Domain decomposition and expanded mixed method for parabolic partial differential equations

A fourth-order kernel-free boundary integral method for implicitly defined surfaces in three space dimensions

Relying on the classical connection between backward stochastic differential equations and nonlinear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional semilinear parabolic PDEs. This scheme is inspired by the approach coming from machine learning and developed using deep neural networks in Han et al. (2018, Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci., 115, 8505–8510. Our algorithm is based on a Picard iteration scheme in which a sequence of linear-quadratic optimization problem is solved by means of stochastic gradient descent algorithm. In the framework of a linear specification of the approximation space, we manage to prove a convergence result for our scheme, under some smallness condition. In practice, in order to be able to treat high-dimensional examples, we employ sparse-grid approximation spaces. In the case of periodic coefficients and using pre-wavelet basis functions, we obtain an upper bound on the global complexity of our method. It shows, in particular, that the curse of dimensionality is tamed in the sense that in order to achieve a root mean squared error of order $\varepsilon $, for a prescribed precision $\varepsilon $, the complexity of the Picard algorithm grows polynomially in $\varepsilon ^{-1}$ up to some logarithmic factor $|\!\log (\varepsilon )|$, whose exponent grows linearly with respect to the PDE dimension. Various numerical results are presented to validate the performance of our method, and to compare them with some recent machine learning schemes proposed in E et al. (2017, Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat., 5, 349–380) and Huré et al. (2020, Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comput., 89, 1547–1579).

MATLAB Distributed Computing Server (MDCS) and Parallel Virtual Machine (PVM) software are two types of distributed computing environments. MDCS is recently used in selecting the best network training algorithm and assessing the effect of parallelization. Since it is practical and user friendly, it gives PVM the opportunity to become the communication paradigm of choice. The PVM provides a powerful set of process control and dynamic resource management features. In the distributed parallel computing (DPC), however, both solutions have different strengths and limitations. Based on these concerns, this paper compares the numerical analysis for mathematical modeling of large sparse 2D and 3D of second order parabolic partial differential equations (PDE) on MDCS and PVM based on parallel performance indicators (PPI). The PDE geometry is discretized into a sparse grid structure using the FDM method. In using a method with the highest accuracy, Parallel Alternating Group Explicit (PAGE) scheme was chosen. The parallel strategies focus on the PAGE's convergence speed and various domain de-composition techniques, as well as a block iterative scheme and load balance using fine granular techniques. Furthermore, comparison of distributed computing environments also relies on multiple processors running on Unix-like operating systems with Fedora installed to support large-scale simulations. The analysis and validation of PPI for both communication software is also investigated in this paper. For a sequential algorithm, accuracy, estimation of error, and stability are employed as indicators. As a conclusion, based on the PPI and numerical analysis, the tables and graphs show that compared to MDCS, PVM is a better environment for solving multidimensional parabolic PDE modeling.KeywordsDistributed computingParabolic PDEsMDCSPVM

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives.

: This document considers two adaptive finite element techniques for parabolic partial differential equations (PDEs) that are based on using error estimates to control mesh refinement. One technique is a method of lines approach that uses a Galerkin method to discretize the PDEs in space and implicit multi-step integration in time. Spatial elements are added and deleted in regions of high and low error and are all advanced with the same sequence of varying time steps. The second technique is a local refinement method that uses Galerkin approximations in both space and time. Fine grids of space-time elements are added to coarser grids and the problem is recursively solved in regions of high error. (Author)

We introduce and analyze a frequency-domain method for parabolic partial differential equations. The method is naturally parallelizable. After taking the Fourier transformation of given equations in the space-time domain into the space-frequency domain, we propose to solve an indefinite, complex elliptic problem for each frequency. Fourier inversion will then recover the solution in the space-time domain. Existence and uniqueness as well as error estimates are given. Fourier invertibility is also examined. Numerical experiments are presented.

We attempt to obtain a two‐level implicit finite difference scheme using nine spatial grid points of O(k2 + kh2 + h4) for solving the 2D nonlinear parabolic partial differential equation v1uxx + v2uyy = f(x, y, t, u, ux, uy, u1) where v1 and v2 are positive constants, with Dirichlet boundary conditions. The method, when applied to a linear diffusion‐convection problem, is shown to be unconditionally stable. Computational efficiency and the results of numerical experiments are discussed.

Deep neural networks based temporal-difference methods for high-dimensional parabolic partial differential equations

In Part I of this paper (SIAM J. Numer. Anal., 15 (1978), pp. 1212–1224), novel algorithms were introduced for solving parabolic differential equations in which high-frequency components occurred in the solution and for which $A_0 $-stable methods, exemplified by the classical Crank-Nicolson method, were less than satisfactory. The algorithms presented were based on a simple extrapolation of the Backward Euler method which produced $L_0 $ -stability. In all cases the algorithms presented were second order accurate in time. In the present paper some of the previous algorithms are generalized to higher orders of accuracy. In particular, a more detailed consideration of the second order algorithm is given which leads naturally to methods of orders three and four. The novel algorithms are tested on a heat equation with constant coefficients in which a discontinuity between the initial values and boundary values exists.

For Example: Modelling Processes in Porous Media with Differential Equations.- For the Beginning: The Finite Difference Method for the Poisson Equation.- The Finite Element Method for the Poisson Equation.- The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order.- Grid Generation and a posteriori Error Estimation.- Iterative Methods for Systems of Linear Equations.- The Finite Volume Method.- Discretization Methods for Parabolic Initial Boundary Value Problems.- Iterative Methods for Nonlinear Equations.- Discretization Methods for Convection-Dominated Problems.- Appendices.

We consider the Heston model as an example of a parameterized parabolic partial differential equation. A space-time variational formulation is derived that allows for parameters in the coefficients (for calibration) as well as choosing the initial condition (for option pricing) as a parameter function. A corresponding discretization in space and time amd initial condition is introduced and shown to be stable. Finally, a Reduced Basis Method (RBM) is introduced that is able to use parameter functions also for the initial condition. Corresponding numerical results are shown.

Convergence analysis of constraint energy minimizing generalized multiscale finite element method for a linear stochastic parabolic partial differential equation driven by additive noises

Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions

We analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to the expected value of the mild solution. The complexity of the multilevel estimator is shown to scale log-linearly with respect to the corresponding work to generate a single path of the solution on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. © 2012 Springer Science + Business Media B.V.

Chaos synchronization is a topic of great interest, due to its observation in a huge variety of phenomena of different nature. We study synchronization of two chaotic oscillators in a Master- Slave configuration. The two dynamic systems are coupled via a directed feedback that randomly switches among a finite set of given constant function at a prescribed time rate. And we use Lyapunov stability theory. This paper discussed the using of lag synchronization approach, and provided the equilibrium solutions of a new class of higher order parabolic partial differential equations to be applicable for Lorenz chaotic system in order to minimize the dynamical error of large Lorenz chaotic system