In this paper, we carry out the numerical analysis of a nonsmooth quasilinear elliptic optimal control problem, where the coefficient in the divergence term of the corresponding state equation is not differentiable with respect to the state variable. Despite the lack of differentiability of the nonlinearity in the quasilinear elliptic equation, the corresponding control-to-state operator is of class $C^1$ but not of class $C^2$. Analogously, the discrete control-to-state operators associated with the approximated control problems are proven to be of class $C^1$ only. By using an explicit second-order sufficient optimality condition, we prove a priori error estimates for a variational approximation, a piecewise constant approximation, and a continuous piecewise linear approximation of the continuous optimal control problem. The numerical tests confirm these error estimates.
A Nonlinear Discretization Theory for Meshfree Collocation Methods applied to Quasilinear Elliptic Equations
Fachbereich Mathematik und Informatik, Universität Marburg（马尔堡大学数学与计算机科学系）；Institut für Numerische und Angewandte Mathematik, Universität Göttingen（哥廷根大学数值与应用数学研究所）
We couple the L1 discretization for Caputo derivative in time with spectral Galerkin method in space to devise a scheme that solves quasilinear subdiffusion equations. Both the diffusivity and the source are allowed to be nonlinear functions of the solution. We prove method's stability and convergence with spectral accuracy in space. The temporal order depends on solution's regularity in time. Further, we support our results with numerical simulations that utilize parallelism for spatial discretization. Moreover, as a side result we find asymptotic exact values of error constants along with th…
This paper analyzes a time-stepping discontinuous Galerkin method for fractional diffusion-wave problems. This method uses piecewise constant functions in the temporal discretization and continuous piecewise linear functions in the spatial discretization. Nearly optimal convergence rate with respect to the regularity of the solution is established when the source term is nonsmooth, and nearly optimal convergence rate $ \ln(1/τ)(\sqrt{\ln(1/h)} h^2+τ)$ is derived under appropriate regularity assumption on the source term. Convergence is also established without smoothness assumption on the init…
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs), which are related to the FBSDEs by the Pardoux-Peng theory. The algorithms rely on a learning process by minimizing the pathwise difference between two discrete stochastic processes, defined by the time discretization of the FBSDEs and the DNN representation of the PDE solutions, respectively. The proposed algorithms are shown to generate DNN solutions for a…
Convergence analysis of the Magnus-Rosenbrock type method for the finite element discretization of semilinear non-autonomous parabolic PDEs with nonsmooth initial data
Western Norway University of Applied Sciences（西挪威应用科学大学）；University of Cape Town（开普敦大学）；Technische Universität Chemnitz（开姆尼茨工业大学）
In this paper we consider the numerical approximation of a general second order semi-linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using finite element method for space discretization and the exponential Rosenbrock-Euler method for time discretization, we provide a rigorous convergence proof in space and time under only the standard Lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solution. This is in contrast to very restrictive assumptions made in the literature, where the autho…
Trigonometric integrators for quasilinear wave equations
Freie Universität Berlin（柏林自由大学）；Duke University（杜克大学）；UNC-Chapel Hill（北卡罗来纳大学教堂山分校）；Université Paris-Sud（巴黎南大学）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）
In this work, the z-transform is presented to analyze time-discrete solutions for Volterra integrodifferential equations (VIDEs) with nonsmooth multi-term kernels in the Hilbert space, and this class of continuous problem was first considered and analyzed by Hannsgen and Wheeler (SIAM J Math Anal 15 (1984) 579-594). This work discusses three cases of kernels $β_q(t)$ included in the integrals for the multi-term VIDEs, from which we use corresponding numerical techniques to approximate the solution of multi-term VIDEs in different cases. Firstly, for the case of $β_1(t), β_2(t) \in \mathrm{L}_1…
Optimal finite element error estimates and Newton convergence for a quasilinear elliptic problem with mixed boundary conditions
University of Bucharest（布加勒斯特大学）；“Gheorghe Mihoc – Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy（罗马尼亚科学院格奥尔基·米霍克-凯乌斯·雅各布数学统计与应用数学研究所）
This article discusses nonconforming finite element methods for convex minimization problems and systematically derives dual mixed formulations. Duality relations lead to simple error estimates that avoid an explicit treatment of nonconformity errors. A reconstruction formula provides the discrete solution of the dual problem via a simple postprocessing procedure which implies a strong duality relation and is of interest in a posteriori error estimation. The framework applies to differentiable and nonsmooth problems, examples include $p$-Laplace, total-variation regularized, and obstacle probl…
In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems. The method is based on a space-time variational formulation. We provide a residual-based a-posteriori error bound on a space-time level and the corresponding efficiently computable estimator for the certification of the method. We use the Empirical Interpolation method (EIM) to guarantee the efficient offline-online computational procedure. The error of the EIM method is then rigorously incorporated into the certification procedure. The Petrov-Galerkin finite element discretization allows to ben…
A Multigrid Optimization Algorithm for the Numerical Solution of Quasilinear Variational Inequalities Involving the $p$-Laplacian
Escuela Politécnica Nacional（厄瓜多尔国立理工学院）；Research Center on Mathematical Modeling (MODEMAT)（数学建模研究中心）；Departamento de Matemática - Escuela Politécnica Nacional（厄瓜多尔国立理工学院数学系）
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）
This paper considers the numerical analysis of a semilinear fractional diffusion equation with nonsmooth initial data. A new Grönwall's inequality and its discrete version are proposed. By the two inequalities, error estimates in three Sobolev norms are derived for a spatial semi-discretization and a full discretization, which are optimal with respect to the regularity of the solution. A sharp temporal error estimate on graded temporal grids is also rigorously established. In addition, the spatial accuracy $\scriptstyle O(h^2(t^{-α} + \ln(1/h)\!)\!) $ in the pointwise $ \scriptstyle L^2(Ω) $-n…
Sharp Error Estimates for a Fully Discrete Approximation of Semilinear Stochastic Convection--Diffusion Equations with Additive Noise and Nonsmooth Initial Data
A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis
Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche（意大利国家研究委员会计算应用研究所“毛罗·皮切内”）；Laboratoire J. A. Dieudonné, UMR CNRS 7351, Université de Nice-Sophia Antipolis（尼斯索菲亚安蒂波利斯大学）；Project Team COFFEE, INRIA Sophia Antipolis（INRIA索菲亚安蒂波利斯COFFEE项目组）
In this paper we propose a primal-dual dynamical approach to the minimization of a structured convex function consisting of a smooth term, a nonsmooth term, and the composition of another nonsmooth term with a linear continuous operator. In this scope we introduce a dynamical system for which we prove that its trajectories asymptotically converge to a saddle point of the Lagrangian of the underlying convex minimization problem as time tends to infinity. In addition, we provide rates for both the violation of the feasibility condition by the ergodic trajectories and the convergence of the objec…
This paper is devoted to the error analysis of a time-spectral algorithm for fractional diffusion problems of order $α$ ($0 < α< 1$). The solution regularity in the Sobolev space is revisited, and new regularity results in the Besov space are established. A time-spectral algorithm is developed which adopts a standard spectral method and a conforming linear finite element method for temporal and spatial discretizations, respectively. Optimal error estimates are derived with nonsmooth data. Particularly, a sharp temporal convergence rate $1+2α$ is shown theoretically and numerically.
Identifying the discontinuous diffusion coefficient in an elliptic equation with observation data of the gradient of the solution is an important nonlinear and ill-posed inverse problem. Models with total variational (TV) regularization have been widely studied for this problem, while the theoretically required nonsmoothness property of the TV regularization and the hidden convexity of the models are usually sacrificed when numerical schemes are considered in the literature. In this paper, we show that the favorable nonsmoothness and convexity properties can be entirely kept if the well-known …
Folding grid value vectors of size $2^L$ into $L$th order tensors of mode sizes $2\times \cdots\times 2$, combined with low-rank representation in the tensor train format, has been shown to lead to highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Standard choices of the underlying basis, such as piecewise multilinear finite elements on uniform tensor …
