Solution Of Parabolic Partial Differential Equations Research Articles

Parabolic PDE-constrained optimal control under uncertainty with entropic risk measure using quasi-Monte Carlo integration

We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of a parabolic PDE with a random thermal diffusion coefficient, steered by a control function. To account for the presence of uncertainty in the optimal control problem, the objective function is composed with a risk measure. We focus on two risk measures, both involving high-dimensional integrals over the stochastic variables: the expected value and the (nonlinear) entropic risk measure. The high-dimensional integrals are computed numerically using specially designed QMC methods and, under moderate assumptions on the input random field, the error rate is shown to be essentially linear, independently of the stochastic dimension of the problem—and thereby superior to ordinary Monte Carlo methods. Numerical results demonstrate the effectiveness of our method.

A data-driven model reduction method for parabolic inverse source problems and its convergence analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems with uncertain data efficiently. Our method consists of offline and online stages. In the offline stage, we explore the low-dimensional structures in the solution space of parabolic partial differential equations (PDEs) in the forward problems with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problems extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in parabolic inverse source problems, which is referred to as the POD method. Moreover, we design an a posteriori algorithm to find the optimal regularization parameter in the optimization problem using the proposed POD method without knowing the noise level. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD method in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD method for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Numerical results show that the POD method provides considerable computational savings over the finite element method while maintaining the same accuracy.

A Data-Driven Model Reduction Method for Parabolic Inverse Source Problems and its Convergence Analysis

In this paper, we propose a data-driven model reduction method to solve parabolic inverse source problems efficiently. Our method consists of offline and online stages. In the off-line stage, we explore the low-dimensional structures in the solution space of the parabolic partial differential equations (PDEs) in the forward problem with a given class of source functions and construct a small number of proper orthogonal decomposition (POD) basis functions to achieve significant dimension reduction. Equipped with the POD basis functions, we can solve the forward problem extremely fast in the online stage. Thus, we develop a fast algorithm to solve the optimization problem in the parabolic inverse source problems, which is referred to as the POD algorithm in this paper. Under a weak regularity assumption on the solution of the parabolic PDEs, we prove the convergence of the POD algorithm in solving the forward parabolic PDEs. In addition, we obtain the error estimate of the POD algorithm for parabolic inverse source problems. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method. Our numerical results show that the POD algorithm provides considerable computational savings over the finite element method.

An algorithm for probabilistic solution of parabolic PDEs

The aim of this work is to approximate the trajectory solution of parabolic partial differential equations (PDEs) by the probabilistic method. This method is based on the representation of Feynman-Kac and Monte Carlo methods. As an alternative to classical Monte Carlo, here we employ quasi–Monte Carlo methods and propose some solutions to the problem of using this alternative through a more efficient algorithm than the classics.

Solution of Parabolic Partial Differential Equations Via Non-Polynomial Cubic Spline Technique

The discovery of parabolic partial differential equation (PDE) has made a profound impact on the scientific, engineering and technological community. A vast amount of research has been conducted to find the solution of parabolic PDEs. In this research, we introduced a novel technique to find the numerical solution of the fourth order PDEs. The novel technique is based upon the polynomial cubic spline method (PCSM) used along with Adomian decomposition method (ADM). The constraint of the alternative variables was decomposed by ADM to achieve successive approximation. Additionally, a numerical test problem of parabolic PDEs was solved by the proposed technique to check its viability.
 Copyright (c) The Author

Optimal error estimate of the finite element approximation of second order semilinear non-autonomous parabolic PDEs

In this work, numerical approximation of the second order non-autonomous semilinear parabolic partial differential equations (PDEs) is investigated using the classical finite element method. To the best of our knowledge, only the linear case is investigated in the literature. Using an approach based on evolution operator depending on two parameters, we obtain the error estimate of the semi-discrete scheme based on finite element method toward the mild solution of semilinear non-autonomous PDEs under polynomial growth and one-sided Lipschitz conditions of the nonlinear term. Our convergence rate is obtained with general non-smooth initial data, and is similar to that of the autonomous case. Such convergence result is very important in numerical analysis. For instance, it is one step forward for numerical approximation of non-autonomous stochastic partial differential equations with the finite element method.

A VOLATILITY-OF-VOLATILITY EXPANSION OF THE OPTION PRICES IN THE SABR STOCHASTIC VOLATILITY MODEL

We propose a new type of asymptotic expansion for the transition probability density function (or heat kernel) of certain parabolic partial differential equations (PDEs) that appear in option pricing. As other, related methods developed by Costanzino, Hagan, Gatheral, Lesniewski, Pascucci, and their collaborators, among others, our method is based on the computation of the truncated asymptotic expansion of the heat kernel with respect to a “small” parameter. What sets our method apart is that our small parameter is possibly different from the time to expiry and that the resulting commutator calculations go beyond the nilpotent Lie algebra case. In favorable situations, the terms of this asymptotic expansion can quickly be computed explicitly leading to a “closed-form” approximation of the solution, and hence of the option price. Our approximations tend to have much fewer terms than the ones obtained from short time asymptotics, and are thus easier to generalize. Another advantage is that the first term of our expansion corresponds to the classical Black-Scholes model. Our method also provides equally fast approximations of the derivatives of the solution, which is usually a challenge. A full theoretical justification of our method seems very difficult at this time, but we do provide some justification based on the results of (Siyan, Mazzucato, and Nistor, NWEJ 2018). We therefore mostly content ourselves to demonstrate numerically the efficiency of our method by applying it to the solution of the mean-reverting SABR stochastic volatility model PDE, commonly referred to as the [Formula: see text]SABR PDE, by taking the volatility of the volatility parameter [Formula: see text] (vol-of-vol) as a small parameter. For this PDE, we provide extensive numerical tests to gauge the performance of our method. In particular, we compare our approximation to the one obtained using Hagan’s formula and to the one obtained using a new, adaptive finite difference method. We provide an explicit asymptotic expansion for the implied volatility (generalizing Hagan’s formula), which is what is typically needed in concrete applications. We also calibrate our model to observed market option price data. The resulting values for the parameters [Formula: see text], [Formula: see text], and [Formula: see text] are realistic, which provides more evidence for the conjecture that the volatility is mean-reverting.

A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

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.

Numerical simulation and applications of the convection–diffusion–reaction equation with the radial basis function in a finite-difference mode

This paper develops two local mesh-free methods for designing stencil weights and spatial discretization, respectively, for parabolic partial differential equations (PDEs) of convection–diffusion–reaction type. These are known as the radial-basis-function generated finite-difference method and the Hermite finite-difference method. The convergence and stability of these schemes are investigated numerically using some examples in two and three dimensions with regularly and irregularly shaped domains. Then we consider the numerical pricing of European and American options under the Heston stochastic volatility model. The European option leads to the solution of a two-dimensional parabolic PDE, and the price of the American option is given by a linear complementarity problem with a two-dimensional parabolic PDE of convection–diffusion–reaction type. Then we use the operator splitting method to perform time-stepping after space discretization. The resulting linear systems of equations are well conditioned and sparse, and by numerical experiments we show that our numerical technique is fast and stable with respect to the change in the shape parameter of the radial basis function. Finally, numerical results are provided to illustrate the quality of approximation and to show how well our approach converges with the results presented in the literature.

Input-to-State Stability for a Class of One-Dimensional Nonlinear Parabolic PDEs with Nonlinear Boundary Conditions

The aim of this paper is to introduce a weak maximum principle--based approach for studying the input-to-state stability (ISS) with respect to boundary disturbances and states in certain classes for a class of one-dimensional nonlinear parabolic partial differential equations (PDEs) with nonlinear boundary conditions. To tackle the difficulties in ISS analysis due to, in particular, the nonlinear terms on the boundary, we establish first several maximum estimates for the solutions of linear parabolic PDEs with different nonlinear boundary conditions by means of the weak maximum principle. Then, using the technique of splitting and combining maximum estimates for the solutions of linear parabolic PDEs and the Lyapunov method, we establish ISS estimates for nonlinear parabolic PDEs with nonlinear boundary conditions. Two examples of specific parabolic equations with nonlinear boundary conditions are provided to illustrate the developed approach.

A class of numerical algorithms based on cubic trigonometric B-spline functions for numerical simulation of nonlinear parabolic problems

In this work, the authors developed two new B-spline collocation algorithms based on cubic trigonometric B-spline functions to find approximate solutions of nonlinear parabolic partial differential equations (PDEs) with Dirichlet and Neumann boundary conditions. In the first algorithm, cubic trigonometric B-spline functions are directly used for approximate solutions of parabolic PDEs with Neumann boundary conditions. But, the Dirichlet boundary conditions cannot be handled directly by cubic trigonometric B-spline functions. Then, a modification is made in cubic trigonometric B-spline functions to handle the Dirichlet boundary conditions and the second algorithm is developed with the help of modified cubic trigonometric B-spline functions. The proposed algorithms reduced the parabolic problem into a system of first-order nonlinear ordinary differential equations (ODEs) in time variable. Then, strong stability-preserving-Runge–Kutta3 (SSP-RK3) scheme is used to solve the obtained system. Some well-known parabolic problems are solved to check the accuracy and efficiency of the proposed algorithms. The algorithms can be extended to solve multidimensional problems arising as model equations in physical, chemical and biophysical phenomenon.

On Multilevel Picard Numerical Approximations for High-Dimensional Nonlinear Parabolic Partial Differential Equations and High-Dimensional Nonlinear Backward Stochastic Differential Equations

Parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) are key ingredients in a number of models in physics and financial engineering. In particular, parabolic PDEs and BSDEs are fundamental tools in the state-of-the-art pricing and hedging of financial derivatives. The PDEs and BSDEs appearing in such applications are often high-dimensional and nonlinear. Since explicit solutions of such PDEs and BSDEs are typically not available, it is a very active topic of research to solve such PDEs and BSDEs approximately. In the recent article [E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. Linear scaling algorithms for solving high-dimensional nonlinear parabolic differential equations. arXiv:1607.03295 (2017)] we proposed a family of approximation methods based on Picard approximations and multilevel Monte Carlo methods and showed under suitable regularity assumptions on the exact solution for semilinear heat equations that the computational complexity is bounded by $O( d \, \epsilon^{-(4+\delta)})$ for any $\delta\in(0,\infty)$, where $d$ is the dimensionality of the problem and $\epsilon\in(0,\infty)$ is the prescribed accuracy. In this paper, we test the applicability of this algorithm on a variety of $100$-dimensional nonlinear PDEs that arise in physics and finance by means of numerical simulations presenting approximation accuracy against runtime. The simulation results for these 100-dimensional example PDEs are very satisfactory in terms of accuracy and speed. In addition, we also provide a review of other approximation methods for nonlinear PDEs and BSDEs from the literature.

An Ensemble Algorithm for Numerical Solutions to Deterministic and Random Parabolic PDEs

In this paper, we develop an ensemble-based time-stepping algorithm to efficiently find numerical solutions to a group of linear, second-order parabolic partial differential equations (PDEs). Particularly, the PDE models in the group could be subject to different diffusion coefficients, initial conditions, boundary conditions, and body forces. The proposed algorithm leads to a single discrete system for the group with multiple right-hand-side vectors by introducing an ensemble average of the diffusion coefficient functions and using a new semi-implicit time integration method. The system could be solved more efficiently than multiple linear systems with a single right-hand-side vector. We first apply the algorithm to deterministic parabolic PDEs and derive a rigorous error estimate that shows the scheme is first-order accurate in time and is optimally accurate in space. We then extend it to find stochastic solutions of parabolic PDEs with random coefficients and put forth an ensemble-based Monte Carlo met...

Numerical solution of parabolic partial differential equations using adaptive gird Haar wavelet collocation method

In this paper, we applied the adaptive grid Haar wavelet collocation method (AGHWCM) for the numerical solution of parabolic partial differential equations (PDEs). The approach of AGHWCM for the numerical solution of parabolic PDEs is mentioned, the obtained numerical results, error analysis are presented in figures and tables. This shows that, the AGHWCM gives better accuracy than the HWCM and FDM. Some of the test problems are taken for demonstrating the validity and applicability of the AGHWCM.

Error Analysis of a Space-Time Finite Element Method for Solving PDEs on Evolving Surfaces

In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations (PDEs) posed on evolving hypersurfaces in Rd, d =2 , 3. The method employs discontinuous piecewise linear in time-continuous piecewise linear in space finite elements and is based on a space-time weak formulation of a surface PDE problem. Trial and test surface finite element spaces consist of traces of standard volumetric elements on a space-time manifold resulting from the evolution of a surface. We prove first order convergence in space and time of the method in an energy norm and second order convergence in a weaker norm. Furthermore, we derive regularity results for solutions of parabolic PDEs on an evolving surface, which we need in a duality argument used in the proof of the second order convergence estimate.

On the Accuracy of a Finite-Difference Method for Parabolic Partial Differential Equations with Discontinuous Boundary Conditions

Although the numerical solution of parabolic partial differential equations (PDEs) is widely documented, the effect of discontinuous boundary conditions on numerical accuracy is not. This article employs the Keller box finite-difference method to study the effect of such discontinuities when solving the linear one-demensional transient heat equation. We demonstrate that this formally second-order-accurate scheme can lose accuracy, but that an analytical understanding of the behavior of the solution helps in providing an accuracy-restoring formulation. Benchmark computations are presented that will provide guidance in the numerical solution of nonlinear parabolic PDEs for which there are no closed-form analytical solutions.

Order‐reduction of parabolic PDEs with time‐varying domain using empirical eigenfunctions

A novel methodology for the order‐reduction of parabolic partial differential equation (PDE) systems with time‐varying domain is explored. In this method, a mapping functional is obtained, which relates the time‐evolution of the solution of a parabolic PDE with time‐varying domain to a fixed reference domain, while preserving space invariant properties of the initial solution ensemble. Subsequently, the Karhunen–Loève decomposition is applied to the solution ensemble on fixed spatial domain resulting in a set of optimal eigenfunctions. Further, the low dimensional set of empirical eigenfunctions is mapped on the original time‐varying domain by an appropriate mapping, resulting in the basis for the construction of the reduced‐order model of the parabolic PDE system with time‐varying domain. This methodology is used in three representative cases, one‐ and two‐dimensional (1‐D and 2‐D) models of nonlinear reaction‐diffusion systems with analytically defined domain evolutions, and the 2‐D model of the Czochralski crystal growth process with nontrivial geometry. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4142–4150, 2013

Error-controlled global sensitivity analysis of ordinary differential equations

We propose a novel strategy for global sensitivity analysis of ordinary differential equations. It is based on an error-controlled solution of the partial differential equation (PDE) that describes the evolution of the probability density function associated with the input uncertainty/variability. The density yields a more accurate estimate of the output uncertainty/variability, where not only some observables (such as mean and variance) but also structural properties (e.g., skewness, heavy tails, bi-modality) can be resolved up to a selected accuracy. For the adaptive solution of the PDE Cauchy problem we use the Rothe method with multiplicative error correction, which was originally developed for the solution of parabolic PDEs. We show that, unlike in parabolic problems, conservation properties necessitate a coupling of temporal and spatial accuracy to avoid accumulation of spatial approximation errors over time. We provide convergence conditions for the numerical scheme and suggest an implementation using approximate approximations for spatial discretization to efficiently resolve the coupling of temporal and spatial accuracy. The performance of the method is studied by means of low-dimensional case studies. The favorable properties of the spatial discretization technique suggest that this may be the starting point for an error-controlled sensitivity analysis in higher dimensions.

Computer simulation of materials using parallel architectures

Algorithms are designed to implement molecular dynamics (MD) and quantum molecular dynamics (QMD) simulations on emerging concurrent architectures. For systems with finite-range interactions, a domain-decomposition algorithm is used to implement the multiple-time-step (MTS) approach to MD simulations on distributed-memory multiple instruction multiple data (MIMD) machines. Parallel algorithms are also designed for MD simulations of bulk Coulombic systems. The performances of these algorithms are tested on the Intel iPSC/860 system. The computational complexities of these algorithms are O( N) and parallel efficiencies close to 0.9. The core computational kernel of the QMD approach consists of solutions of parabolic partial differential equations (PDE) such as the time-dependent Schrödinger equation or time-dependent Kohn-Sham equation. This problem is coupled with another computationally intensive problem, i.e., solution of elliptic PDEs (the Poisson equation) for the long-range electron-electron interaction. We have designed parallel algorithms for both problems on SIMD (single instruction multiple data) machines. In the past two years, we have used the parallel computer architectures in our Concurrent Computing Laboratory for Materials Simulations (CCLMS) to carry out MD and QMD simulations on network glasses, ceramic composites, nanophase materials, solid C 60 and graphitic tubules, and quantum transport in nanoscale devices. Structural transformation, intermediate-range order and dynamic behavior of SiO 2 glass at high pressures are investigated with molecular dynamics. At high densities, the height of the first sharp diffraction peak is considerably diminished, its position changes from 1.6 to 2.2 Å −1, and a new peak appears at 2.85 Å -1. At twice the normal density, SiO bond length increases, SiO coordination changes from 4 to 6 and OSiO bond-angle changes from 109° to 90°. This is a tetrahedral to octahedral transformation, which was reported recently by Meade, Hem1ey, and Mao. Molecular dynamics simulations of porous silica, in the density range 2.2-0.1 g/cm 3, are carried out on a 41 472 particle system using a MIMD computer. The internal surface area, pore surface-to-volume ratio, pore-size distribution, fractal dimension, correlation length and mean-particle size are determined as a function of the density. Structural transition between a condensed amorphous phase and a low-density porous phase is characterized by these quantities. Various dissimilar porous structures with different fractal dimensions are obtained by controlling the preparation schedule and temperature.

A cohomological Conley index for maps on metric spaces

We define a cohomological Conley index of an isolated invariant set of a time-discrete semidynamical system on a metric space. We prove the homotopy invariance and the additivity properties of the index. For applications to time-periodic parabolic partial differential equations (PDEs) the reader is referred to the authors' paper: Nontrivial full bounded solutions of time-periodic semilinear parabolic PDEs ( Proc. Roy. Soc. Edinburgh, to appear).