Figure - available from: Numerische Mathematik
This content is subject to copyright. Terms and conditions apply.
Source publication
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 th...
Citations
... QMC methods have become a popular tool in the numerical treatment of uncertainties in partial differential equation (PDE) models with random or uncertain inputs. Studied topics include elliptic eigenvalue problems [1,2], optimal control [3,4], various diffusion problems [5,6,7], parametric operator equations [8,9] as well as elliptic PDEs with random or lognormal coefficients [10,11,12,13,14,15,16,17]. A common and sought-after advantage of these applications are the faster-than-Monte Carlo convergence rates of QMC methods and-under some moderate conditions-this convergence can be shown to be independent of the dimensionality of the associated integration problems. QMC methods are particularly well-suited to large-scale uncertainty quantification problems since it is typically easy to parallelize the computation over QMC point sets. ...
The theoretical development of quasi-Monte Carlo (QMC) methods for uncertainty quantification of partial differential equations (PDEs) is typically centered around simplified model problems such as elliptic PDEs subject to homogeneous zero Dirichlet boundary conditions. In this paper, we present a theoretical treatment of the application of randomly shifted rank-1 lattice rules to electrical impedance tomography (EIT). EIT is an imaging modality, where the goal is to reconstruct the interior conductivity of an object based on electrode measurements of current and voltage taken at the boundary of the object. This is an inverse problem, which we tackle using the Bayesian statistical inversion paradigm. As the reconstruction, we consider QMC integration to approximate the unknown conductivity given current and voltage measurements. We prove under moderate assumptions placed on the parameterization of the unknown conductivity that the QMC approximation of the reconstructed estimate has a dimension-independent, faster-than-Monte Carlo cubature convergence rate. Finally, we present numerical results for examples computed using simulated measurement data.
... In the last decade there has been a lot of research on open-loop control problems under uncertainty [1,14,22,26]. In these works controls are developed that are optimal with respect to given performance measures, which allow for different levels of risk aversion [19,34]. ...
... The performance measures typically involve high-dimensional integrals over the space of uncertain parameters, resulting in computationally challenging problems. Strategies to reduce the computational burden include, for instance, (multilevel) Monte Carlo methods [27,30,39], (multilevel) quasi-Monte Carlo methods [14,18,24], sparse grids [4,20], and variants of the stochastic gradient descent algorithm [11,25]. We point out that quasi-Monte Carlo methods are particularly well-suited, since they retain the convexity structure of the optimal control problem while achieving faster convergence rates as compared to Monte Carlo methods. ...
... The QMC approximation of integrals with Banach space-valued integrands was first studied in [14], where the error analysis is presented for randomly shifted lattice rules. Higher-order QMC methods have been studied in [24] in a Hilbert space setting. ...
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--Lo\`eve 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.
... In order to derive a dimension-truncation error bound for this problem, we employ the Taylor series approach used in [9,12] together with a change of variable technique. ...
... Many studies in uncertainty quantification for PDEs with random coefficients exploit the parametric structure of the problem, using a Neumann series expansion to obtain dimension truncation error bounds. However, there have been several recent studies suggesting a Taylor series approach which is based on the parametric regularity of the problem [9,11,12]. The Taylor series approach can be used to obtain dimension truncation rates for non-affine parametric PDE problems, but a limitation of the aforementioned papers is that the parametric regularity bound needs to be of product-and-order dependent (POD) form in order for the Taylor-based approach to yield useful results. ...
We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja et al. (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.
... In the context of open-loop control, this problem has been studied recently, for (spatial discretizations of) parabolic equations, for finite time-horizon, in [15,25,20], and for infinite time-horizon receding horizon control in [2]. ...
... Many of these studies have been carried out under the assumption of the so-called "affine and uniform setting" as in (2). Examples include the source problem for elliptic PDEs with random coefficients [29,8,27,28,10], spectral eigenvalue problems under uncertainty [12,13,14], Bayesian inverse problems [11,7,19], domain uncertainty quantification [18], PDE-constrained optimization under uncertainty [15,16], and many others. When the input random field is modified to involve a composition with a periodic function as in (6), the regularity bound naturally changes, as we have encountered in [21,20,17]. ...
We describe a fast method for solving elliptic partial differential equations (PDEs) with uncertain coefficients using kernel interpolation at a lattice point set. By representing the input random field of the system using the model proposed by Kaarnioja, Kuo, and Sloan (SIAM J.~Numer.~Anal.~2020), in which a countable number of independent random variables enter the random field as periodic functions, it was shown by Kaarnioja, Kazashi, Kuo, Nobile, and Sloan (Numer.~Math.~2022) that the lattice-based kernel interpolant can be constructed for the PDE solution as a function of the stochastic variables in a highly efficient manner using fast Fourier transform (FFT). In this work, we discuss the connection between our model and the popular ``affine and uniform model'' studied widely in the literature of uncertainty quantification for PDEs with uncertain coefficients. We also propose a new class of weights entering the construction of the kernel interpolant -- \emph{serendipitous weights} -- which dramatically improve the computational performance of the kernel interpolant for PDE problems with uncertain coefficients, and allow us to tackle function approximation problems up to very high dimensionalities. Numerical experiments are presented to showcase the performance of the serendipitous weights.
... Concerning control and stabilization, which are well-studied for deterministic infinite-dimensional systems, and stochastic systems with the stochastic terms appearing in an affine manner, there is little research on infinite-dimensional systems under uncertainty. Among them, we can mention [2,7,17] in the context of controllability results and [11,16,18,19] for optimal control problems. To our knowledge, RHC has not yet been studied for control systems with uncertainty inputs. ...
Stabilization of a class of time-varying parabolic equations with uncertain input data using Receding Horizon Control (RHC) is investigated. The diffusion coefficient and the initial function are prescribed as random fields. We consider both cases, uniform and log-normal distributions of the diffusion coefficient. The controls are chosen to be finite-dimensional and enter into the system as a linear combination of finitely many indicator functions (actuators) supported in open subsets of the spatial domain. Under suitable regularity assumptions, we study the expected (averaged) stabilizability of the RHC-controlled system with respect to the number of actuators. An upper bound is also obtained for the failure probability of RHC in relation to the choice of the number of actuators and parameters in the equation.
... In the non-affine setting, using Taylor series makes it possible to derive dimension truncation error rates by exploiting the parametric regularity of the problem, whereas the Neumann series approach relies fundamentally on the parametric structure of the model. The Taylor series approach was first applied in [8], and motivated the authors in [11] and [12] to derive dimension truncation error rates for sufficiently smooth, Banach space valued integrands, and with parameters following a generalized β-Gaussian distribution. An overview of the various dimension truncation error bounds studied in the literature is given in Table 1. ...
... As the integration lattice, we use in both cases an off-the-shelf rank-1 lattice rule [19, lattice-39101-1024-1048576.3600] and use the same random shift for each value of ϑ. As the reference solution, we use the PDE solution corresponding to s ′ = 2 11 . The numerical results for dimensions s ∈ {2 k : k = 1, . . . ...
Parametric mathematical models such as 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 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.
... A second problem class is PDE-constrained optimization. Specifically, the optimal control problem (OCP) of parametric PDEs under an entropic risk measure [7], where the state variable satisfies a parametric PDE constraint [9]. ...
... Due to convexity of R and α 2 > 0, the functional J is strongly convex so that (12) is a well-posed minimization problem [9,12]. ...
... A similar result can be given for the OCP, based on the results in [8,9]. ...
We propose a novel a-posteriori error estimation technique where the target quantities of interest are ratios of high-dimensional integrals, as occur e.g. in PDE constrained Bayesian inversion and PDE constrained optimal control subject to an entropic risk measure. We consider in particular parametric, elliptic PDEs with affine-parametric diffusion coefficient, on high-dimensional parameter spaces. We combine our recent a-posteriori Quasi-Monte Carlo (QMC) error analysis, with Finite Element a-posteriori error estimation. The proposed approach yields a computable a-posteriori estimator which is reliable, up to higher order terms. The estimator's reliability is uniform with respect to the PDE discretization, and robust with respect to the parametric dimension of the uncertain PDE input.
... In order to derive a dimension-truncation error bound for this problem, we employ the Taylor series approach used in [8,11] together with a change of variable technique. ...
... Many studies in uncertainty quantification for PDEs with random coefficients exploit the parametric structure of the problem, using a Neumann series expansion to obtain dimension truncation error bounds. However, there have been several recent studies suggesting a Taylor series approach which is based on the parametric regularity of the problem [8,10,11]. The Taylor series approach can be used to obtain dimension truncation rates for non-affine parametric PDE problems, but a limitation of the aforementioned papers is that the parametric regularity bound needs to be of product-and-order dependent (POD) form in order for the Taylor-based approach to yield useful results. ...
We consider uncertainty quantification for the Poisson problem subject to domain uncertainty. For the stochastic parameterization of the random domain, we use the model recently introduced by Kaarnioja, Kuo, and Sloan (SIAM J. Numer. Anal., 2020) in which a countably infinite number of independent random variables enter the random field as periodic functions. We develop lattice quasi-Monte Carlo (QMC) cubature rules for computing the expected value of the solution to the Poisson problem subject to domain uncertainty. These QMC rules can be shown to exhibit higher order cubature convergence rates permitted by the periodic setting independently of the stochastic dimension of the problem. In addition, we present a complete error analysis for the problem by taking into account the approximation errors incurred by truncating the input random field to a finite number of terms and discretizing the spatial domain using finite elements. The paper concludes with numerical experiments demonstrating the theoretical error estimates.
... The use of Taylor series makes it possible to exploit the underlying parametric regularity of the model problem in order to derive dimension truncation rates, as opposed to Neumann series which is heavily dependent on the parametric structure of the PDE problem. A similar Taylor series approach-motivated by the paper [11]-was used to derive a dimension truncation rate for a smooth nonlinear quantity of interest subject to an affine parametric parabolic PDE in [19]. ...
... The PhD thesis [8] provides some numerical experiments suggesting that the dimension truncation error rates for certain non-affine parametric PDE problems are actually significantly better than the theoretical bounds derived using Neumann series. Both the numerical and theoretical results in [19] indicate that the use of a smooth nonlinear quantity of interest does not deteriorate the dimension truncation rate for an affine parametric PDE problem. Furthermore, it is known in the context of lognormal parameterizations for diffusion coefficients of parametric elliptic PDEs that the use of special Matérn covariances can yield even exponentially convergent dimension truncation errors (cf. ...
... The steps are completely analogous to Theorem 4.1 in the special case α j = 0 for all j ≥ 1 and by restricting the domain of integration to [−1, 1] N . In the special case Θ |ν| = (|ν| + r 1 )!, r 1 ∈ N 0 , the proof works as in [19,Theorem 6.2]. ...
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.
