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

Transcription

1 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

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

3 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.

4 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.

5 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.

6 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).

7 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).

8 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.

9 WHE: The solution process can be expanded in terms of Wiener-Hermite functionals (Wiener-Ito integrals) as [Imamura 1965], Or: Where: (0) (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) u ( t, x; x, x, x ) H ( x, x, x ; ) dx dx dx (3) (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

10 n th -order Hermite functionals (Orthonormal set) taes the form : Recurrence relation: n n 2 2 xi n i 1 i ( ) ( 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

11 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

12 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

13 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)

14 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).

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

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

17 Linearization technique, [e.g. Failla 2012]. Not considered in our wor. WHEP (WHE with Perturbation) Assume small nonlinearity Thiebaux (1971), Magdy El-Tawil, ( ) 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

18 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

19 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 (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

20 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) L( x ) [ x ] [ x ( t; t )] dt (0) (0) 2 (1) 2 (0) R L( x ) 2 x x ( t ) ; x (0; t ) 0 (1) (0) (1) (1) ; x (0) 1 ; x (0) 0 1 x 0 In this simple case, analytic solution is available for comparison.

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

22 1-D diffusion model: Comparison between WHE and WHEP solutions WHE is the limit of WHEP with infinite no. of corrections.

23 Variance, comparison between WHE and WHEP

24 Convergence of Picard approximations (WHE):

25 Expectation (Different Orders) using WHE:

26 Variance (Different Orders) using WHE:

27 Convergence (Different Orders) using WHE:

28 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

29 Resulting deterministic set of equations (3 rd order WHE)

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

31 : 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 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

32 Heat equation output:

33 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

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

35 Picard s approximations:

36 Using FVM scheme:

37 Use FVM with Grunwald-Letniov (GL) approximation:

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

39 Compare numerical & analytical:

40 Stochastic response can be obtained as:

41 Comparison with MC simulations:

42 Variance:

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

44 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) (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)

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

46 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 (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 , (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: 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 AIP Conference Proceedings, Volume 1558, pp (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), (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 , (2014).

47 Thans Any Questions?