Solution of Stochastic Nonlinear PDEs Using Wiener-Hermite Expansion of High Orders

Similar documents
Benha University Faculty of Science Department of Mathematics. (Curriculum Vitae)

The Chebyshev Collection Method for Solving Fractional Order Klein-Gordon Equation

Introduction to Computational Stochastic Differential Equations

Performance Evaluation of Generalized Polynomial Chaos

CRANK-NICOLSON FINITE DIFFERENCE METHOD FOR SOLVING TIME-FRACTIONAL DIFFUSION EQUATION

ON THE NUMERICAL SOLUTION FOR THE FRACTIONAL WAVE EQUATION USING LEGENDRE PSEUDOSPECTRAL METHOD

White Noise Functional Solutions for Wick-type Stochastic Fractional KdV-Burgers-Kuramoto Equations

ON THE SOLUTIONS OF NON-LINEAR TIME-FRACTIONAL GAS DYNAMIC EQUATIONS: AN ANALYTICAL APPROACH

EFFICIENT SHAPE OPTIMIZATION USING POLYNOMIAL CHAOS EXPANSION AND LOCAL SENSITIVITIES

Application of Taylor-Padé technique for obtaining approximate solution for system of linear Fredholm integro-differential equations

Final Report: DE-FG02-95ER25239 Spectral Representations of Uncertainty: Algorithms and Applications

Research Article He s Variational Iteration Method for Solving Fractional Riccati Differential Equation

An Efficient Numerical Method for Solving. the Fractional Diffusion Equation

SOLUTION OF FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD

Stochastic Integration and Stochastic Differential Equations: a gentle introduction

Contents. 1 Preliminaries 3. Martingales

Solution of Linear System of Partial Differential Equations by Legendre Multiwavelet Andchebyshev Multiwavelet

Safety Envelope for Load Tolerance and Its Application to Fatigue Reliability Design

Sequential Monte Carlo Samplers for Applications in High Dimensions

PC EXPANSION FOR GLOBAL SENSITIVITY ANALYSIS OF NON-SMOOTH FUNCTIONALS OF UNCERTAIN STOCHASTIC DIFFERENTIAL EQUATIONS SOLUTIONS

Estimating functional uncertainty using polynomial chaos and adjoint equations

Regularization by noise in infinite dimensions

Rough paths methods 1: Introduction

Beyond Wiener Askey Expansions: Handling Arbitrary PDFs

Stochastic Collocation Methods for Polynomial Chaos: Analysis and Applications

Harnack Inequalities and Applications for Stochastic Equations

Introduction to multiscale modeling and simulation. Explicit methods for ODEs : forward Euler. y n+1 = y n + tf(y n ) dy dt = f(y), y(0) = y 0

Malliavin Calculus in Finance

Handbook of Stochastic Methods

M E M O R A N D U M. Faculty Senate approved November 1, 2018

PARTIAL DIFFERENTIAL EQUATIONS

Stochastic differential equation models in biology Susanne Ditlevsen

Polynomial chaos expansions for sensitivity analysis

Table of Contents [ntc]

The Modified Adomian Decomposition Method for. Solving Nonlinear Coupled Burger s Equations

On Solution of Nonlinear Cubic Non-Homogeneous Schrodinger Equation with Limited Time Interval

Interest Rate Models:

Numerical Optimization Algorithms

Adomian Decomposition Method with Laguerre Polynomials for Solving Ordinary Differential Equation

A NEW SOLUTION OF SIR MODEL BY USING THE DIFFERENTIAL FRACTIONAL TRANSFORMATION METHOD

Introduction to Uncertainty Quantification in Computational Science Handout #3

An Empirical Chaos Expansion Method for Uncertainty Quantification

Hypocoercivity and Sensitivity Analysis in Kinetic Equations and Uncertainty Quantification October 2 nd 5 th

Solving nonlinear fractional differential equation using a multi-step Laplace Adomian decomposition method

Discretization of SDEs: Euler Methods and Beyond

Fast Numerical Methods for Stochastic Computations

Approximation of BSDEs using least-squares regression and Malliavin weights

Regularity of the density for the stochastic heat equation

Uncertainty Quantification for multiscale kinetic equations with random inputs. Shi Jin. University of Wisconsin-Madison, USA

Adaptive timestepping for SDEs with non-globally Lipschitz drift

δ Substituting into the differential equation gives: x i+1 x i δ f(t i,x i ) (3)

HOMOTOPY PERTURBATION METHOD TO FRACTIONAL BIOLOGICAL POPULATION EQUATION. 1. Introduction

Lecture #8: Quantum Mechanical Harmonic Oscillator

Lecture 6: Multiple Model Filtering, Particle Filtering and Other Approximations

Numerical Integration of SDEs: A Short Tutorial

An Analytic Study of the (2 + 1)-Dimensional Potential Kadomtsev-Petviashvili Equation

Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals

EFFICIENT SPECTRAL COLLOCATION METHOD FOR SOLVING MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS BASED ON THE GENERALIZED LAGUERRE POLYNOMIALS

Stochastic Integration (Simple Version)

Series Solution of Weakly-Singular Kernel Volterra Integro-Differential Equations by the Combined Laplace-Adomian Method

MATHEMATICAL METHODS INTERPOLATION

UNIVERSITY OF CALIFORNIA, SAN DIEGO. An Empirical Chaos Expansion Method for Uncertainty Quantification

Handbook of Stochastic Methods

Lecture 1: Pragmatic Introduction to Stochastic Differential Equations

Workshop on Heterogeneous Computing, 16-20, July No Monte Carlo is safe Monte Carlo - more so parallel Monte Carlo

New computational method for solving fractional Riccati equation

ON MODELING THE GROUNDWATER FLOW WITHIN A CONFINED AQUIFER

Approximation of Top Lyapunov Exponent of Stochastic Delayed Turning Model Using Fokker-Planck Approach

EQUATION LANGEVIN. Physics, Chemistry and Electrical Engineering. World Scientific. With Applications to Stochastic Problems in. William T.

Adomian Decomposition Method for Solving the Kuramoto Sivashinsky Equation

. Frobenius-Perron Operator ACC Workshop on Uncertainty Analysis & Estimation. Raktim Bhattacharya

Gaussian Process Approximations of Stochastic Differential Equations

1. Introductory Examples

Derivation of SPDEs for Correlated Random Walk Transport Models in One and Two Dimensions

MODIFIED LAGUERRE WAVELET BASED GALERKIN METHOD FOR FRACTIONAL AND FRACTIONAL-ORDER DELAY DIFFERENTIAL EQUATIONS

A PRACTICAL SPLITTING METHOD FOR STIFF SDES WITH APPLICATIONS TO PROBLEMS WITH SMALL NOISE

ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process

Uncertainty Quantification and hypocoercivity based sensitivity analysis for multiscale kinetic equations with random inputs.

A path integral approach to the Langevin equation

THE DIFFERENTIAL TRANSFORMATION METHOD AND PADE APPROXIMANT FOR A FORM OF BLASIUS EQUATION. Haldun Alpaslan Peker, Onur Karaoğlu and Galip Oturanç

Solution of Conformable Fractional Ordinary Differential Equations via Differential Transform Method

Multi-Factor Finite Differences

Biological population model and its solution by reduced differential transform method

Connections between score matching, contrastive divergence, and pseudolikelihood for continuous-valued variables. Revised submission to IEEE TNN

ON FRACTIONAL ORDER CANCER MODEL

Application of Homotopy Perturbation Method in Nonlinear Heat Diffusion-Convection-Reaction

Random Vibrations & Failure Analysis Sayan Gupta Indian Institute of Technology Madras

Stochastic Spectral Approaches to Bayesian Inference

Stochastic Modeling of Flow-Structure Interactions Using Generalized Polynomial Chaos

Gaussian Process Approximations of Stochastic Differential Equations

Algorithms for Uncertainty Quantification

BOOK REVIEW. Review by Denis Bell. University of North Florida

On rough PDEs. Samy Tindel. University of Nancy. Rough Paths Analysis and Related Topics - Nagoya 2012

Adaptation of Taylor s Formula for Solving System of Differential Equations

A stochastic particle system for the Burgers equation.

Control, Stabilization and Numerics for Partial Differential Equations

Pathwise volatility in a long-memory pricing model: estimation and asymptotic behavior

Uncertainty Quantification in Computational Science

The variational homotopy perturbation method for solving the K(2,2)equations

State Space Representation of Gaussian Processes

Transcription:

Solution of Stochastic Nonlinear PDEs Using Wiener-Hermite Expansion of High Orders Dr. Mohamed El-Beltagy 1,2 Joint Wor with Late Prof. Magdy El-Tawil 2 1 Effat University, Engineering College, Electrical & Comp. Engr. Dept., Saudi Arabia 2 Cairo University, Engineering Faculty, Eng. Math. & Physics Dept., Egypt UQAW 2016 Thuwal - KAUST January 2016

Introduction Solution methods Wiener-Hermite Expansion Automation & generalization Solution of the equivalent system Applications White-noise of higher dimension General nonlinearity Future wor

Stochastic partial differential equations (SPDEs) are important tools in modeling complex phenomena. SPDEs arise in many physics and engineering applications. Developing efficient numerical methods for simulating SPDEs is a very important while challenging research topic. Applications: Wave propagation, Diffusion through heterogeneous random media, randomly forced Burgers and Navier- Stoes equations, Materials science, Chemistry, Biology, and other areas.

The large structures and dominant dynamics are governed by deterministic physical laws, while the unresolved small scales, microscopic effects, and other uncertainties can be naturally modeled by stochastic processes. The resulting equations are usually PDEs with either random coefficients, random initial conditions, random forcing or mixed. It is important to be able to study their statistical characteristics, e.g., mean, variance, and higher order moments.

Solution Methods: 1- Monte Carlo (MC) methods: handle different sources of randomness, easy to implement, use existing code. But, Slow convergence rate. 2- Numerical Methods: e.g. Euler-Maruyama 3- Stochastic Spectral Methods: Wiener (1938) suggested expansion based on Polynomial Chaos (pure chaos = Brownian motion = Wiener process). Cameron and Martin (1947) developed more explicit and intuitive formulation for the Wiener-Hermite expansion as a function of a set of random variables. Wiener (1958) suggested more detailed orthogonal expansion based on Hermite polynomials.

Based on Wiener's ideas, Meecham and his team developed the theory of Wiener-Hermite Expansion (WHE) for turbulence in a fundamental series of papers (1964, 1965, 1968). The basis of WHE is the derivative of the Brownian motion (ideal random function = white-noise). Meecham team studied Burger s turbulence with 2 nd order WHE. Orsaze (1967) and Crow & Canavan (1970) showed that 2 nd order WHE is not sufficient to study turbulence. Higher-order WHE is required or, Lagrangian (time-dependent) basis should be adopted (Crow 1968, Bodner 1969, Imamura 1972).

WHE is used in many applications e.g. Finance and turbulence (no need for models) 2 nd order Wiener-Hermite expansion cannot represent the dynamics of the Burgers turbulence. Including higher order terms was technically very difficult. The Wiener s formalism for studying turbulence phenomena had been largely abandoned/neglected. With the modern technology, we can re-visit the WHE and develop higher-order solutions of SPDE. In this wor, we study a numerical method based on the Wiener-Hermite expansion (WHE) for solving SPDEs driven by Brownian motion forcing, Using WHE of high orders. WHE is also called Wiener-Ito Expansion (WIE).

Model Equation: n L u( t, x) u f ( t, x) g( t, x) N( x) ; ( t, x) (0, T] R with the proper set of initial and boundary conditions. N(x)=dB(x;w)/dx is the white-noise (derivative of the Brownian motion). Later, we will generalize it to be multi-dimensional. L is a linear (integer or fractional order) operator. f(t,x) is a deterministic forcing, g(t,x) is a deterministic envelop function. u n is the nonlinear term, Many applications contain nonlinear terms in this polynomial type (e.g. Duffing oscillator u 3, radiation losses in heat equation u 4 ). Nonlinear analytic functions can be written/approx. in terms of polynomial type.

WHE: The solution process can be expanded in terms of Wiener-Hermite functionals (Wiener-Ito integrals) as [Imamura 1965], Or: Where: (0) (1) (1) 1 1 1 u( t, x; ) u ( t, x) u ( t, x; x ) H ( x ; ) dx u ( t, x; x, x ) H ( x, x ; ) dx dx (2) (2) 1 2 1 2 1 2 u ( t, x; x, x, x ) H ( x, x, x ; ) dx dx dx (3) (3) 1 2 3 1 2 3 1 2 3, ;,, u( t, x; w) u..., H x, x,..., x ; w d u ( ) 0 R ( ) ( ) t x x1 x2 x 1 2 t x x x x, ; 1, 2,..., d... dx 1dx 2 dx ; K th deterministic ernel of u(t,x) R ; K-dim integration w.r.t. the disposable variables x 1, x 2,..., x

n th -order Hermite functionals (Orthonormal set) taes the form : Recurrence relation: n n 2 2 xi n i 1 i 1 1 1 ( ) ( xi ) 2 2 ( n) n/2 H ( x 1, x 2,..., x n ) (0) e e 1 ( x ) H ( x, x,..., x ) H ( x, x,..., x ) H ( x ) ( n) ( n1) (1) 1 2 n 1 2 n1 n n1 ( n2) H x n x 1 n x 2 n x n 2 n m x n m 1 (,,..., ) ( ) Statistical Properties of the solution process. (0) E u( t, x; w) u 2 2 (1) (2) Var[ u( t, x)] u ( t, x; x ) dx 2 u ( t, x; x, x ) dx dx... M 2 ( ) Var u( t, x; w)! u d 1 R 1 1 1 2 1 2

Automated WHE: m ( ) ( ) u u H d 0 R, m m m ( ) ( ) ( ) ( ) R R 0 0 Using Multinomial Theorem for (u n ) [M. EL-Beltagy 2013]. To get: Lu H d u H d f ( t, x) g( t, x) N( x) m n m ( ) ( ) ( i ) ( i ) u H d f i i c u H d R 0 f i 0 ( ) ( ) ( i) ( i) f i R 0 f i0 m L u H d c u H d f ( t, x) g( t, x) N( x) R i n i f i f

Multiply by H (j) and Use Orthogonality: R j L u H H d c u E d m i ( j) ( j) ( j) ( i) f j j f f z f z R i0 ( j) ( j) H f ( t, x) g( t, x) H N( x) ; 0 j m Where the averages is defined as: Statistical Properties. E u( x, t; w) m j ( j) ( i) f i0 E H H u (0) m 2 j f 2 ( ) ( ) ( ) Var[ u( t, x; w)] u H H d 1 R 2

Averages: Imamura [1965] formula: m j ( j) ( i) f i0 E H H i 2 i 1 [ (0)] x ; ix j Automation (Recursive technique) [M. EL-Beltagy 2013]. 1 m n m distnict exogamous i 1 ( ni ) exogamous pairs H pairings i 1 m 0 ; j f m n i 1 i n i Even Odd x 1 x 4 x 2 x 6 x 3 x 5 x 1 x 5 x 2 x 4 x 3 x 6 x 1 x 5 x 2 x 6 x 3 x 4 x 1 x 6 x 2 x 4 x 3 x 5 x x x x x x H ( x, x, x ) H ( x, x, x ) x x x x x x (3) (3) 1 2 3 4 5 6 1 4 2 5 3 6 1 6 2 5 3 4

Integral Reductions [El-Beltagy 2013]: using the averages to reduce the integrals: R z m NC p () i hg p j z i0 p0 exog. pairs I u x x d This can be automated to generate the resulting system using Mathematics Marup Language (e.g. MML 2.0) [M. EL-Beltagy 2013] Typical Output (n=2, m=1 i.e. first-order WHE).

Typical Output (n=2, m=2; second-order WHE). We have a set of coupled nonlinear integro-differential equations Very difficult to solve

Higher Order WHE (n=2; m=3 third-order WHE):

Linearization technique, [e.g. Failla 2012]. Not considered in our wor. WHEP (WHE with Perturbation) Assume small nonlinearity Thiebaux (1971), Magdy El-Tawil, (1993-2013) Increases (but simplify) the number of equations Decouples the equivalent system Has a clear sequence (procedure) of the solution Automation (M. El-Beltagy 2013) Picard-lie Successive Approximation (Jahedi & Ahmadi 1983): Iterative, efficient, but has a restrict condition of convergence Other methods Newton s method (under trial) Suggestions are welcome

WHEP: Substitute to get: ( j) ( b u NC ( j) i ( j) ui i0 j! L u f ( t, x) g( t, x) ( x x ) ; 0 j m 0 j0 j1 1 j! L u c D E ; 0 j m, 1 b NC j) ( j) j f f, b1 f f m NC p ( j) ( i) hg Df, b 1 c g u p d z z R var i0 p0 Statistical Properties: (0) (, ) u E u x t m NC 2 i ( ) i 1 R i0 Var[ u( t, x)] (!) u d

Picard-lie Successive Approximation: (0) (0) (0) 2 (1) 2 L( u ) f ( t, x) u [ u ] 3 [ u ( t 1 1)] dt 1 R 2 1 1 0 (1) 2 L( u ) g( t, x) ( t t1) 3 u ( t1) u [ u ( t )] dt 1 R 1 1 Convergence condition: Statistical properties: u u ( j) ( j) 1 E x() t x (0) m 2 ( ) Var[ x( t)]! x d 1 2 R

2 x ( t) a( t) x( t) x ( t) N( t), t 0, x(0) x0 The generated WHEP equations (n=2, m=1, NC=1): : L( x ) 0 (0) (0) 0 0 L( x ) ( t t ) ; x (0; t ) 0 (1) (1) 0 1 0 L( x ) [ x ] [ x ( t; t )] dt (0) (0) 2 (1) 2 (0) 1 0 0 1 1 1 R L( x ) 2 x x ( t ) ; x (0; t ) 0 (1) (0) (1) (1) 1 0 0 1 1 ; x (0) 1 ; x (0) 0 1 x 0 In this simple case, analytic solution is available for comparison.

Solve using the FVM technique. In this simple case, analytic solution is available for comparison.

1-D diffusion model: Comparison between WHE and WHEP solutions WHE is the limit of WHEP with infinite no. of corrections.

Variance, comparison between WHE and WHEP

Convergence of Picard approximations (WHE):

Expectation (Different Orders) using WHE:

Variance (Different Orders) using WHE:

Convergence (Different Orders) using WHE:

Heat Equation with nonlinear losses: u t u x 2 2 ( t) u c f ( t) N( t) 2 With the proper set of initial and boundary conditions. Apply the above technique to get the set of equations (3 rd order WHE) Apply also the perturbation technique (NC=2) to get the following set of equations. Let: L 2 () t t x 2

Resulting deterministic set of equations (3 rd order WHE)

Resulting deterministic set of equations (2 nd order, 3 corrections WHEP)

: The equivalent deterministic system can be written in the following model form for any ( v u j ) ;0 j m L( v) F( v x The Picard s iteration taes the form: (0) ( m), u,, u ) ; v(0, x; 1, x j ) v0 v L F( v, u,, u ) Where is the iteration number. 1 1 (0) ( m) We can apply the FVM with the semi-implicit Cran-Nicolson scheme. Cran-Nicolson scheme has the advantage that it is unconditionally stable. t 0.5 rv (1 r) v 0.5rv 0.5 rv (1 r) v 0.5rv F x 1 1 1 i 1 i i 1 i 1 i i 1 i 2 x /2 i r t / x (0) ( m) F i x x i x /2 F ( v, u,, u ) dx

Heat equation output:

Duffing Oscillator with viscoelastic damping: 2 3 x t c ( ) ( ; ) Dt x t x t 0x t f0 t g0 t N t w Caputo definition: We shall assume, initially, a quiescent system 0 j s s t 1 x Dt xt ds ; j 1 j ; j 1 t x(0) (0) 0 x 0

Using the above outlined procedure to get: For M=1 (first-order WHE)

Picard s approximations:

Using FVM scheme:

Use FVM with Grunwald-Letniov (GL) approximation:

Analytical solution using fractional Green s functions [Podlubny 1999] with 3 terms:

Compare numerical & analytical:

Stochastic response can be obtained as:

Comparison with MC simulations:

Variance:

White-noise of Higher dimension: Typical output: ( ) H x1 t1 x2 t2 x t (,,,,...,, ) m ( ) ( ) u u H d 2 0 2 R L( u ) f ( t, x) (0) 0 (1) 0 1 1 2 N( t, x) L( u ) g( t, x) ( x x ) ( t t ) L( u ) [ u ] [ u ( x, t )] dx dt (0) (0) 2 (1) 2 1 0 0 1 1 1 1 R (1) (0) (1) 1 0 0 1, 1 L( u ) 2 u u ( x t )

For analytic functions: Typical output: L u( t, x) h( u) f ( t, x) g( t, x) N( x) ; ( t, x) (0, T ] R L( u ) f ( tx, ) e. g. h( u) a a u a u (0) 0 (1) 0 1 L( u ) g( t, x) ( x x ) (1) (0) (1) 1 2 0 0 1 2 4 0 2 4 (0) 4 (0) 2 (1) 2 [ u0 ] 6[ u0 ] [ u0 ( x1 )] dx 1 1 (0) (0) 2 (1) 2 R L( u1 ) a0 a2[ u0 ] [ u0 ( x1 )] dx1 a4 (1) 2 (1) 2 R 3 [ u0 ( x1 )] [ u0 ( x2)] dx1dx 2 2 R L( u ) 2 a u u ( x ) a 4[ u ] u ( x ) 12 u u ( x ) [ u ( x )] dx 1 R (0) 3 (1) (0) (1) (1) 2 4 0 0 1 0 0 1 0 2 2

Analysis of WHE, WHEP WHE with fractional-noise basis Re-study Burger s turbulence with higher-order WHE And more...

References: 1- M. El-Beltagy, Magdy. A. El-Tawil, Toward a Solution of a Class of Non-Linear Stochastic perturbed PDEs using Automated WHEP Algorithm, Appl. Math. Modelling, Volume 37, Issues 12 13, Pages 7174 7192 (2013), 2- M. El-Beltagy and A. Al-Johani, "Stochastic Response of Duffing Oscillator with Fractional or Variable-Order Damping", Journal of Fractional Calculus and Applications, Vol. 4(2), pp. 357-366, (2013) 3- M. El-Beltagy and A. Al-Johany, "Numerical Approximation of Higher-Order Solutions of the Quadratic Nonlinear Stochastic Oscillatory Equation Using WHEP Technique", Journal of Applied Mathematics, Vol. (2013), ID: 685137. 4- A. Al-Johani and M. El-Beltagy, "Numerical Solution of Stochastic Nonlinear Differential Equations Using Wiener-Hermite Expansion", 11TH INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2013: ICNAAM 2013. AIP Conference Proceedings, Volume 1558, pp. 2099-2102 (2013). 5- M. Hamed, M. El-Tawil, B. El-Desouy, M. El-Beltagy, "Solution of Nonlinear Stochastic Langevin's Equation Using WHEP, Picard and HPM Methods", Applied Mathematics, 5 (3), 398-412 (2014). 6- M. A. El-Beltagy and N. A. Al-Mulla, "Solution of the Stochastic Heat Equation with Non- Linear Losses Using Wiener-Hermite Expansion", Journal of Applied Mathematics, Volume 2014, Article ID 843714, http://dx.doi.org/10.1155/2014/843714 (2014).

Thans Any Questions?

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback