Sebastian Krumscheid

Arved Bartuska, André Gustavo Carlon, Luis Espath et al.

Nested integration of the form $\int f\left(\int g(\bs{y},\bs{x})\di{}\bs{x}\right)\di{}\bs{y}$, characterized by an outer integral connected to an inner integral through a nonlinear function $f$, is a challenging problem in various fields, such as engineering and mathematical finance. The available numerical methods for nested integration based on Monte Carlo (MC) methods can be prohibitively expensive owing to the error propagating from the inner to the outer integral. Attempts to enhance the efficiency of these approximations using the quasi-MC (QMC) or randomized QMC (rQMC) method have focused on either the inner or outer integral approximation. This work introduces a novel nested rQMC method that simultaneously addresses the approximation of the inner and outer integrals. The method leverages the unique nested integral structure to offer a more efficient approximation mechanism. As the primary contribution, we derive asymptotic error bounds for the bias and variance of our estimator, along with the regularity conditions under which these bounds can be attained. Incorporating Owen's scrambling techniques, we address integrands exhibiting infinite variation in the Hardy--Krause sense, enabling theoretically sound error estimates. Moreover, we derive a truncation scheme for applications in the context of expected information gain estimation. We verify the estimator quality through numerical experiments by comparing the computational efficiency of the nested rQMC method against standard nested MC estimation to highlight the computational savings and enhanced applicability of the proposed approach.

Arved Bartuska, André Gustavo Carlon, Luis Espath et al.

Nested integration problems arise in various scientific and engineering applications, including Bayesian experimental design, financial risk assessment, and uncertainty quantification. These nested integrals take the form $\int f\left(\int g(\boldsymbol{y},\boldsymbol{x})\mathrm{d}\boldsymbol{x}\right)\mathrm{d}\boldsymbol{y}$, for nonlinear $f$, making them computationally challenging, particularly in high-dimensional settings. Although widely used for single integrals, traditional Monte Carlo (MC) methods can be inefficient when encountering complexities of nested integration. This work introduces a novel multilevel estimator, combining deterministic and randomized quasi-MC (rQMC) methods to handle nested integration problems efficiently. In this context, the inner number of samples and the discretization accuracy of the inner integrand evaluation constitute the level. We provide a comprehensive theoretical analysis of the estimator, deriving error bounds demonstrating significant reductions in bias and variance compared with standard methods. The proposed estimator is particularly effective in scenarios where the integrand is evaluated approximately, as it adapts to different levels of resolution without compromising precision. We verify the performance of our method via numerical experiments, focusing on estimating the expected information gain of experiments. When applied to Gaussian noise in the experiment, a truncation scheme ensures finite error bounds. The results reveal that the proposed multilevel rQMC estimator outperforms existing MC and rQMC approaches, offering a substantial reduction in computational costs and offering a powerful tool for practitioners dealing with complex, nested integration problems across various domains.

Vinh Hoang, Sebastian Krumscheid, Holger Rauhut et al.

We propose a novel deterministic purification method to improve adversarial robustness by mapping a potentially adversarial sample toward a nearby sample that lies close to a mode of the data distribution, where classifiers are more reliable. We design the method to be deterministic to ensure reliable test accuracy and to prevent the degradation of effective robustness observed in stochastic purification approaches when the adversary has full knowledge of the system and its randomness. We employ a score model trained by minimizing the expected reconstruction error of noise-corrupted data, thereby learning the structural characteristics of the input data distribution. Given a potentially adversarial input, the method searches within its local neighborhood for a purified sample that minimizes the expected reconstruction error under noise corruption and then feeds this purified sample to the classifier. During purification, sharpness-aware minimization is used to guide the purified samples toward flat regions of the expected reconstruction error landscape, thereby enhancing robustness. We further show that, as the noise level decreases, minimizing the expected reconstruction error biases the purified sample toward local maximizers of the Gaussian-smoothed density; under additional local assumptions on the score model, we prove recovery of a local maximizer in the small-noise limit. Experimental results demonstrate significant gains in adversarial robustness over state-of-the-art methods under strong deterministic white-box attacks.

Sebastian Krumscheid

We consider the inference problem for parameters in stochastic differential equation models from discrete time observations (e.g. experimental or simulation data). Specifically, we study the case where one does not have access to observations of the model itself, but only to a perturbed version which converges weakly to the solution of the model. Motivated by this perturbation argument, we study the convergence of estimation procedures from a numerical analysis point of view. More precisely, we introduce appropriate consistency, stability, and convergence concepts and study their connection. It turns out that standard statistical techniques, such as the maximum likelihood estimator, are not convergent methodologies in this setting, since they fail to be stable. Due to this shortcoming, we introduce and analyse a novel inference procedure for parameters in stochastic differential equation models which turns out to be convergent. As such, the method is particularly suited for the estimation of parameters in effective (i.e. coarse-grained) models from observations of the corresponding multiscale process. We illustrate these theoretical findings via several numerical examples.

Jan Giesselmann, Kiwoong Kwon, Sebastian Krumscheid

We develop reliable a posteriori error estimators for fully discrete Runge-Kutta discontinuous Galerkin approximations of nonlinear convection-diffusion systems endowed with a convex entropy in multiple spatial dimensions on the flat torus T^d, with a focus on the convection-dominated regime. In order to use the relative entropy method, we reconstruct the numerical solution via tensor-product Smoothness-Increasing Accuracy-Conserving (SIAC) filtering which has superconvergence properties. We then derive reliable a posteriori error estimators for the difference between the entropy weak solution and the reconstruction, with constants that are uniform in the vanishing viscosity limit. Our numerical experiments show that the a posteriori error bounds converge with the same order as the error of the reconstructed numerical solution.

Luis Espath, Sebastian Krumscheid, Raúl Tempone et al.

In this study, we demonstrate that the norm test and inner product/orthogonality test presented in \cite{Bol18} are equivalent in terms of the convergence rates associated with Stochastic Gradient Descent (SGD) methods if $ε^2=θ^2+ν^2$ with specific choices of $θ$ and $ν$. Here, $ε$ controls the relative statistical error of the norm of the gradient while $θ$ and $ν$ control the relative statistical error of the gradient in the direction of the gradient and in the direction orthogonal to the gradient, respectively. Furthermore, we demonstrate that the inner product/orthogonality test can be as inexpensive as the norm test in the best case scenario if $θ$ and $ν$ are optimally selected, but the inner product/orthogonality test will never be more computationally affordable than the norm test if $ε^2=θ^2+ν^2$. Finally, we present two stochastic optimization problems to illustrate our results.

Truong-Vinh Hoang, Sebastian Krumscheid, Hermann G. Matthies et al.

This paper presents the machine learning-based ensemble conditional mean filter (ML-EnCMF) -- a filtering method based on the conditional mean filter (CMF) previously introduced in the literature. The updated mean of the CMF matches that of the posterior, obtained by applying Bayes' rule on the filter's forecast distribution. Moreover, we show that the CMF's updated covariance coincides with the expected conditional covariance. Implementing the EnCMF requires computing the conditional mean (CM). A likelihood-based estimator is prone to significant errors for small ensemble sizes, causing the filter divergence. We develop a systematical methodology for integrating machine learning into the EnCMF based on the CM's orthogonal projection property. First, we use a combination of an artificial neural network (ANN) and a linear function, obtained based on the ensemble Kalman filter (EnKF), to approximate the CM, enabling the ML-EnCMF to inherit EnKF's advantages. Secondly, we apply a suitable variance reduction technique to reduce statistical errors when estimating loss function. Lastly, we propose a model selection procedure for element-wisely selecting the applied filter, i.e., either the EnKF or ML-EnCMF, at each updating step. We demonstrate the ML-EnCMF performance using the Lorenz-63 and Lorenz-96 systems and show that the ML-EnCMF outperforms the EnKF and the likelihood-based EnCMF.