arXiv subjects

Niklas Baumgarten

Publications and source records attributed to Niklas Baumgarten.

6 recordsLinked to original sources

arXiv2026

Optimized Multilevel Sampling Methods under Resource Constraints

We present recent developments in multilevel sampling methods under resource constraints. Over the past 15 years, multilevel methods have become widely used for uncertainty quantification. However, scaling them to high-dimensional problems and high-performance computing (HPC) environments remains challenging. In this work, we discuss two algorithms designed to address these issues: the budgeted Multilevel Monte Carlo (MLMC) method and the Multilevel Stochastic Gradient Descent (MLSGD) method. We demonstrate their effectiveness on HPC systems under consideration of the available computational resources for applications in forward uncertainty quantification (UQ) and optimal control (OC) under uncertainty.

math.NA↗

arXiv2026

Risk-averse Optimization in Random Materials: Algorithmic Advances and HPC Acceleration

We summarize our advances in the algorithmic development and hardware utilization for risk-averse optimization problems in random materials. This includes risk-averse optimization using the entropic risk measure, as well as recently developed sampling techniques for random materials, that are interoperable with the optimization framework. Furthermore, we discuss recent progress in the efficient utilization of modern hybrid hardware architectures for these methods to solve three-dimensional partial differential equations.

math.NA↗

arXiv2026

Multilevel Stochastic Gradient Descent for Risk-Averse PDE-Constrained Optimization

We present recent advances in applying and analyzing multilevel stochastic gradient descent algorithms to risk-averse, three-dimensional PDE-constrained optimization problems. The algorithm uses adaptive multilevel Monte Carlo gradient estimates, provides parallel scalability as well as improved convergence rates and computational complexity compared to standard batched stochastic gradient descent methods. We study the method in computationally demanding settings using three-dimensional elliptic diffusion problems and large risk-aversion parameters.

math.OC↗

arXiv2025

A Budgeted Multi-Level Monte Carlo Method for Full Field Estimates of Multi-PDE Problems

We present a high-performance budgeted multi-level Monte Carlo method for estimates on the entire spatial domain of multi-PDE problems with random input data. The method is designed to operate optimally within memory and CPU-time constraints and eliminates the need for a priori knowledge of the problem's regularity and the algorithm's potential memory demand. To achieve this, we build on the budgeted multi-level Monte Carlo framework and enhance it with a sparse multi-index update algorithm operating on a dynamically assembled parallel data structure to enable estimates of the full field solution. We demonstrate numerically and provide mathematical proof that this update algorithm allows computing the full spatial domain estimates at the same CPU-time cost as a single quantity of interest, and that the maximum memory usage is similar to the memory demands of the deterministic formulation of the problem despite solving the stochastic formulation in parallel. We apply the method to a sequence of interlinked PDE problems, ranging from a stochastic partial differential equation for sampling random fields that serve as the diffusion coefficient in an elliptic subsurface flow problem, to a hyperbolic PDE describing mass transport in the resulting flux field.

math.NA↗

arXiv2025

Multilevel Stochastic Gradient Descent for Optimal Control Under Uncertainty

We present a multilevel stochastic gradient descent method for the optimal control of systems governed by partial differential equations under uncertain input data. The gradient descent method used to find the optimal control leverages a parallel multilevel Monte Carlo method as stochastic gradient estimator. As a result, we achieve precise control over the stochastic gradient's bias, introduced by numerical approximation, and its sampling error, arising from the use of incomplete gradients, while optimally managing computational resources. We show that the method exhibits linear convergence in the number of optimization steps while avoiding the cost of computing the full gradient at the highest fidelity. Numerical experiments demonstrate that the method significantly outperforms the standard (mini-) batched stochastic gradient descent method in terms of convergence speed and accuracy. The method is particularly well-suited for high-dimensional control problems, taking advantage of parallel computing resources and a distributed multilevel data structure. Additionally, we evaluate and implement different step size strategies, optimizer schemes, and budgeting techniques. The method's performance is studied using a two-dimensional elliptic subsurface diffusion problem with log-normal coefficients and Mat\'ern covariance.

math.OC↗

arXiv2023

A Fully Parallelized and Budgeted Multi-Level Monte Carlo Method and the Application to Acoustic Waves

We present a novel variant of the multi-level Monte Carlo method that effectively utilizes a reserved computational budget on a high-performance computing system to minimize the mean squared error. Our approach combines concepts of the continuation multi-level Monte Carlo method with dynamic programming techniques following Bellman's optimality principle, and a new parallelization strategy based on a single distributed data structure. Additionally, we establish a theoretical bound on the error reduction on a parallel computing cluster and provide empirical evidence that the proposed method adheres to this bound. We implement, test, and benchmark the approach on computationally demanding problems, focusing on its application to acoustic wave propagation in high-dimensional random media.

math.NA↗