending on the log-densities of Ornstein-Uhlenbeck forward processes are a popular sampling tool. In Berner et al. [2022] the authors point out that these log-densities can be obtained by solution of a \textit{Hamilton-Jacobi-Bellman} (HJB) equation known from stochastic optimal control. While this HJB equation is usually treated with indirect methods such as policy iteration and unsupervised training of black-box architectures like Neural Networks, we propose instead to solve the HJB equation by direct time integration, using compressed polynomials represented in the Tensor Train (TT) format for spatial discretization. Crucially, this method is sample-free, agnostic to normalization constants and can avoid the curse of dimensionality due to the TT compression. We provide a complete derivation of the HJB equation's action on Tensor Train polynomials and demonstrate the performance of the proposed time-step-, rank- and degree-adaptive integration method on a nonlinear sampling task in 20 dimensions.

Robert Gruhlke, Matei Hanu, Claudia Schillings et al.

We introduce a gradient-free framework for Bayesian Optimal Experimental Design (BOED) in sequential settings, aimed at complex systems where gradient information is unavailable. Our method combines Ensemble Kalman Inversion (EKI) for design optimization with the Affine-Invariant Langevin Dynamics (ALDI) sampler for efficient posterior sampling-both of which are derivative-free and ensemble-based. To address the computational challenges posed by nested expectations in BOED, we propose variational Gaussian and parametrized Laplace approximations that provide tractable upper and lower bounds on the Expected Information Gain (EIG). These approximations enable scalable utility estimation in high-dimensional spaces and PDE-constrained inverse problems. We demonstrate the performance of our framework through numerical experiments ranging from linear Gaussian models to PDE-based inference tasks, highlighting the method's robustness, accuracy, and efficiency in information-driven experimental design.

Neil K. Chada, Claudia Schillings, Xin T. Tong et al.

One fundamental problem when solving inverse problems is how to find regularization parameters. This article considers solving this problem using data-driven bilevel optimization, i.e. we consider the adaptive learning of the regularization parameter from data by means of optimization. This approach can be interpreted as solving an empirical risk minimization problem, and we analyze its performance in the large data sample size limit for general nonlinear problems. We demonstrate how to implement our framework on linear inverse problems, where we can further show the inverse accuracy does not depend on the ambient space dimension. To reduce the associated computational cost, online numerical schemes are derived using the stochastic gradient descent method. We prove convergence of these numerical schemes under suitable assumptions on the forward problem. Numerical experiments are presented illustrating the theoretical results and demonstrating the applicability and efficiency of the proposed approaches for various linear and nonlinear inverse problems, including Darcy flow, the eikonal equation, and an image denoising example.

Claudia Schillings, Andrew Stuart

We present an analysis of ensemble Kalman inversion, based on the continuous time limit of the algorithm. The analysis of the dynamical behaviour of the ensemble allows us to establish well-posedness and convergence results for a fixed ensemble size. We will build on the results presented in [26] and generalise them to the case of noisy observational data, in particular the influence of the noise on the convergence will be investigated, both theoretically and numerically. We focus on linear inverse problems where a very complete theoretical analysis is possible.