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# 03/17/2015 SLC talk

the slides for the talk i give at SLC in 2015's spring

the slides for the talk i give at SLC in 2015's spring

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## Weitere Verwandte Inhalte

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### 03/17/2015 SLC talk

1. 1. Numerical Methods for SPDEs driven by L´evy Jump Processes: Probabilistic and Deterministic Approaches Mengdi Zheng, George Em Karniadakis (Brown University) 2015 SIAM Conference on Computational Science and Engineering March 17, 2015
2. 2. Contents Motivation Introduction L´evy process Dependence structure of multi-dim pure jump process Generalized Fokker-Planck (FP) equation Overdamped Langevin equation driven by 1D TαS process by MC and PCM (probabilistic methods) by FP equation (deterministic method, tempered fractional PDE) Diﬀusion equation driven by multi-dimensional jump processes SPDE w/ 2D jump process in LePage’s rep SPDE w/ 2D jump process by L´evy copula SPDE w/ 10D jump process in LePage’s rep (ANOVA decomposition) Future work 2 of 25
3. 3. Section 1: motivation Figure : We aim to develop gPC method (probabilistic) and generalized FP equation (deterministic) approach for UQ of SPDEs driven by non-Gaussian L´evy processes. 3 of 25
4. 4. Section 2.1: L´evy processes Deﬁnition of a L´evy process Xt (a continuous random walk): Independent increments: for t0 < t1 < ... < tn, random variables (RVs) Xt0 , Xt1 − Xt0 ,..., Xtn−1 − Xtn−1 are independent; Stationary increments: the distribution of Xt+h − Xt does not depend on t; RCLL: right continuous with left limits; Stochastic continuity: ∀ > 0, limh→0 P(|Xt+h − Xt| ≥ ) = 0; X0 = 0 P-a.s.. Decomposition of a L´evy process Xt = Gt + Jt + vt: a Gaussian process (Gt), a pure jump process (Jt), and a drift (vt). Deﬁnition of the jump: Jt = Jt − Jt− . Deﬁnition of the Poisson random measure (an RV): N(t, U) = 0≤s≤t I Js ∈U, U ∈ B(Rd 0 ), ¯U ⊂ Rd 0 . (1) 4 of 25
5. 5. Section 2.2: Pure jump process Jt L´evy measure ν: ν(U) = E[N(1, U)], U ∈ B(Rd 0 ), ¯U ⊂ Rd 0 . 3 ways to describe dependence structure between components of a multi-dimensional L´evy process: Figure : We will discuss the 1st (LePage) and the 3rd (L´evy copula) methods here. 5 of 25
6. 6. Section 2.2: LePage’s multi-d jump processes Example 1: d-dim tempered α-stable processes (TαS) in spherical coordinates (”size” and ”direction” of jumps): L´evy measure (dependence structure): νrθ(dr, dθ) = σ(dr, θ)p(dθ) = ce−λr dr r1+α p(dθ) = ce−λr dr r1+α 2πd/2 dθ Γ(d/2) , r ∈ [0, +∞], θ ∈ Sd . Series representation by Rosinksi (simulation)1 : L(t) = +∞ j=1 j [( αΓj 2cT )−1/α ∧ ηj ξ 1/α j ] (θj1, θj2, ..., θjd )I{Uj ≤t}, for t ∈ [0, T]. P( j = 0, 1) = 1/2, ηj ∼ Exp(λ), Uj ∼ U(0, T), ξj ∼U(0, 1). {Γj } are the arrival times in a Poisson process with unit rate. (θj1, θj2, ..., θjd ) is uniformly distributed on the sphereSd−1 . 1 J. Ros´ınski, On series representations of inﬁnitely divisible random vectors, Ann. Probab., 18 (1990), pp. 405–430.6 of 25
7. 7. Section 2.2: dependence structure by L´evy copula Example 2: 2-dim jump process (L1, L2) w/ TαS components2 (L++ 1 , L++ 2 ), (L+− 1 , L+− 2 ), (L−+ 1 , L−+ 2 ), and (L−− 1 , L−− 2 ) Figure : Construction of L´evy measure for (L++ 1 , L++ 2 ) as an example 2 J. Kallsen, P. Tankov, Characterization of dependence of multidimensional L´evy processes using L´evy copulas, Journal of Multivariate Analysis, 97 (2006), pp. 1551–1572.7 of 25
8. 8. Section 2.2: dependence structure by (L´evy copula) Example 2 (continued): Simulation of (L1, L2) ((L++ 1 , L++ 2 ) as an example) by series representation L++ 1 (t) = +∞ j=1 1j ( αΓj 2(c/2)T )−1/α ∧ ηj ξ 1/α j I[0,t](Vj ), L++ 2 (t) = +∞ j=1 2j U ++(−1) 2 F−1 (Wi U++ 1 ( αΓj 2(c/2)T )−1/α ∧ ηj ξ 1/α j ) I[0,t](Vj ) F−1 (v2|v1) = v1 v − τ 1+τ 2 − 1 −1/τ . {Vi } ∼Uniform(0, 1) and {Wi } ∼Uniform(0, 1). {Γi } is the i-th arrival time for a Poisson process with unit rate. {Vi }, {Wi } and {Γi } are independent. 8 of 25
9. 9. Section 2.3: generalized Fokker-Planck (FP) equations For an SODE system du = C(u, t) + dL(t), where C(u, t) is a linear operator on u. Let us assume that the L´evy measure of the pure jump process L(t) has the symmetry ν(x) = ν(−x). The generalized FP equation for the joint PDF satisﬁes3: ∂P(u, t) ∂t = − ·(C(u, t)P(u, t))+ Rd −{0} ν(dz) P(u+z, t)−P(u, t) . (2) 3 X. Sun, J. Duan, Fokker-Planck equations for nonlinear dynamical systems driven by non-Gaussian L´evy processes. J. Math. Phys., 53 (2012), 072701.9 of 25
10. 10. Section 3: overdamped Langevin eqn driven by 1D TαS process We solve: dx(t; ω) = −σx(t; ω)dt + dLt(ω), x(0) = x0. L´evy measure of Lt is: ν(x) = ce−λ|x| |x|α+1 , 0 < α < 2 FP equation as a tempered fractional PDE (TFPDE) When 0 < α < 1, D(α) = c α Γ(1 − α) ∂ ∂t P(x, t) = ∂ ∂x σxP(x, t) −D(α) −∞Dα,λ x P(x, t)+x Dα,λ +∞P(x, t) When 1 < α < 2, D(α) = c α(α−1) Γ(2 − α) ∂ ∂t P(x, t) = ∂ ∂x σxP(x, t) +D(α) −∞Dα,λ x P(x, t)+x Dα,λ +∞P(x, t) −∞Dα,λ x and x Dα,λ +∞ are left and right Riemann-Liouville tempered fractional derivatives4 . 4 M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Studies in Mathematics Vol. 43, 2012.10 of 25
11. 11. Section 3: PCM V.s. TFPDE in moment statistics 0 0.2 0.4 0.6 0.8 1 10 4 10 3 10 2 10 1 10 0 t err2nd fractional density equation PCM/CP 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 10 3 10 2 10 1 10 0 t err2nd fractional density equation PCM/CP Figure : err2nd versus time by: 1) TFPDEs; 2) PCM. Problem: α = 0.5, c = 2, λ = 10, σ = 0.1, x0 = 1 (left); α = 1.5, c = 0.01, λ = 0.01, σ = 0.1, x0 = 1 (right). For PCM: Q = 50 (left); Q = 30 (right). For density approach: t = 2.5e − 5, 2000 points on [−12, 12], IC is δD 40 (left); t = 1e − 5, 2000 points on [−20, 20], i.c. given by δG 40 (right). 11 of 25
12. 12. Section 3: MC V.s. TFPDE in density 4 2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x(T = 0.5) densityP(x,t) histogram by MC/CP density by fractional PDEs 4 2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x(T=1) densityP(x,t) histogram by MC/CP density by fractional PDEs Figure : Zoomed in plots of P(x, T) by TFPDEs and MC at T = 0.5 (left) and T = 1 (right): α = 0.5, c = 1, λ = 1, x0 = 1 and σ = 0.01 (left and right). In MC: s = 105 , 316 bins, t = 1e − 3 (left and right). In the TFPDEs: t = 1e − 5, and Nx = 2000 points on [−12, 12] in space (left and right). 12 of 25
13. 13. Section 4: heat equation w/ multi-dim jump process We solve :    du(t, x; ω) = µ∂2u ∂x2 dt + d i=1 fi (x)dLi (t; ω), x ∈ [0, 1] u(t, 0) = u(t, 1) = 0 boundary condition u(0, x) = u0(x) initial condition, (3) L(t; ω), {Li (t; ω), i = 1, ..., d} are mutually dependent. fk(x) = √ 2sin(πkx), x ∈ [0, 1], k = 1, 2, 3, ... is a set of orthonormal basis functions on [0, 1]. By u(x, t; ω) = +∞ i=1 ui (t; ω)fi (x) and Galerkin projection onto {fi (x)}, we obtain an SODE system, where Dmm = −(πm)2:    du1(t) = µD11u1(t)dt + dL1, du2(t) = µD22u2(t)dt + dL2, ... dud (t) = µDdd ud (t)dt + dLd , (4) 13 of 25
14. 14. Section 4.1: SPDEs driven by multi-d jump processes Figure : An illustration of probabilistic and deterministic methods to solve the moment statistics of SPDEs driven by multi-dim L´evy processes. 14 of 25
15. 15. Section 4.2: FP eqn when Lt (2D) is in LePage’s rep When the L´evy measure of Lt is given by νrθ(dr, dθ) = ce−λr dr r1+α 2πd/2dθ Γ(d/2) , for r ∈ [0, +∞], θ ∈ Sd The generalized FP equation for the joint PDF P(u, t) of solutions in the SODE system is: ∂P(u,t) ∂t = − d i=1 µDii (P + ui ∂P ∂ui ) − c α Γ(1 − α) Sd−1 Γ(d/2)dσ(θ) 2πd/2 r Dα,λ +∞P(u + rθ, t) , where θ is a unit vector on the unit sphere Sd−1. x Dα,λ +∞ is the right Riemann-Liouville Tempered Fractional (TF) derivative. Later, for d = 10, we will use ANOVA decomposition to obtain equations for marginal distributions from this FP equation. 15 of 25
16. 16. Section 4.2: simulation if Lt (2D) is in LePage’s rep Figure : FP vs. MC/S: joint PDF P(u1, u2, t) of SODEs system from FP Equation (3D contour) and by MC/S (2D contour), horizontal and vertical slices at the peak of density. t = 1 , c = 1, α = 0.5, λ = 5, µ = 0.01, NSR = 16.0% at t = 1. 16 of 25
17. 17. Section 4.2: simulation when Lt (2D) is in LePage’s rep 0.2 0.4 0.6 0.8 1 10 −10 10 −8 10 −6 10 −4 10 −2 l2u2(t) t PCM/S Q=5, q=2 PCM/S Q=10, q=2 TFPDE NSR 4.8% 0.2 0.4 0.6 0.8 1 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 l2u2(t) t PCM/S Q=10, q=2 PCM/S Q=20, q=2 TFPDE NSR 6.4% Figure : FP vs. PCM: L2 error norm in moments obtained by PCM and FP equation. α = 0.5, λ = 5, µ = 0.001 (left and right). c = 0.1 (left); c = 1 (right). In FP: initial condition is given by δG 2000, RK2 scheme. 17 of 25
18. 18. Section 4.3: FP eqn if Lt (2D) is from L´evy copula The L´evy measure of Lt is given by L´evy copula on each corners (++, +−, −+, −−) dependence structure is described by the Clayton family of copulas with correlation length τ on each corner The generalized FP eqn is : ∂P(u,t) ∂t = − · (C(u, t)P(u, t)) + +∞ 0 dz1 +∞ 0 dz2ν++(z1, z2)[P(u + z, t) − P(u, t)] + +∞ 0 dz1 0 −∞ dz2ν+−(z1, z2)[P(u + z, t) − P(u, t)] + 0 −∞ dz1 +∞ 0 dz2ν−+(z1, z2)[P(u + z, t) − P(u, t)] + 0 −∞ dz1 0 −∞ dz2ν−−(z1, z2)[P(u + z, t) − P(u, t)] 18 of 25
19. 19. Section 4.3: FP eqn if Lt (2D) is from L´evy copula Figure : FP vs. MC: P(u1, u2, t) of SODE system from FP eqn (3D contour) and by MC/S (2D contour). t = 1 , c = 1, α = 0.5, λ = 5, µ = 0.005, τ = 1, NSR = 30.1% at t = 1. 19 of 25
20. 20. Section 4.3: if Lt (2D) is from L´evy copula 0.2 0.4 0.6 0.8 1 10 −5 10 −4 10 −3 10 −2 t l2u2(t) TFPDE PCM/S Q=1, q=2 PCM/2 Q=2, q=2 NSR 6.4% 0.2 0.4 0.6 0.8 1 10 −3 10 −2 10 −1 10 0 t l2u2(t) TFPDE PCM/S Q=2, q=2 PCM/S Q=1, q=2 NSR 30.1% Figure : FP vs. PCM: L2 error of the solution for heat equation α = 0.5, λ = 5, τ = 1 (left and right). c = 0.05, µ = 0.001 (left). c = 1, µ = 0.005 (right). In FP: I.C. is given by δG 1000, RK2 scheme. 20 of 25
21. 21. Section 4.3: FP eqn if Lt is in LePage’s rep by ANOVA The unanchored analysis of variance (ANOVA) decomposition is 5: P(u, t) ≈ P0(t) + 1≤j1≤d Pj1 (uj1 , t) + 1≤j1<j2≤d Pj1,j2 (uj1 , uj2 , t) +... + 1≤j1<j2...<jκ≤d Pj1,j2,...,jκ (uj1 , uj2 , ..., uκ, t) κ is the eﬀective dimension P0(t) = Rd P(u, t)du Pi (ui , t) = Rd−1 du1...dui−1dui+1...dud P(u, t) − P0(t) = pi (ui , t) − P0(t) Pij (xi , xj , t) = Rd−1 du1...dui−1dui+1...duj−1duj+1...dud P(u, t) −Pi (ui , t) − Pj (uj , t) − P0(t) = pij (x1, x2, t) − pi (x1, t) − pj (x2, t) + P0(t) 5 M. Bieri, C. Schwab, Sparse high order FEM for elliptic sPDEs, Tech. Report 22, ETH, Switzerland, (2008).21 of 25
22. 22. Section 4.3: FP eqn if Lt is in LePage’s rep by ANOVA When the L´evy measure of Lt is given by νrθ(dr, dθ) = ce−λr dr r1+α 2πd/2dθ Γ(d/2) , for r ∈ [0, +∞], θ ∈ Sd (for 0 < α < 1) ∂pi (ui ,t) ∂t = − d k=1 µDkk pi (xi , t) − µDii xi ∂pi (xi ,t) ∂xi −cΓ(1−α) α Γ( d 2 ) 2π d 2 2π d−1 2 Γ( d−1 2 ) π 0 dφsin(d−2)(φ) r Dα,λ +∞pi (ui +rcos(φ), t) ∂pij (ui ,uj ,t) ∂t = − d k=1 µDkk pij −µDii ui ∂pij ∂ui −µDjj uj ∂pij ∂uj −cΓ(1−α) α Γ( d 2 ) 2π d 2 2π d−2 2 Γ(d−2 2 ) π 0 dφ1 π 0 dφ2sin8(φ1)sin7(φ2) r Dα,λ +∞pij (ui + rcosφ1, uj + rsinφ1cosφ2, t) 22 of 25
23. 23. Section 4.3: FP eqn if Lt is in LePage’s rep by ANOVA 0 0.2 0.4 0.6 0.8 1 −2 0 2 4 6 8 10 12 x E[u(x,T=1)] E[uPCM ] E[u 1D−ANOVA−FP ] E[u 2D−ANOVA−FP ] 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 x 10 −4 T L2 normofdifferenceinE[u] ||E[u 1D−ANOVA−FP −E[u PCM ]|| L 2 ([0,1]) /||E[u PCM ]|| L 2 ([0,1]) ||E[u 2D−ANOVA−FP −E[u PCM ]|| L 2 ([0,1]) /||E[u PCM ]|| L 2 ([0,1]) Figure : 1D-ANOVA-FP V.s. 2D-ANOVA-FP in 10D: the mean (left) for the solution of heat eqn at T = 1. The L2 norms of diﬀerence in E[u](right). c = 1, α = 0.5, λ = 10, µ = 10−4 . I.C. of ANOVA-FP: MC/S data at t0 = 0.5, s = 1 × 104 . NSR ≈ 18.24% at T = 1. 23 of 25
24. 24. Section 4.3: FP eqn if Lt is in LePage’s rep by ANOVA 0 0.2 0.4 0.6 0.8 1 0 20 40 60 80 100 120 x E[u 2 (x,T=1)] E[u 2 PCM ] E[u2 1D−ANOVA−FP ] E[u2 2D−ANOVA−FP ] 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 T L2 normofdifferenceinE[u 2 ] ||E[u2 1D−ANOVA−FP −E[u2 PCM ]|| L 2 ([0,1]) /||E[u2 PCM ]|| L 2 ([0,1]) ||E[u2 2D−ANOVA−FP −E[u2 PCM ]|| L 2 ([0,1]) /||E[u2 PCM ]|| L 2 ([0,1]) Figure : 1D-ANOVA-FP V.s. 2D-ANOVA-FP in 10D: the 2nd moment (left) for heat eqn.The L2 norms of diﬀerence in E[u2 ] (right). c = 1, α = 0.5, λ = 10, µ = 10−4 . I.C. of ANOVA-FP: MC/S data at t0 = 0.5, s = 1 × 104 .NSR ≈ 18.24% at T = 1. 24 of 25
25. 25. Future work multiplicative noise (now we have additive noise) nonlinear SPDE (now we have linear SPDE) higher dimensions (we computed up to < 20 dimensions) thanks! 25 of 25