Solving second order non-linear parabolic PDEs using generalized finite difference method (GFDM)

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present the implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance.

Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs it has also been proved mathematically that deep neural networks overcome the curse of dimensionality in the numerical approximation of solutions of such linear PDEs. The key contribution of this article is to rigorously prove this for the first time for a class of nonlinear PDEs. More precisely, we prove in the case of semilinear heat equations with gradient-independent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced full history recursive multilevel Picard approximations for semilinear PDEs.

In this work, we explore a methodology to compute the empirical eigenfunctions for the order-reduction of nonlinear parabolic partial differential equations (PDEs) system with time-varying domain. The idea behind this method is to obtain the mapping functional, which relates the time-evolution scalar physical property solution ensemble of the nonlinear parabolic PDE with the time-varying domain to a fixed reference domain, while preserving space invariant properties of the raw solution ensemble. Subsequently, the Karhunen-Lo'eve decomposition is applied to the solution ensemble with fixed spatial domain resulting in a set of optimal eigenfunctions that capture the most energy of data. Further, the low dimensional set of empirical eigenfunctions is mapped (“pushed-back”) on the time-varying domain by an appropriate mapping resulting in the basis for the construction of the reduced-order model of the parabolic PDEs with time-varying domain. Finally, this methodology is applied in the representative cases of calculation of empirical eigenfunctions in the case of one and two dimensional model of nonlinear reaction-diffusion parabolic PDE systems with analytically defined domain evolutions. In particular, the design of both mappings which relate the raw data and function spaces transformations from the time-varying to time-invariant domain are designed to preserve dynamic features of the scalar physical property and we provide comparisons among reduced and high order fidelity models.

This study presents an observer‐based output feedback compensator design for a linear parabolic partial differential equation (PDE), where a finite number of actuators and sensors are active over partial areas and at specified points in the spatial domain, respectively. In the proposed design method, a Luenberger‐type PDE observer is first constructed by using the non‐collocated pointwise observation to exponentially track the PDE state. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing a Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for definite integrals, sufficient conditions on the exponential stability of the resulting closed‐loop coupled PDEs are presented in terms of standard linear matrix inequalities. Furthermore, both open‐loop and closed‐loop well‐posedness analysis results are also established by the ‐semigroup method. Numerical simulation results are presented to show the effectiveness of the proposed design method.

This paper presents an observer-based dynamic feedback control design for a linear parabolic partial differential equation (PDE) system, where a finite number of actuators and sensors are active over part thereof and at specified points in the spatial domain, respectively. In the proposed design method, a Luenberger-type PDE observer is first constructed by using the non-collocated pointwise measurements to exponentially tract the state of the PDE system. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed. By employing Lyapunov direct method, integration by parts, Wirtinger's inequality and first mean value theorem for integrals, a sufficient condition on exponential stability of the resulting closed-loop system is presented in term of standard linear matrix inequalities (LMIs). Numerical simulation results are presented to show the effectiveness of the proposed design method.

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the "curse of dimensionality" related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). We assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001-29 October 2020.

This work proposes a hybrid control methodology that integrates feedback and switching for fault-tolerant constrained control of linear parabolic partial differential equations (PDEs) for which the spectrum of the spatial differential operator can be partitioned into a finite “slow” set and an infinite stable “fast” complement. Modal decomposition techniques are initially used to derive a finite-dimensional system (set of ordinary differential equations (ODEs) in time) that captures the dominant dynamics of the PDE. The ODE system is then used as the basis for the integrated synthesis, via Lyapunov techniques, of a stabilizing nonlinear feedback controller together with a switching law that orchestrates the switching between the admissible control actuator con- figurations, based on their constrained regions of stability, in a way that respects actuator constraints and maintains closed-loop stability in the event of actuator failure. Precise conditions that guarantee stability of the constrained closed-loop PDE system under actuator switching are provided. The proposed method is applied to the problem of stabilizing an unstable, spatially unifrom steady-state of a linear parabolic PDE under constraints and actuator failures.

In this paper, we extend the reduced-basis approximations developed earlier for linear elliptic and parabolic partial differential equations with affine parameter dependence to problems in- volving (a) nonaffine dependence on the parameter, and (b) nonlinear dependence on the field variable. The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational decomposition. We first review the coefficient function approximation procedure: the essential ingredients are (i) a good collateral reduced-basis approximation space, and (ii) a stable and inexpensive interpolation procedure. We then apply this approach to linear nonaffine and nonlinear elliptic and parabolic equations; in each instance, we discuss the reduced-basis approximation and the associated offline-online computational procedures. Numeri- cal results are presented to assess our approach.

The H ∞ control problem is considered for linear parabolic partial differential equation (PDE) systems with completely unknown system dynamics. We propose a model-free policy iteration (PI) method for learning the H ∞ control policy by using measured system data without system model information. First, a finite-dimensional system of ordinary differential equation (ODE) is derived, which accurately describes the dominant dynamics of the parabolic PDE system. Based on the finite-dimensional ODE model, the H ∞ control problem is reformulated, which is theoretically equivalent to solving an algebraic Riccati equation (ARE). To solve the ARE without system model information, we propose a least-square based model-free PI approach by using real system data. Finally, the simulation results demonstrate the effectiveness of the developed model-free PI method.

In the optimal control theory, the Hamiltonian formalism is a famous one to find an optimal solution. However, when the performance index is complicated or for a degenerate case with a non-convexity of the Hamiltonian function with respect to the control force the Hamiltonian method does not work to find the solution. In this paper we will address this important issue via a quite different approach, which uses the optimal control problem of nonlinear Duffing oscillator as a demonstrative example. The optimally controlled vibration problem of nonlinear oscillator is recast into a nonlinear inverse problem by identifying the unknown heat source in a nonlinear parabolic partial differential equation (PDE). Then through a semi-discretization of the resultant PDE, the inverse problem is further reformulated to be a system of n-dimensional ODEs with n unknown pointwise sources, which allows a Lie-group adaptive method (LGAM) to recover the point-wise sources. The present method has three-fold advantages: it can easily minimize a complicated performance index to find an optimal control force of the nonlinear vibration system, it is effective for highly nonlinear optimal control problem, and it does not resort on the classical Hamiltonian formulation, which provides only a necessary condition, but not a sufficient condition, for the optimality of the control law. Numerical examples show that the LGAM may find a better performance than the classical one.

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

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

A fault detection and isolation scheme is addressed for a class of linear distributed parameter systems (DPS) described by partial differential equations (PDE). In contrast to using ordinary differential equations (ODE) for describing DPS, in this paper, a filter based observer based on the linear PDE representation is proposed with an output measurement. A fault is declared active when the magnitude of the detection residual exceeds a predefined threshold. Upon detection, an actuator and a sensor fault isolation estimators are activated to identify the fault type when their isolation residual is below a predefined threshold and the other is above the threshold. When both actuator and sensor fault isolation estimator residuals are above their isolation thresholds, a state fault is considered to have occurred. Upon isolation, the magnitude of the fault parameter is identified. Finally, the performance of the fault detection and isolation scheme is demonstrated on a heat reactor system which is represented by linear parabolic PDEs.

9R3. Mathematics of Wave Propagation. - JL Davis (consultant). Princeton UP, Princeton. 2000. 395 pp. ISBN 0-691-02643-2. $49.50.Reviewed by LN Sankar (Sch of Aerospace Eng, Georgia Inst of Tech, Atlanta GA 30332-0150).There are a number of textbooks and references that have been published on wave propagation in fluids. One may also find numerous books that individually address the propagation of waves in solids, surface waves in water, and tidal waves. There are very few books that treat all of these related phenomena in a clear, unified manner. This book does this challenging job admirably well. The first chapter deals with the physics of propagating waves. Many classical problems such as the transverse oscillations of a string, acoustic waves in a duct, and compression waves in a bar are discussed. Topics such as the Doppler effect, dispersion, and group velocity are also covered. The mathematical aspects of these problems are not emphasized in this chapter. The second chapter deals with the theory behind hyperbolic partial differential equations. The method of characteristics is discussed and illustrated with increasingly complex model problems: 1D advection equation, 2D wave equation, and a system of first order linear and nonlinear partial differential equations, etc. Chapter 3 deals with how the partial differential equations developed in Chapter 2 may be solved. Classical mathematical techniques such as the separation of variables and Laplace transforms, as well as numerical methods that solve ODEs or algebraic equations along characteristic lines are covered. Numerous applications ranging from the vibration of a rectangular membrane to the propagation of current in an LC circuit are discussed. Having established the common mathematical foundation and the solution techniques for these problems, this book moves on to individual topics: wave propagation in viscous and inviscid fluids, elastic, viscoelastic and thermoelastic solids, and water waves. Each of these chapters may be read individually without reference to others. All the governing equations are derived from first principles, making it easier for the reader to understand and appreciate the physics behind these problems. This book ends with variational calculus based approaches (eg, Hamilton’s variational principle, Hamilton-Jacobi theory) to deriving the governing equations. Asymptotic approaches that rely on expansion of the solution for large wave numbers or frequency are also briefly discussed. There are numerous worked-out examples throughout the book. There are also exercises at the end of most of the chapters. These features make this work a suitable textbook for a senior-level or a graduate-level course on the theory of waves. In summary, Mathematics of Wave Propagation is an excellent book that covers seemingly diverse wave phenomena in a unified, coherent manner. Students and practicing engineers and physicists will find this book a useful addition to their collections.

An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burgers’ type in two dimensions. These nonlinear PDEs are essential because they describe various mechanisms in engineering and physics. The nonlinear convective and advective processes are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization, so that one discrete equation leads to the accuracy of orders three or four, depending upon the choice of the grid network. The scheme is used for solving celebrated nonlinear PDEs, such as the nondegenerate convection–diffusion equation, the generalized Burgers–Huxley equation, the Buckley–Leverett equation, and the Burgers–Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.