Semilinear Partial Differential Equations Research Articles

Analytic and Gevrey class regularity for parametric semilinear reaction-diffusion problems and applications in uncertainty quantification

We investigate a class of parametric elliptic semilinear partial differential equations of second order with homogeneous essential boundary conditions, where the coefficients and the right-hand side (and hence the solution) may depend on a parameter. This model can be seen as a reaction-diffusion problem with a polynomial nonlinearity in the reaction term. The efficiency of various numerical approximations across the entire parameter space is closely related to the regularity of the solution with respect to the parameter. We show that if the coefficients and the right-hand side are analytic or Gevrey class regular with respect to the parameter, the same type of parametric regularity is valid for the solution. The key ingredient of the proof is the combination of the alternative-to-factorial technique from our previous work [1] with a novel argument for the treatment of the power-type nonlinearity in the reaction term. As an application of this abstract result, we obtain rigorous convergence estimates for numerical integration of semilinear reaction-diffusion problems with random coefficients using Gaussian and Quasi-Monte Carlo quadrature. Our theoretical findings are confirmed in numerical experiments.

An asymptotic expansion of the solution of a semi-linear partial differential equation implied by a nonlinear Feynman–Kac formula

This paper introduces an asymptotic expansion for the smooth solution of a semi-linear partial differential equation. Our scheme is based on Itô’s formula, Taylor’s expansion, nonlinear Feynman–Kac formula and some algebras.

Solution theory to semilinear stochastic equations of Schrödinger type on curved spaces I: operators with uniformly bounded coefficients

We study the Cauchy problem for Schrödinger type stochastic semilinear partial differential equations with uniformly bounded variable coefficients, depending on the space variables. We give conditions on the coefficients, on the drift and diffusion terms, on the Cauchy data, and on the spectral measure associated with the noise, such that the Cauchy problem admits a unique function-valued mild solution in the sense of Da Prato and Zabczyc.

On a reaction diffusion problem with a moving impulse on boundary

Abstract We study an asymptotic problem of a semilinear partial differential equation (PDE) with Neumann boundary condition, periodic coefficients and highly oscillating drift and nonlinear terms. Our analysis focuses on the double limiting behavior of the PDE-solution perturbed by ε (viscosity parameter) and δ (scaling coefficient) both tending to zero. To do so, we state basic properties of the large deviations principle (LDP) and we express the logarithmic asymptotic of the PDE-solution. Particularly, we provide it for the case when ε converges more quickly than δ.

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.

Second-Order Estimates for Transition Layers and a Curvature Estimate for the Parabolic Allen–Cahn

Abstract The parabolic Allen–Cahn equation is a semilinear partial differential equation that is closely linked to the mean curvature flow by a singular perturbation. Motivated by the work of Wang–Wei [ 21] and Chodosh–Mantoulidis [ 3] in the elliptic setting, we initiate the corresponding regularity theory for parabolic Allen–Cahn flows. In particular, we establish an improved convergence property of parabolic Allen–Cahn flows to the mean curvature flow: if the phase-transition level sets converge in $C^{2}$, then they converge in $C^{2,\theta }$ as well. As an application, we obtain a curvature estimate for the parabolic Allen–Cahn equation, which can be seen as a diffused version of Brakke’s [ 1] and White’s [ 24] regularity theorems for mean curvature flow.

Well-posedness results for a new class of stochastic spatio-temporal SIR-type models driven by proportional pure-jump Lévy noise

This paper provides a first attempt to incorporate the massive discontinuous changes in the spatio-temporal dynamics of epidemics. Namely, we propose an extended class of epidemic models, governed by coupled stochastic semilinear partial differential equations, driven by pure-jump Lévy noise. Based on the considered type of incidence functions, by virtue of semigroup theory, a truncation technique and Banach fixed point theorem, we prove the existence and pathwise uniqueness of mild solutions, depending continuously on the initial datum. Moreover, by means of a regularization technique, based on the resolvent operator, we acquire that mild solutions can be approximated by a suitable converging sequence of strong solutions. With this result at hand, for positive initial states, we derive the almost-sure positiveness of the obtained solutions. Finally, we present the outcome of several numerical simulations, in order to exhibit the effect of the considered type of stochastic noise, in comparison to Gaussian noise, which has been used in the previous literature. Our established results lay the ground-work for investigating other problems associated with the new proposed class of epidemic models, such as asymptotic behavior analyses, optimal control as well as identification problems, which primarily rely on the existence and uniqueness of biologically feasible solutions.

Numerical Finite Difference Scheme For A Two-Dimensional Time-Fractional Semi Linear Partial Differential Equations

Numerical Finite Difference Scheme For A Two-Dimensional Time-Fractional Semi Linear Partial Differential Equations

Mean-field neural networks: Learning mappings on Wasserstein space

We study the machine learning task for models with operators mapping between the Wasserstein space of probability measures and a space of functions, like e.g. in mean-field games/control problems. Two classes of neural networks based on bin density and on cylindrical approximation, are proposed to learn these so-called mean-field functions, and are theoretically supported by universal approximation theorems. We perform several numerical experiments for training these two mean-field neural networks, and show their accuracy and efficiency in the generalization error with various test distributions. Finally, we present different algorithms relying on mean-field neural networks for solving time-dependent mean-field problems, and illustrate our results with numerical tests for the example of a semi-linear partial differential equation in the Wasserstein space of probability measures.

Ergodic properties of a semilinear partial differential equation

Semiflows on some spaces of continuous functions generated by a semilinear partial differential equation are studied. Sufficient conditions for the existence of invariant measures with strong ergodic properties are given. These properties imply chaotic behaviour of the semiflows, that is, the existence of dense trajectories and that each trajectory is unstable. In the proofs we use the Lévy d-parameter Brownian motion and we show that these semiflows are isomorphic with translation along the radii.

A Note on Existence of Solutions to Control Problems of Semilinear Partial Differential Equations

In this paper, we study optimal control problems of semilinear elliptic and parabolic equations. A tracking cost functional, quadratic in the control and state variables, is considered. No control constraints are imposed. We prove that the corresponding state equations are well posed for controls in . However, it is well known that in the framework the mappings involved in the control problem are not Frechet differentiable in general, which makes any analysis of the optimality conditions challenging. Nevertheless, we prove that every optimal control belongs to , and consequently standard optimality conditions are available.

A stochastic gradient method for a class of nonlinear PDE-constrained optimal control problems under uncertainty

The study of optimal control problems under uncertainty plays an important role in scientific numerical simulations. This class of optimization problems is strongly utilized in engineering, biology and finance. In this paper, a stochastic gradient method is proposed for the numerical resolution of a nonconvex stochastic optimization problem on a Hilbert space. We show that, under suitable assumptions, strong or weak accumulation points of the iterates produced by the method converge almost surely to stationary points of the original optimization problem. Measurability and convergence rates of a stationarity measure are handled, filling a gap for applications to nonconvex infinite dimensional stochastic optimization problems. The method is demonstrated on an optimal control problem constrained by a class of elliptic semilinear partial differential equations (PDEs) under uncertainty.

Gevrey regularity of the solutions of some inhomogeneous semilinear partial differential equations with variable coefficients

Gevrey regularity of the solutions of some inhomogeneous semilinear partial differential equations with variable coefficients

In silico studies of bacterial quorum sensing during population dynamics: simulations by using COMSOL Multiphysics

In the present study, computing techniques are designated and employed in order to analyze “quorum sensing” as a special case of cell-to-cell bacterial communication attributed to the Pseudomonas bacterial genus. One of the challenges consists in predicting the concentration of key substances characterizing the “quorum level” during bacterial population dynamics. To estimate relevant characteristics of bacterial communication, we applied a deterministic approach. The mathematical model is formalized as an initial-boundary value problem for a system of semilinear partial differential equations supplemented by the procedure to specify the multiphase dynamics of bacterial populations. The finite element solution of the problem is obtained by COMSOL Multiphysics. The comparative numerical analysis for various types of space approximation of bacterial density is performed. A series of computational experiments were conducted to estimate changes in concentrations of chemical compounds during the Pseudomonas putida population dynamics.

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 reduced order modeling approach with Petrov–Galerkin projection based on hybrid snapshot simulation in Semilinear-PDEs

This paper proposes an efficient and economic framework to generate reduced order models (ROMs) with Petrov–Galerkin projection based on hybrid snapshot simulation to extraordinarily enhance the speedup and enlighten the representation of snapshot data in Semilinear Partial Differential Equations (Semilinear-PDEs). The total snapshot simulation is divided into multiple temporal intervals with the alternations in each interval between two types of models: the full order model (FOM) and the local ROM based on Petrov–Galerkin projection (ROM-PG). During the snapshot training process, the simulation can determine the model type on-the-fly by the switch criteria introduced in Bai and Wang (2022) and continuously update the subspace by incremental singular value decomposition (iSVD) to preserve the subspace representation. In numerical experiments, we consider two typical Semilinear-PDEs to illustrate the proposed method successfully in finite volume element discretization: Semilinear-Parabolic Equation (Semilinear-Parabolics) and Semilinear-Convection–Diffusion Equation (Semilinear-ConvDiffs). It is found that the subspace updating is directly related to the POD numbers, the parameter μ (μp in Semilinear-Parabolics or μc in Semilinear-ConvDiffs). The tolerance of criteria also significantly affects the performance of computational acceleration. The quantitative analyses of the criteria and the brief discussion of stability about the framework are also performed to clarify the model switch and stability condition, respectively, due to no detailed discussion in the previous work (Bai and Wang 2022).

Error estimation for the time to a threshold value in evolutionary partial differential equations

We develop an a posteriori error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals of the solution. Taylor’s theorem and an adjoint-based a posteriori analysis is used to derive computable and accurate error estimates in the case of semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.

Fractional backward SDEs with locally monotone coefficient and application to PDEs

Abstract In this work, we will try to weaken the hypothesis imposed by Hu and Peng. We will be concerned with finding the solution of locally monotone BSDEs associated to fBm. As an auxiliary step, we study the existence and uniqueness of a solution to the monotone backward SDEs associated to fBm. Then we connect these two kinds of fractional backward SDEs with the corresponding semilinear partial differential equations (PDEs for short).

Large time behaviour of semilinear stochastic partial differential equations perturbed by a mixture of Brownian and fractional Brownian motions

We study the trajectorywise blowup behaviour of a semilinear partial differential equation that is driven by a mixture of multiplicative Brownian and fractional Brownian motion, modelling different types of random perturbations. The linear operator is supposed to have an eigenfunction of constant sign, and we show its influence, as well as the influence of its eigenvalue and of the other parameters of the equation, on the occurrence of a blowup in finite time of the solution. We give estimates for the probability of finite time blowup and of blowup before a given fixed time. Essential tools are the mild and weak form of an associated random partial differential equation.

On a novel approach for the investigation and approximation of solutions to the systems of higher order nonlinear PDEs

We study a boundary value problem for a system of the third order semi-linear partial differential equations with nonlocal boundary conditions. We establish sufficient conditions of existence, uniqueness, regularity and sign-preserving property of solutions of the studied problem and construct an iterative method for its approximation.