Igor Cialenco. Department of Applied Mathematics Illinois Institute of Technology, USA joint with N.

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1 Parameter Estimation for Stochastic Navier-Stokes Equations Igor Cialenco Department of Applied Mathematics Illinois Institute of Technology, USA joint with N. Glatt-Holtz (IU) Asymptotical Statistics of Stochastic Processes VIII Universite du Maine, Le Mans, March, 211 Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 1

2 Chicago, Summer 21 c Ig.C.

3 The Problem SPDE du(t,x) = θau(t,x)dt+fu(t,x)dt+σm(u(t,x))dw(t,x) u in some suitable Hilbert spaces and stochastic basis (Ω,F,F,P) A - linear operator F, M - some (could be nonlinear) operators W(t, x) - cylindrical Brownian motion u is known for all x G, t [,T] - continuous observations θ, σ - parameters (scalars), unknown Goal: Find estimators ˆθ(ω), ˆσ(ω), ω Ω, for parameters θ, σ by observing a single outcome u = u(ω) over a finite time horizon [,T]. Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 3

4 Stochastic ODE The Drift Girsanov Th, maximize Log-Likelihood Ratio w.r.t. θ du = θu(t)dt+σu(t)dw(t), θt = 1 t t du(s) u(s) = 1 u(t) log t u() σ2 2 Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 4

5 dy = y(t)dt+σdw(t), t T; σ > y t = σ 2 t σ = Stochastic ODE Volatility y T T ; In practice: σ2 N k=1 ( ) 2 y( kt N ) y((k 1)T N ) T the drift θ - approximated, the volatility σ - exactly Regular model 1) dp θ dq exists; 2) has a special form (LAN) Same procedure for all Find MLE by maximizing likelihood ratio WHY? Singular model otherwise Individual approach In particular, if P θ1 P θ2 for θ 1 θ 2, then one may find θ exactly Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 5

6 The Heat Equation (simulated by Euler) du = ν u xx dt+ ε σ γ dw, T=1, ν=.1, γ=, ε= What do we have for SPDEs? Mostly Singular Models We will try to understand the singularity and explore it to find the exact θ.

7 SPDE drift additive noise: Huebner-Khasminskii-Rozovskii 92, 95 Bayesian: Bishwal ( 2) Several parameters: Huebner ( 97) Discrete-time observations: Piterbarg-Rozovskii ( 97) q = 2(m 1 2m) d 1 θ(t)-random: Lototsky ( 4) Small noise: Huebner ( 97), Ibragimov-Khasminskii ( 98, 99) almost commutative case or almost diagonalizable model- Rozovskii-Lototsky ( 97), Lototsky ( 1) additive fractional noise: IgC, Lototsky, Pospisil ( 9) multiplicative noise: IgC and Lototsky ( 8), IgC ( 1) nonlinear SPDE: IgC and Glatt-Holtz ( 11) Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 7

8 SPDE General Framework du(t) = θau(t)dt+f(u)dt+σdw(t), U() = U ( A) a linear, selfadjoint, positive-defined (think Laplace) in H with eigenvalues {Φ k } k 1 CONS in H F maybe nonlinear σdw(t) = k 1 σ kφ k dw k (t), W k,k N ind. Brownian Motions σ known θ - parameter/scalar of interest U(t) H is observed/measured/known for all t [,T] but one ω Ω; Observe one path during t [,T] Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 8

9 SPDE General Framework Formal Procedure to Derive an Estimator Project the full system down to N dimensions P N (H) = H N R N du N = (θau N +Ψ N )dt+p N σdw, U() = U Formally treat Ψ N = P N F(U) as an external and known quantity (independent of θ) Assume that P N σ is invertible on H N and take G = (P N σ) 1 Consider P N,T ν ( ) = P(U N ), the measure on C([,T];R N ) generated by U N. For a reference values θ, apply (formally) Girsanov Theorem and get the likelihood ratio (Radon-Nikodym derivative) dp N,T θ /dp N,T θ Maximize the Log-Likelihood Ratio ˆθ(ω) := argmax θ dp N,T θ /dp N,T θ (ω) Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 9

10 SPDE General Framework dp N,T θ dp N,T θ [ =exp 1 2 (θ θ ) AU N,G 2 du N (t) (θ 2 θ 2 ) AU N,G 2 AU N dt ] (θ θ ) AU N,G 2 ψ N dt, θ N = AUN,G 2 du N + AUN,G 2 P N F(U) dt GAUN 2 dt Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 1

11 SPDE General Framework Motivated by MLE type estimator θ N = ˇθ N = AUN,G 2 du N + AUN,G 2 P N F(U) dt GAUN 2 dt AUN,G 2 du N + AUN,G 2 F N (U N ) dt GAUN 2 dt θ N = AUN,G 2 du N GAUN 2 dt Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 11

12 Successfully applied to: Stochastic Navier Stokes Equations, 2D, additive noise Stochastic Reaction-Diffusion Equation, additive noise Stochastic Fractional Burgers Equation, additive noise Work in progress to derive general conditions on A,F,G that guarantee consistency and asymptotic normality of ˇθ N and θ N as number of modes N ˇθ N = θ + + AUN,G 2 N j=1 σ j(u)φ j dw j (t) GAUN 2 dt AUN,G 2 (F N (U) F N (U N )) dt GAUN 2 dt θ N = θ + + AUN,G 2 N j=1 σ j(u)φ j dw j (t) GAUN 2 dt AUN,G 2 F N (U) dt GAUN 2 dt

13 Stochastic Navier Stokes, 2D du +((U )U ν U)dt = k λ γ k Φ kdw k U =, U() = U Models a viscous 2D incompressible fluid. U velocity field, ν kinematic viscosity Periodic or Dirichlet boundary conditions Φ k, λ k λ 1 k eigenfunctions, eigenvalues of the Stokes operator A (P N σ) 1 = A γ on H N = Span{Φ 1,...,Φ N } When γ > 1,! Strong, Pathwise solutions i.e. U C([, ),H 1 ) L 2 loc ([, )H2 ). Higher regularity for larger γ. Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 13

14 Stochastic Navier Stokes, 2D Put G = A α γ, for some α R (determined later). Then, ˇν N = = A1+2α U N,dU N +P N B(U N )dt A1+α U N 2 dt N k=1 λ 1+2α [ k u k(t)du k (t)+ u k(t)b k (t)dt ] N k=1 λ 2(1+α) k u k(t) 2 dt ˆν N = A1+2α U N,dU N = A1+α U N 2 dt = N k=1 λ1+2α k N k=1 λ1+2α k N k=1 λ2(1+α) k [u 2 k (T) u2 k () Tλ 2γ T u k(t) 2 dt 2 N k=1 λ2(1+α) k k ] u k(t)du k (t) u k(t) 2 dt Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 14

15 Stochastic Navier Stokes, 2D Main Result Theorem (Ig.C. & N. Glatt-Holtz 211) Let U = U(ω,t,x) is a solution of the 2D SNSE with Periodic or Dirichlet BCs. If γ > 1 (also γ 1 < 1/4 in the Dirichlet case) then: (i) Consistency: If α > γ 1, then N ˇν lim N = ν, N ˆν lim N ν in Probability. (ii) Asymptotic normality (rate N): If α > γ 1 2, then N( ν N ν) ( d 2ν(α γ +1) 2 ) η N, λ 1 T(α γ +1/2) Note: In particular we can take α = γ. This corresponds to formal MLE. Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 15

16 Sketch of the proof A Splitting Argument ˆν N ν = A1+2α γ(ūn +R N ), k Φ kdw k +A γ P N B(U)dt A1+α (ŪN +R N ) 2 dt = E A1+α Ū N 2 dt A1+α (ŪN +R N ) 2 dt A1+2α γ (ŪN +R N ), k Φ kdw k +A γ P N B(U)dt E A1+αŪN 2 dt Decompose U = Ū +R = linear + nonlinear dū +νaūdt = σdw, Ū() =. t R+νAR = B(U), R() = U. Find explicit and exact rates for moments of the linear part R is more regular in comparison to Ū Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 16

17 Sketch of the proof A Splitting Argument N δ 1 A1+α (ŪN +R N ) 2 dt E 1 T A1+αŪN 2 dt A1+2α γ U N, N k=1 Φ kdw k A1+αŪN 2 dt N δ 2 A1+2α γ R N, N k=1 Φ kdw k A1+αŪN 2 dt N δ 3 A1+2α U N,P N (B(U)) A1+αŪN 2 dt N A1+2α γ U N, N k=1 Φ kdw k E 2γ(α γ +1) 2 T A1+α U N N(, 2 dt λ 1 T(α γ +1/2) ) Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 17

18 1 The Heat Equation (simulated by Euler) du = ν u xx dt+ ε σ γ dw, T=1, ν=.2, γ=.1, ε= du = νu xx dt+dw(t,x), u(,x) = u, u(t,) = u(t,1) =

19 The Burgers Equation (simulated by Euler) du = ν u xx dt β u u x dt + ε σ γ dw, T=1, nu=.2, γ=.1, ε= x t du = νu xx dt uu x dt+dw(t,x), u(,x) = u, u(t,) = u(t,1) =

20 The Burgers Equation (simulated by Euler) du = ν u xx dt β u u x dt + ε σ γ dw, T=1, nu=.2, γ=.1, ε= du = νu xx dt uu x dt+dw(t,x), u(,x) = u, u(t,) = u(t,1) =

21 .23 Heat Equation du = ν u xx dt + σ γ dw, ν=.2, T=1, α=.1, γ= nucheck, Method 1, u(s)du(s) nucheck, Method 2,.5(u 2 (T) u 2 () T λ 2γ nucheckexp, Method 1, u(s)du(s) nucheckexp, Method 2,.5(u 2 (T) u 2 () T λ 2γ True Parameter du = νu xx dt+dw(t,x), u(,x) = u, u(t,) = u(t,1) =

22 Burgers Equation du = ν u xx dt β u u x dt + σ γ dw, ν=.2, T=1, α=.1, γ=.1 nucheck, Method 1, u(s)du(s) nucheck, Method 2,.5(u 2 (T) u 2 () T λ 2γ nucheckexp, Method 1, u(s)du(s) nucheckexp, Method 2,.5(u 2 (T) u 2 () T λ 2γ True Parameter du = νu xx dt uu x dt+dw(t,x), u(,x) = u, u(t,) = u(t,1) =

23 x nuhat for α=γ=.1 nuhat for α = du = νu xx dt uu x dt+dw(t,x), u(,x) = u, u(t,) = u(t,1) =

24 Future Work

25 Future Work... a lot

26 Thank You! The end of the talk But not of the story... Ig.Cialenco, Applied Math, IIT Le Mans, March 22 Slide # 25