Semilinear Elliptic Partial Differential Equations Research Articles

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs.

The Gevrey class implicit mapping theorem with application to UQ of semilinear elliptic PDEs

This paper is concerned with a regularity analysis of parametric operator equations with a perspective on uncertainty quantification. We study the regularity of mappings between Banach spaces near branches of isolated solutions that are implicitly defined by a residual equation. Under [Formula: see text]-Gevrey assumptions on the residual equation, we establish [Formula: see text]-Gevrey bounds on the Fréchet derivatives of the locally defined data-to-solution mapping. This abstract framework is illustrated in a proof of regularity bounds for a semilinear elliptic partial differential equation with parametric and random field input.

Analytic regularity and solution approximation for a semilinear elliptic partial differential equation in a polygon

In an open, bounded Lipschitz polygon Ω⊂R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R}^2$$\\end{document}, we establish weighted analytic regularity for a semilinear, elliptic PDE with analytic nonlinearity and subject to a source term f which is analytic in Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}. The boundary conditions on each edge of ∂Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega $$\\end{document} are either homogeneous Dirichlet or homogeneous Neumann BCs. The presently established weighted analytic regularity of solutions implies exponential convergence of various approximation schemes: hp-finite elements, reduced order models via Kolmogorov n-widths of solution sets in H1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1(\\Omega )$$\\end{document}, quantized tensor formats and certain deep neural networks.

On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs

We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation (PDE) which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc(ℝ). This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. The convergence of the resulting algorithm is proven in the function space setting. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero. The paper concludes with numerical experiments which confirm the theoretical findings.

Semilinear optimal control with Dirac measures

Abstract The purpose of this work is to study an optimal control problem for a semilinear elliptic partial differential equation with a linear combination of Dirac measures as a forcing term; the control variable corresponds to the amplitude of such singular sources. We analyze the existence of optimal solutions and derive first- and, necessary and sufficient, second-order optimality conditions. We develop a solution technique that discretizes the state and adjoint equations with continuous piecewise linear finite elements; the control variable is already discrete. We analyze the convergence properties of discretizations and obtain, in two dimensions, an a priori error estimate for the underlying approximation of an optimal control variable.

Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.

L∞-Convergence Analysis of a Finite Element Linear Schwarz Alternating Method for a Class of Semi-Linear Elliptic PDEs

In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for a class of semi-linear elliptic partial differential equations, in the context of linear iterations and non-matching grids. More precisely, making use of the subsolution-based concept, we prove that finite element Schwarz iterations converge, in the maximum norm, to the true solution of the PDE. We also give numerical results to validate the theory. This work introduces a new approach and generalizes the one in [14] as it encompasses a larger class of problems.

Error Estimates for Fractional Semilinear Optimal Control on Lipschitz Polytopes

We adopt the integral definition of the fractional Laplace operator and analyze solution techniques for fractional, semilinear, and elliptic optimal control problems posed on Lipschitz polytopes. We consider two strategies of discretization: a semidiscrete scheme where the admissible control set is not discretized and a fully discrete scheme where such a set is discretized with piecewise constant functions. As an instrumental step, we derive error estimates for finite element discretizations of fractional semilinear elliptic partial differential equations (PDEs) on quasi-uniform and graded meshes. With these estimates at hand, we derive error bounds for the semidiscrete scheme and improve the ones that are available in the literature for the fully discrete scheme.

Normalized Wolfe-Powell-type local minimax method for finding multiple unstable solutions of nonlinear elliptic PDEs

The local minimax method (LMM) proposed by Li and Zhou (2001) and Li and Zhou (2002) is an efficient method to solve nonlinear elliptic partial differential equations (PDEs) with certain variational structures for multiple solutions. The steepest descent direction and the Armijo-type step-size search rules are adopted in Li and Zhou (2002) and play a significant role in the performance and convergence analysis of traditional LMMs. In this paper, a new algorithm framework of the LMMs is established based on general descent directions and two normalized (strong) Wolfe-Powell-type step-size search rules. The corresponding algorithm framework named as the normalized Wolfe-Powell-type LMM (NWP-LMM) is introduced with its feasibility and global convergence rigorously justified for general descent directions. As a special case, the global convergence of the NWP-LMM algorithm combined with the preconditioned steepest descent (PSD) directions is also verified. Consequently, it extends the framework of traditional LMMs. In addition, conjugate gradient-type (CG-type) descent directions are utilized to speed up the NWP-LMM algorithm. Finally, extensive numerical results for several semilinear elliptic PDEs are reported to profile their multiple unstable solutions and compared for different algorithms in the LMM’s family to indicate the effectiveness and robustness of our algorithms. In practice, the NWP-LMM combined with the CG-type direction indeed performs much better than its known LMM companions.

A posteriori error estimates of two-grid weak Galerkin methods for semilinear elliptic differential equations

In this paper, we investigate the residual-based a posteriori error estimates of two-grid weak Galerkin (WG) methods for second order semilinear elliptic partial differential equations (PDEs). First, we propose two different two-grid weak Galerkin methods for the model problem and then establish a posteriori error estimators of the two-grid weak Galerkin methods. Theoretical analysis is given to prove the reliability and efficiency of the error estimators. We mainly study the lowest order case of WG element (Pj(T),Pℓ(∂T),RTj(T)) with j=ℓ=0[19]. Numerical experiments are provided to confirm the theoretical results.

A probabilistic approach to Neumann problems for elliptic PDEs with nonlinear divergence terms

By a probabilistic method, we prove the existence and uniqueness of weak solutions to Neumann problems for a class of semi-linear elliptic partial differential equations with nonlinear singular divergence terms, which can only be understood in distributional sense. This leads to the further study on a new class of infinite horizon backward stochastic differential equations, which involves integrals with respect to a forward–backward martingale and a singular continuous increasing process.

Error Estimates for a Pointwise Tracking Optimal Control Problem of a Semilinear Elliptic Equation

We consider a pointwise tracking optimal control problem for a semilinear elliptic partial differential equation. We derive the existence of optimal solutions and obtain first order optimality conditions. We also obtain necessary and sufficient second order optimality conditions. To approximate the solution of the aforementioned optimal control problem we devise a finite element technique that approximates the solution to the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. We analyze convergence properties and prove that the error approximation of the control variable converges with rate $\mathcal{O}(h|\log h|)$ when measured in the $L^2$-norm.

Stability in Affine Optimal Control Problems Constrained by Semilinear Elliptic Partial Differential Equations

This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.

Optimization with learning-informed differential equation constraints and its applications

Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.

Convergence, stability analysis, and solvers for approximating sublinear positone and semipositone boundary value problems using finite difference methods

Positone and semipositone boundary value problems are semilinear elliptic partial differential equations (PDEs) that arise in reaction–diffusion models in mathematical biology and the theory of nonlinear heat generation. Under certain conditions, the problems may have multiple positive solutions or even nonexistence of a positive solution. We develop analytic techniques for proving admissibility, stability, and convergence results for simple finite difference approximations of positive solutions to sublinear problems. We also develop guaranteed solvers that can detect nonuniqueness for positone problems and nonexistence for semipositone problems. The admissibility and stability results are based on adapting the method of sub- and supersolutions typically used to analyze the underlying PDEs. The new convergence analysis technique directly shows that all pointwise limits of finite difference approximations are solutions to the boundary value problem eliminating the possibility of false algebraic solutions plaguing the convergence of the methods. Most known approximation methods for positone and semipositone boundary value problems rely upon shooting techniques; hence, they are restricted to one-dimensional problems and/or radial solutions. The results in this paper will serve as a foundation for approximating positone and semipositone boundary value problems in higher dimensions and on more general domains using simple approximation methods. Numerical tests for known applied problems with multiple positive solutions are provided. The tests focus on approximating certain positive solutions as well as generating discrete bifurcation curves that support the known existence and uniqueness results for the PDE problem.

On the no-gap second-order optimality conditions for a non-smooth semilinear elliptic optimal control

This work is concerned with second-order necessary and sufficient optimality conditions for optimal control of a non-smooth semilinear elliptic partial differential equation, where the nonlinearity is the non-smooth max-function and thus the associated control-to-state operator is, in general, not Gâteaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the Gâteaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the set of all zeros of the corresponding state. The second one is a structural assumption on the sets of all points at which the values of the interested state are ‘close’ to the non-differentiability point of the max-function. We then derive a ‘no-gap’ theory of second-order optimality conditions in terms of a second-order generalized derivative of the cost functional, i.e. for which the only change between necessary and sufficient second-order optimality conditions are between a strict and non-strict inequality.

Homogenization of periodic elliptic degenerate PDEs with non-linear Neumann boundary condition

In this paper, a semi-linear elliptic partial differential equation (PDE) with non linear Neumann boundary condition and rapidly oscillating coefficients is homogenized. The novelty of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of R d . Our fully probabilistic method is based on the connection between PDEs and backward stochastic differential equations (BSDEs) with random terminal time and the weak convergence of a class of diffusion processes.

Study of an Elliptic Partial Differential Equation Modeling the Ocean Flow in Arctic Gyres

We study the ocean flow in Arctic gyres using a recent model for gyres derived in spherical coordinates on the rotating sphere. By projecting this model onto the plane using the Mercator projection, we obtain a semi-linear elliptic partial differential equation in an unbounded domain, difficulty which is then overcome by projecting the PDE onto the unit disk via a conformal map. We then study existence, regularity and uniqueness of solutions for constant and linear vorticity functions.

Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs

In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant.

ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization

The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach.