Functional error estimators for the adaptive discretization of inverse problems
Faculty of Mathematics, Universität Duisburg-Essen（杜伊斯堡-埃森大学数学学院）；Institute of Mathematics, Alpen-Adria Universität Klagenfurt（克拉根福阿尔卑斯-亚得里亚大学数学研究所）；Institute of Mathematics, Universität Würzburg（维尔茨堡大学数学研究所）
Christian Clason, Barbara Kaltenbacher, Daniel Wachsmuth
So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for partial differential equations, not only for quadratic Hilbert space regularization terms but also for nonsmooth Banach space penalties. Examples include the measure-space norm (i.e., sparsity regularization) or the indicator function of an $L^\infty$ ball (i.e., Ivanov regularization). The error estimators can be written in terms of residuals in the optimality system that can then be estimated by conventional techniques, thus leading to explicit estimators. This is illustrated by means of an elliptic inverse source problem with the above-mentioned penalties, and numerical results are provided for the case of sparsity regularization.
We present an equilibration-based a posteriori error estimator for Nédélec element discretizations of the magnetostatic problem. The estimator is obtained by adding a gradient correction to the estimator for Nédélec elements of arbitrary degree presented in [J. Gedicke, S. Geevers, and I. Perugia. An equilibrated a posteriori error estimator for arbitrary-order Nédélec elements for magnetostatic problems. Journal of Scientific Computing, 83:1-23, 2020]. This new estimator is proven to be reliable, with reliability constant 1, and efficient, with an efficiency constant that is independent of th…
This paper introduces a discretization-accurate stopping criterion of symmetric iterative methods for solving systems of algebraic equations resulting from the finite element approximation. The stopping criterion consists of the evaluations of the discretization and the algebraic error estimators, that are based on the respective duality error estimator and the difference of two consecutive iterates. Iterations are terminated when the algebraic estimator is of the same magnitude as the discretization estimator. Numerical results for multigrid $V(1,1)$-cycle and symmetric Gauss-Seidel iterative…
The present work aims at the application of finite element discretizations to a class of equilibrium problems involving moving constraints. Therefore, a Moreau--Yosida based regularization technique, controlled by a parameter, is discussed and, using a generalized $Γ$-convergence concept, a priori convergence results are derived. The latter technique is applied to the discretization of the regularized problems and is used to prove the convergence to the orginal equilibrium problem, when both -- regularization and discretization -- are imposed simultaneously. In addition, a primal-dual gap tech…
A posteriori error estimators for hierarchical B-spline discretizations
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Istituto di Matematica Applicata e Tecnologie Informatiche ‘E. Magenes’ (CNR)（应用数学与信息技术研究所（意大利国家研究委员会））；Universidad Nacional del Litoral（国立里奥内格拉斯大学）；Consejo Nacional de Investigaciones Científicas y Técnicas（国家科学研究委员会）
We consider the ill-posed inverse problem of identifying a nonlinearity in a time-dependent PDE model. The nonlinearity is approximated by a neural network, and needs to be determined alongside other unknown physical parameters and the unknown state. Hence, it is not possible to construct input-output data pairs to perform a supervised training process. Proposing an all-at-once approach, we bypass the need for training data and recover all the unknowns simultaneously. In the general case, the approximation via a neural network can be realized as a discretization scheme, and the training with n…
We derive an equilibrated a posteriori error estimator for the space (semi) discretization of the scalar wave equation by finite elements. In the idealized setting where time discretization is ignored and the simulation time is large, we provide fully-guaranteed upper bounds that are asymptotically constant-free and show that the proposed estimator is efficient and polynomial-degree-robust, meaning that the efficiency constant does not deteriorate as the approximation order is increased. To the best of our knowledge, this work is the first to derive provably efficient error estimates for the w…
We consider finite element discretizations of Maxwell's equations coupled with a non-local hydrodynamic Drude model that accurately accounts for electron motions in metallic nanostructures. Specifically, we focus on a posteriori error estimation and mesh adaptivity, which is of particular interest since the electromagnetic field usually exhibits strongly localized features near the interface between metals and their surrounding media. We propose a novel residual-based error estimator that is shown to be reliable and efficient. We also present a set of numerical examples where the estimator dri…
Robust recovery-type a posteriori error estimators for streamline upwind/Petrov Galerkin discretizations for singularly perturbed problems
Chongqing Jiaotong University（重庆交通大学）；Texas A&M International University（德州农工国际大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Wayne State University（韦恩州立大学）
Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Network Tikhonov Regularization), which contains a trained neural network as regularizer in generalized Tikhonov regularization. The existing analysis of NETT considers fixed operator and fixed regularizer and analyzes the convergence as the noise level in the data approaches zero. In this paper, we extend the frameworks and analysis considerably to ref…
In order to characterize the fluctuation between the ergodic limit and the time-averaging estimator of a full discretization in a quantitative way, we establish a central limit theorem for the full discretization of the parabolic stochastic partial differential equation. The theorem shows that the normalized time-averaging estimator converges to a normal distribution with the variance being the same as that of the continuous case, where the scale used for the normalization corresponds to the temporal strong convergence order of the considered full discretization. A key ingredient in the proof …
In this paper we analyze a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem. A posteriori estimators for the nearly and perfectly compressible elasticity spectral problems are proposed. With a post-process argument, we are able to prove reliability and efficiency for the proposed estimators. The numerical method is based in Raviart-Thomas elements to approximate the pseudostress and piecewise polynomials for the displacement. We illustrate our results with numerical tests.
We develop a novel a posteriori error estimator for the $L^2$ error committed by the finite element discretization of the solution of the fractional Laplacian. Our a posteriori error estimator takes advantage of the semi-discretization scheme using rational approximations which allow to reformulate the fractional problem into a family of non-fractional parametric problems. The estimator involves applying the implicit Bank-Weiser error estimation strategy to each parametric non-fractional problem and reconstructing the fractional error through the same rational approximation used to compute the…
In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we propose and analyze a posteriori error estimators for an optimal control problem involving the stationary Navier--Stokes equations; control constraints are also considered. We devise two strategies of discretization: a semi discrete scheme where the control variable is not discretized -- the so-called variational discrezation approach -- and a fully discrete scheme where the control is discretized with piecewise quadratic functions. For each solution solution technique, we design an a posteriori error est…
The scalar, one-dimensional advection equation and heat equation are considered. These equations are discretized in space, using a finite difference method satisfying summation-by-parts (SBP) properties. To impose the boundary conditions, we use a penalty method called simultaneous approximation term (SAT). Together, this gives rise to two semi-discrete schemes where the discretization matrices approximate the first and the second derivative operators, respectively. The discretization matrices depend on free parameters from the SAT treatment. We derive the inverses of the discretization matric…
We consider the Virtual Element method (VEM) introduced by Beirão da Veiga, Lovadina and Vacca in 2016 for the numerical solution of the steady, incompressible Navier-Stokes equations; the method has arbitrary order $k \geq 2$ and guarantees divergence-free velocities. For such discretization, we develop a residual-based a posteriori error estimator, which is a combination of standard terms in VEM analysis (residual terms, data oscillation, and VEM stabilization), plus some other terms originated by the VEM discretization of the nonlinear convective term. We show that a linear combination of t…
Error Analysis for a Nonconforming Finite Element Discretization of the 3D-CCPF Model
Institut de Mathématiques et de Sciences Physiques (IMSP)（数学与物理科学研究所）；Faculté des Sciences et Techniques (FAST) Université d’Abomey-Calavi (UAC)（阿波美-卡拉维大学科学技术学院）
We study a posteriori error analysis of linear-quadratic boundary control problems under bilateral box constraints on the control which acts through a Neumann type boundary condition. We adopt the hybridizable discontinuous Galerkin method as discretization technique, and the flux variables, the scalar variables and the boundary trace variables are all approximated by polynomials of degree k. As for the control variable, it is discretized by the variational discretization concept. Then an efficient and reliable a posteriori error estimator is introduced, and we prove that the error estimator p…
In this article, we provide stability estimates for the finite element discretization of a class of inverse parameter problems of the form $-\nabla\cdot(μS) = \g f$ in a domain $Ω$ of $\R^d$. Here $μ$ is the unknown parameter to recover, the matrix valued function $S$ and the vector valued distribution $\g f$ are known. As uniqueness is not guaranteed in general for this problem, we prove a Lipschitz-type stability estimate in an hyperplane of $L^2(Ω)$. This stability is obtained through an adaptation of the so-called discrete \emph{inf-sup} constant or LBB constant to a large class of first-o…
This paper suggests a framework for the learning of discretizations of expensive forward models in Bayesian inverse problems. The main idea is to incorporate the parameters governing the discretization as part of the unknown to be estimated within the Bayesian machinery. We numerically show that in a variety of inverse problems arising in mechanical engineering, signal processing and the geosciences, the observations contain useful information to guide the choice of discretization.
We present an a posteriori error analysis for one-dimensional random hyperbolic systems of conservation laws. For the discretization of the random space we consider the Non-Intrusive Spectral Projection method, the spatio-temporal discretization uses the Runge--Kutta Discontinuous Galerkin method. We derive an a posteriori error estimator using smooth reconstructions of the numerical solution, which combined with the relative entropy stability framework yields computable error bounds for the space-stochastic discretization error. Moreover, we show that the estimator admits a splitting into a s…
For the first time we introduce an error estimator for the numerical approximation of the equations describing the dynamics of sea ice. The idea of the estimator is to identify different error contributions coming from spatial and temporal discretization as well as from the splitting in time of the ice momentum equations from further parts of the coupled system. The novelty of the error estimator lies in the consideration of the splitting error, which turns out to be dominant with increasing mesh resolution. Errors are measured in user specified functional outputs like the total sea ice extent…
A posteriori error estimates for mixed finite element discretization of the multigroup Neutron Simplified Transport equations with Robin boundary condition
POEMS, CNRS, INRIA, ENSTA, Institut Polytechnique de Paris（POEMS, 法国国家科学研究中心, 法国国家信息与自动化研究所, ENSTA巴黎综合理工学院, 巴黎理工学院）；Université Paris-Saclay, CEA, Service d’Études des Réacteurs et de Mathématiques Appliquées（巴黎萨克雷大学, 法国原子能和替代能源委员会, 反应堆研究与数学应用处）
Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems. Due to the typical ill-posedness of inverse problems, the reduced linear systems are often ill-posed, especially when their scales are large. This brings great computational difficulty. Particularly, a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution. Therefore, regularization methods should be …
Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems
We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows to introduce a probability measure on standard piecewise linear FEM. We present a posteriori error estimators based uniquely on probabilistic information. A series of numerical experiments illustrates the potential of the RM-FEM for error estimation and validates our analysis. We furthermore demonstrate how employing the RM-FEM enhances the quality of the sol…
Numerical modeling and open-source implementation of variational partition-of-unity localizations of space-time dual-weighted residual estimators for parabolic problems
This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle $< π/2 $, that is the resolvent set of this operator contains $ \{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< θ\}$ for some $ π/2 < θ< π$. The relationship between the time fractional order $α\in (0, 1)$ and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as $ α$ approaches $ 1 $. Then an inverse prob…
We present a novel algorithm based on the ensemble Kalman filter to solve inverse problems involving multiscale elliptic partial differential equations. Our method is based on numerical homogenization and finite element discretization and allows to recover a highly oscillatory tensor from measurements of the multiscale solution in a computationally inexpensive manner. The properties of the approximate solution are analysed with respect to the multiscale and discretization parameters, and a convergence result is shown to hold. A reinterpretation of the solution from a Bayesian perspective is pr…
We study the existence of solutions for Darcy's problem coupled with the heat equation under singular forcing; the right-hand side of the heat equation corresponds to a Dirac measure. The studied model allows thermal diffusion and viscosity depending on the temperature. We propose a finite element solution technique and analyze its convergence properties. In the case that the thermal diffusion is constant, we propose an a posteriori error estimator and investigate reliability and efficiency properties. We illustrate the theory with numerical examples.
In this work, we consider pressurized phase-field fracture problems in nearly and fully incompressible materials. To this end, a mixed form for the solid equations is proposed. To enhance the accuracy of the spatial discretization, a residual-type error estimator is developed. Our algorithmic advancements are substantiated with several numerical tests that are inspired from benchmark configurations. Therein, a primal-based formulation is compared to our newly developed mixed phase-field fracture method for Poisson ratios approaching $ν\to 0.5$. Finally, for $ν= 0.5$, we compare the numerical r…
