arXiv subjects

Philipp A. Guth

Publications and source records attributed to Philipp A. Guth.

16 recordsLinked to original sources

arXiv2026

Dynamic output-feedback stabilization of uncertain linear dynamics via digital twins

This work presents a digital twin framework for output-feedback stabilization and parameter identification in uncertain dynamical systems. A virtual model evolves in parallel with the physical process, assimilating measurement data in real time. By design, the digital twin reconstructs the system state and generates a stabilizing feedback, while model parameters are simultaneously inferred from data of the controlled dynamics using a Bayesian approach. Numerical results for the coupled physical-virtual dynamics demonstrate how digital twins can act jointly as observers, parameter estimators, and control agents, ensuring robust performance under uncertainty.

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arXiv2026

On the optimality of dimension truncation error rates for a class of parametric partial differential equations

In uncertainty quantification for parametric partial differential equations (PDEs), it is common to model uncertain random field inputs using countably infinite sequences of independent and identically distributed random variables. The lognormal random field is a prime example of such a model. While there have been many studies assessing the error in the PDE response that occurs when an infinite-dimensional random field input is replaced with a finite-dimensional random field, there do not seem to be any analyses in the existing literature discussing the sharpness of these bounds. This work seeks to remedy the situation. Specifically, we investigate two model problems where the existing dimension truncation error rates can be shown to be sharp.

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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.

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arXiv2025

Quasi-Monte Carlo for partial differential equations with generalized Gaussian input uncertainty

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.

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arXiv2025

Approximation of risk-averse optimal feedback control

The challenge of constructing feedback control laws for risk-averse optimal control of partial differential equations (PDEs) with random coefficients is addressed. The control objective composes a tracking-type cost with the nonlinear entropic risk measure. A sequential quadratic programming scheme is derived that iteratively solves linear quadratic subproblems obtained through second-order Taylor expansions of the objective functional, with each subproblem re-centered at the previous iterate. It is shown that this method converges locally quadratically to the unique risk-averse optimal control. This work provides the first rigorous feedback synthesis for risk-averse objectives subject to PDEs with random coefficients.

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arXiv2024

Quasi-Monte Carlo integration for feedback control under uncertainty

A control in feedback form is derived for linear quadratic, time-invariant optimal control problems subject to parabolic partial differential equations with coefficients depending on a countably infinite number of uncertain parameters. It is shown that the Riccati-based feedback operator depends analytically on the parameters provided that the system operator depends analytically on the parameters, as is the case, for instance, in diffusion problems when the diffusion coefficient is parameterized by a Karhunen--Loève expansion. These novel parametric regularity results allow the application of quasi-Monte Carlo (QMC) methods to efficiently compute an a-priori chosen feedback law based on the expected value. Moreover, under moderate assumptions on the input random field, QMC methods achieve superior error rates compared to ordinary Monte Carlo methods, independently of the stochastic dimension of the problem. Indeed, our paper for the first time studies Banach-space-valued integration by higher-order QMC methods.

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arXiv2024

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

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.

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arXiv2024

Tracking optimal feedback control under uncertain parameters

Optimal control problems of tracking type for a class of linear systems with uncertain parameters in the dynamics are investigated. An affine tracking feedback control input is obtained by considering the minimization of an energy-like functional depending on a finite ensemble of training/sample parameters. It is computed from the nonnegative definite solution of an associated differential Riccati equation. Simulations are presented showing the tracking performance of the computed input for trained as well as untrained parameters.

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arXiv2023

One-shot Learning of Surrogates in PDE-constrained Optimization Under Uncertainty

We propose a general framework for machine learning based optimization under uncertainty. Our approach replaces the complex forward model by a surrogate, which is learned simultaneously in a one-shot sense when solving the optimal control problem. Our approach relies on a reformulation of the problem as a penalized empirical risk minimization problem for which we provide a consistency analysis in terms of large data and increasing penalty parameter. To solve the resulting problem, we suggest a stochastic gradient method with adaptive control of the penalty parameter and prove convergence under suitable assumptions on the surrogate model. Numerical experiments illustrate the results for linear and nonlinear surrogate models.

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arXiv2023

Generalized dimension truncation error analysis for high-dimensional numerical integration: lognormal setting and beyond

Partial differential equations (PDEs) with uncertain or random inputs have been considered in many studies of uncertainty quantification. In forward uncertainty quantification, one is interested in analyzing the stochastic response of the PDE subject to input uncertainty, which usually involves solving high-dimensional integrals of the PDE output over a sequence of stochastic variables. In practical computations, one typically needs to discretize the problem in several ways: approximating an infinite-dimensional input random field with a finite-dimensional random field, spatial discretization of the PDE using, e.g., finite elements, and approximating high-dimensional integrals using cubatures such as quasi-Monte Carlo methods. In this paper, we focus on the error resulting from dimension truncation of an input random field. We show how Taylor series can be used to derive theoretical dimension truncation rates for a wide class of problems and we provide a simple checklist of conditions that a parametric mathematical model needs to satisfy in order for our dimension truncation error bound to hold. Some of the novel features of our approach include that our results are applicable to non-affine parametric operator equations, dimensionally-truncated conforming finite element discretized solutions of parametric PDEs, and even compositions of PDE solutions with smooth nonlinear quantities of interest. As a specific application of our method, we derive an improved dimension truncation error bound for elliptic PDEs with lognormally parameterized diffusion coefficients. Numerical examples support our theoretical findings.

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arXiv2023

Application of dimension truncation error analysis to high-dimensional function approximation in uncertainty quantification

Parametric mathematical models such as parameterizations of 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.

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arXiv2023

Stabilization of uncertain linear dynamics: an offline-online strategy

A strategy is proposed for adaptive stabilization of linear systems, depending on an uncertain parameter. Offline, the Riccati stabilizing feedback input control operators, corresponding to parameters in a finite training set of chosen candidates for the uncertain parameter, are solved and stored in a library. A uniform partition of the infinite time interval is chosen. In each of these subintervals, the input is given by one of the stored parameter dependent Riccati feedback operators. This parameter is updated online, at the end of each subinterval, based on input and output data, where the true data, corresponding to the true parameter, is compared to fictitious data that one would obtain in case the parameter was in a selected subset of the training set. The auxiliary data can be computed in parallel, so that the parameter update can be performed in real time. The focus is put on the case that the unknown parameter is constant and that the free dynamics is time-periodic. The stabilizing performance of the input obtained by the proposed strategy is illustrated by numerical simulations, for both constant and switching parameters.

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arXiv2023

Ensemble Feedback Stabilization of Linear Systems

Stabilization of linear control systems with parameter-dependent system matrices is investigated. A Riccati based feedback mechanism is proposed and analyzed. It is constructed by means of an ensemble of parameters from a training set. This single feedback stabilizes all systems of the training set and also systems in its vicinity. Moreover its suboptimality with respect to optimal feedback for each single parameter from the training set can be quantified.

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arXiv2021

Multilevel Quasi-Monte Carlo for Optimization under Uncertainty

This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo (MLQMC) method to the estimation of the gradient is investigated, with a circulant embedding method used to sample the stochastic field. A novel regularity analysis of the adjoint variable is essential for the MLQMC estimation of the gradient in combination with the samples generated using the CE method. A rigorous cost and error analysis shows that a randomly shifted quasi-Monte Carlo method leads to a faster rate of decay in the root mean square error of the gradient than the ordinary Monte Carlo method, while considering multiple levels substantially reduces the computational effort. Numerical experiments confirm the improved rate of convergence and show that the MLQMC method outperforms the multilevel Monte Carlo method and the single level quasi-Monte Carlo method.

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arXiv2020

Ensemble Kalman filter for neural network based one-shot inversion

We study the use of novel techniques arising in machine learning for inverse problems. Our approach replaces the complex forward model by a neural network, which is trained simultaneously in a one-shot sense when estimating the unknown parameters from data, i.e. the neural network is trained only for the unknown parameter. By establishing a link to the Bayesian approach to inverse problems, an algorithmic framework is developed which ensures the feasibility of the parameter estimate w.r. to the forward model. We propose an efficient, derivative-free optimization method based on variants of the ensemble Kalman inversion. Numerical experiments show that the ensemble Kalman filter for neural network based one-shot inversion is a promising direction combining optimization and machine learning techniques for inverse problems.

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arXiv2019

A quasi-Monte Carlo Method for an Optimal Control Problem Under Uncertainty

We study an optimal control problem under uncertainty, where the target function is the solution of an elliptic partial differential equation with random coefficients, steered by a control function. The robust formulation of the optimization problem is stated as a high-dimensional integration problem over the stochastic variables. It is well known that carrying out a high-dimensional numerical integration of this kind using a Monte Carlo method has a notoriously slow convergence rate; meanwhile, a faster rate of convergence can potentially be obtained by using sparse grid quadratures, but these lead to discretized systems that are non-convex due to the involvement of negative quadrature weights. In this paper, we analyze instead the application of a quasi-Monte Carlo method, which retains the desirable convexity structure of the system and has a faster convergence rate compared to ordinary Monte Carlo methods. In particular, we show that under moderate assumptions on the decay of the input random field, the error rate obtained by using a specially designed, randomly shifted rank-1 lattice quadrature rule is essentially inversely proportional to the number of quadrature nodes. The overall discretization error of the problem, consisting of the dimension truncation error, finite element discretization error and quasi-Monte Carlo quadrature error, is derived in detail. We assess the theoretical findings in numerical experiments.

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