Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.     

On generalized Malliavin calculus

The Malliavin derivative, the divergence operator (Skorokhod integral), and the Ornstein-Uhlenbeck operator are extended from the traditional Gaussian setting to nonlinear generalized functionals of white noise. These extensions are related to the new developments in the theory of stochastic PDEs, in particular elliptic PDEs driven by spatial white noise and quantized nonlinear equations.     

A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

We propose a multiscale multilevel Monte Carlo(MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage,we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis.The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallestscale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.     

Reduction and Equivariant Branching Lemma: Dynamical systems,evolution PDEs,and gauge theories

We review and recast the Equivariant Branching Lemma-which has proved a remarkable tool in linearly equivariant bifurcation theory-and consider its extension to the case of nonlinear (Lie-point) symmetries. This is then applied to gauge theories and gauge theoretic problems, and to nonlinear evolution PDE's; the paper also contains an original setting of Lie-point symmetries for evolution PDEs, modelled on the dynamical systems setting.     

Persistence of Invariant Manifolds for Nonlinear PDEs

We prove that under certain stability and smoothing properties of the semi-groups generated by the partial differential equations that we consider, manifolds left invariant by these flows persist under C ¹ perturbation. In particular, we extend well-known finite-dimensional results to the setting of an infinite-dimensional Hilbert manifold with a semi-group that leaves a submanifold invariant. We then study the persistence of global unstable manifolds of hyperbolic fixed points, and as an application consider the two-dimensional Navier–Stokes equation under a fully discrete approximation.Finally, we apply our theory to the persistence of inertial manifolds for those PDEs that possess them.     

Analysis of the Flow of Magnetoelastic Materials

We consider evolutionary models for magnetoelastic materials. The PDEs modeling the flow of the material were derived from variational principles in a continuum mechanical setting. In this article, we discuss the existence of weak solutions for two models, one with gradient flow type dynamics on the magnetic variable and in the other the dynamics of the magnetization is described by the Landau-Lifshitz-Gilbert equation. We highlight the main differences of the proofs in the two cases. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A deterministic‐control‐based approach to fully nonlinear parabolic and elliptic equations

We show that a broad class of fully nonlinear, second‐order parabolic or elliptic PDEs can be realized as the Hamilton‐Jacobi‐Bellman equations of deterministic two‐person games. More precisely: given the PDE, we identify a deterministic, discrete‐time, two‐person game whose value function converges in the continuous‐time limit to the viscosity solution of the desired equation. Our game is, roughly speaking, a deterministic analogue of the stochastic representation recently introduced by Cheridito, Soner, Touzi, and Victoir. In the parabolic setting with no u‐dependence, it amounts to a semidiscrete numerical scheme whose timestep is a min‐max. Our result is interesting, because the usual control‐based interpretations of second‐order PDEs involve stochastic rather than deterministic control. © 2009 Wiley Periodicals, Inc.     

On the Drinfeld-Sokolov hierarchy of type E_6~((1)) and its topological solution

In this paper we report some explicit evolutionary PDEs of the Drinfeld-Sokolov hierarchy of type E_6~((1)), and show how the unknown functions in these PDEs are related to the tau function. Moreover, for this hierarchy we compute its topological solution of formal series up to a certain degree, whose coefficients of monomials give the Fan-Jarvis-Ruan-Witten invariants for the E_6 simple singularity. Based on such results we also derive several explicit evolutionary PDEs and some low-degree terms of the topological solution for the Drinfeld-Sokolov hierarchy of type F_4~((1)).     

A study on the PDEs with power-law nonlinearity

In this paper, we introduce two types of PDEs with power-law nonlinearity which contain many types of the linear and nonlinear PDEs. For solving the Cauchy problems of the introduced PDEs, we apply an analytic technique, namely the homotopy analysis method (HAM).     

Brownian-time processes: The PDE connection II and the corresponding Feynman-Kac formula

We delve deeper into our study of the connection of Brownian-time processes (BTPs) to fourth-order parabolic PDEs, which we introduced in a recent joint article with W. Zheng. Probabilistically, BTPs and their cousins BTPs with excursions form a unifying class of interesting stochastic processes that includes the celebrated IBM of Burdzy and other new intriguing processes and is also connected to the Markov snake of Le Gall. BTPs also offer a new connection of probability to PDEs that is fundamentally different from the Markovian one. They solve fourth-order PDEs in which the initial function plays an important role in the PDE itself, not only as initial data. We connect two such types of interesting and new PDEs to BTPs. The first is obtained by running the BTP and then integrating along its path, and the second type of PDEs is related to what we call the Feynman-Kac formula for BTPs. A special case of the second type is a step towards a probabilistic solution to linearized Cahn-Hilliard and Kuramoto-Sivashinsky type PDEs, which we tackle in an upcoming paper.

     

Necessary optimality conditions for infinite dimensional state constrained control problems

This paper is concerned with first order necessary optimality conditions for state constrained control problems in separable Banach spaces. Assuming inward pointing conditions on the constraint, we give a simple proof of Pontryagin maximum principle, relying on infinite dimensional neighboring feasible trajectories theorems proved in [20]. Further, we provide sufficient conditions guaranteeing normality of the maximum principle. We work in the abstract semigroup setting, but nevertheless we apply our results to several concrete models involving controlled PDEs. Pointwise state constraints (as positivity of the solutions) are allowed.     

On some classes of projectively flat Finsler metrics with constant flag curvature

In this paper, we give the equivalent PDEs for projectively flat Finsler metrics with constant flag curvature defined by a Euclidean metric and two 1-forms. Furthermore, we construct some classes of new projectively flat Finsler metrics with constant flag curvature by solving these equivalent PDEs.     

The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

In this article we consider partitioned Runge-Kutta (PRK) methods for Hamiltonian partial differential equations (PDEs) and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs.

     

Application of LQR techniques to the adaptive control of quasilinear parabolic PDEs

LQR problems for linear parabolic PDEs have been studied in detail in the literature for the past 3 to 4 decades. The solvability of feedback control problems for a large class of problems is well understood. In recent years numerical methods for the approximation of the corresponding Riccati operators have been developed. These methods are able to calculate the feedback operator directly and thus can compute the solutions to linear problems efficiently. Here we study the applicability of such techniques to the control of quasilinear equations via local linearization in an adaptive control setting. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On modified method of simplest equation for obtaining exact and approximate solutions of nonlinear PDEs: The role of the simplest equation

The modified method of simplest equation is powerful tool for obtaining exact and approximate solutions of nonlinear PDEs. These solutions are constructed on the basis of solutions of more simple equations called simplest equations. In this paper we study the role of the simplest equation for the application of the modified method of simplest equation. We follow the idea that each function constructed as polynomial of a solution of a simplest equation is a solution of a class of nonlinear PDEs. We discuss three simplest equations: the equations of Bernoulli and Riccati and the elliptic equation. The applied algorithm is as follows. First a polynomial function is constructed on the basis of a simplest equation. Then we find nonlinear ODEs that have the constructed function as a particular solution. Finally we obtain nonlinear PDEs that by means of the traveling-wave ansatz can be reduced to the above ODEs. By means of this algorithm we make a first step towards identification of the above-mentioned classes of nonlinear PDEs.     

Suboptimality Estimates for the Semi-Discretized LQR Problem for Parabolic PDEs

The linear quadratic regulator problem (LQR) for parabolic partial differential equations (PDEs) has been understood to be an infinite-dimensional Hilbert space equivalent of the finite-dimensional LQR problem known from mathematical systems theory. The matrix equations from the finite-dimensional case become operator equations in the infinite-dimensional Hilbert space setting. A rigorous convergence theory for the approximation of the infinite-dimensional problem by Galerkin schemes in the space variable has been developed over the past decades. Numerical methods based on this approximation have been proven capable of solving the case of linear parabolic PDEs. Embedding these solvers in a model predictive control (MPC) scheme, also nonlinear systems can be handled. Convergence rates for the approximation in the linear case are well understood in terms of the PDE's solution trajectories, as well as the solution operators of the underlying matrix/operator equations. However, in practice engineers are often interested in suboptimality results in terms of the optimal cost, i.e., evaluation of the quadratic cost functional. In this contribution, we are closing this gap in the theory. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Node adaptive domain decomposition method by radial basis functions

During the last years, there has been increased interest in developing efficient radial basis function (RBF) algorithms to solve partial differential problems of great scale. In this article, we are interested in solving large PDEs problems, whose solution presents rapid variations. Our main objective is to introduce a RBF dynamical domain decomposition algorithm which simultaneously performs a node adaptive strategy. This algorithm is based on the RBFs unsymmetric collocation setting. Numerical experiments performed with the multiquadric kernel function, for two stationary problems in two dimensions are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Forward–backward SDEs with jumps and classical solutions to nonlocal quasilinear parabolic PDEs

We obtain an existence and uniqueness theorem for fully coupled forward–backward SDEs (FBSDEs) with jumps via the classical solution to the associated quasilinear parabolic partial integro-differential equation (PIDE), and provide the explicit form of the FBSDE solution. Moreover, we embed the associated PIDE into a suitable class of non-local quasilinear parabolic PDEs which allows us to extend the methodology of Ladyzhenskaya et al. (1968) to non-local PDEs of this class. Namely, we obtain the existence and uniqueness of a classical solution to both the Cauchy problem and the initial–boundary value problem for non-local quasilinear parabolic second-order PDEs.     

Gradient estimates for nonlinear diffusion semigroups by coupling methods

In this paper, we obtain gradient estimates for certain nonlinear partial differential equations by coupling methods. First, we derive uniform gradient estimates for certain semi-linear PDEs based on the coupling method introduced by Wang in 2011 and the theory of backward SDEs. Then we generalize Wang's coupling to the G-expectation space and obtain gradient estimates for nonlinear diffusion semigroups, which correspond to the solutions of certain fully nonlinear PDEs.