Modeling uncertainties with kinetic equations

Transcription

1 B. Després (LJLL- University Paris VI) thanks to G. Poette (CEA) and D. Lucor (Orsay) B. Perthame (LJLL) E. Trélat (LJLL) Modeling uncertainties with kinetic equations B. Després (LJLL-University Paris VI) thanks to G. Poette (CEA) and D. Lucor (Orsay) B. Perthame (LJLL) E. Trélat (LJLL) /DFG-CNRS Workshop 6 p. / 3

2 Section /DFG-CNRS Workshop 6 p. / 3

3 Need in applied sciences Uncertainty data base: T> dp=w( ω ) dω T= u(; ω ) u (t)=f(u,...) EDP, EDO Uncertainty propagation u(t; ω) Assume the probability law is given dp = dµ(ω), dp = e (ω ω ) σ dω, ω R P high stochastic dimension. A popular idea to use chaos (Wiener 36 ) expansion u(x, t, ω) u N (x, t, ω) = N u n(x, t)p n(ω), p n P n. /DFG-CNRS Workshop 6 p. / 3

4 Conservation laws Goal : model/propagate uncertainties in conservation laws Euler equations (Wiener 38, Lin-Su-Karniadakis 6, Glimm and al 6,...) tρ + x(ρv) =, t(ρv) + x(ρv + p ω) =, p ω = (γ(ω) )ρε t(ρe) + x(ρve + p ωv) =, e = ε + v. Transport of the uncertain variable ω { tu + xf (U, ω) =, t(ρω) + x(ρvω) =. In this presentation : model conservation law with uncertainties in the initial condition { tu + xf (u) =, F : R R, u(x, ω, ) = u (x, ω) /DFG-CNRS Workshop 6 p. 3 / 3

5 Section /DFG-CNRS Workshop 6 p. 4 / 3

6 formulation of conservation laws The kinetic formulation of conservation laws (Perthame-Tadmor 9 ) writes tf ε + a(ξ) xf ε + ε fε = M(uε; ξ), a(ξ) = F (ξ), (Burgers : a(ξ) = ξ), ε u ε(x, t) = f ε(x, ξ, t)dξ, f ε(t = ) = M(u init ; ξ), For simplicity u init. The kinetic variable is ξ. The pseudo-maxwellian is M(u; ξ) = I {<ξ<u}. Under general conditions (Lions-Perthame-Tadmor 94 ), the limit ε + is : u ε(t) u(t) in L x and f ε(t) M(u(t)) in L xξ tu + xf (u) =, F : R R. Moreover the limit is entropic (we will come back to that). /DFG-CNRS Workshop 6 p. 4 / 3

7 and uncertainty Variables : usual variables are the time t and space x, additionally one has the kinetic variable ξ and the uncertain variable ω. Consider tfε N + a(ξ) xfε N uε N (x, ω, t) = + ε f N ε = ε MN (u N ε ; ξ, ω), f N ε (x, ξ, ω, t)dξ, f N ε (t = ) = M N (u init ; ξ, ω), where M N (u N ε ; ξ, ω) is a suitable polynomial approximation of M. Strategy : construct M N, study properties and pass to the limit ε and N. /DFG-CNRS Workshop 6 p. 5 / 3

8 Convolutions with polynomial kernels Convolution M N (uε N ; ξ) = G N ω M(uε N ; ξ) = G N (ω, ω )M(u N ε (ω ); ξ)dµ(ω ) P N ω with kernel G N (ω, ω ) = N n= cnpn(ω)pn(ω ), with p n the orthonormal basis of for the measure dµ(ω) and where c n are appropriate coefficients, and where G N is a Kernel Polynomial Method (Weisse and al, 6) G N, G N (ω, ω )dµ(ω ) =. () Basic example : Take dµ(ω) = dω, on ω I = (, ), and Tchebycheff orthonormal π ω p n(ω) = T n(ω) = cos (n arcos ω), ω. The Fejer kernel G N F c = and c n = N + n, n N N + is defined by the coefficients N f N F (ω) = c µ + c nµ nt n(ω), µ n = f (ω )T n(ω )dµ(ω ) n= I ( ) π f N F (cos t) = f (cos u)k N F (t u)du, K N F (u) = sin(n + ) u π(n + ) sin u. /DFG-CNRS Workshop 6 p. 6 / 3

9 A priori estimates + a(ξ). f ε N uε N (x, ω, t) = tf N ε + ε f N ε f N ε (x, ξ, ω, t)dξ. = ε G N ω M(u N ε ; ξ, ω), L bounds : fε N, uε N U M := sup x,ω u init (x, ω), fε N (x, ξ, ω, t) for ξ U M. Entropy bounds S (ξ) : t S (ξ)f N ε (x, ξ, ω, t)dξdµ(ω) + div a(ξ)s (ξ)f N ε (x, ξ, ω, t)dξdµ(ω) Propagation of classical BV bounds : xfε N dxdξdµ(ω) C init, xuε N dxdµ(ω) C init, tf ε N (t) dxdξdµ(ω) tf ε N () dxdξdµ(ω) C init, ξ fε N (t) dxdξdµ(ω) e ε t ξ fε N () dxdξdµ(ω) + C Plus new BV estimates with respect to ω, using the convolution structure. /DFG-CNRS Workshop 6 p. 7 / 3

10 Convergence for all t (Jackson kernels) The limit is a weak solution N ε : Consider the Jackson kernels. There exists f L loc(d ) such that lim ε fε N f = and L loc (D ) tf + a(ξ) xf = ξ m weak solution where m is a measure, and f = M(u; ξ) with u = fdξ. So (Lions-Perthame-Tadmor 94 ) : u is the solution (a.e.). Strong error bounds Nε : Moreover fε N (t) G N ω f ε(t) C t L xξµ ε π f (cos(t + α) f (cos t) dt Ct Nε. Projected equations ε = fixed : One gets for N+ uε,i(x, N t) = fε,i(x, N ξ, ω, t)dξdµ(ω), fε,i(x, N ω, t) = fε N (x, ξ, ω, t)t i (ω)dξ tu N ε,i + x a(ξ)f N ε,idξ = h N (i)u N ε,i, h N (i) > for i >. /DFG-CNRS Workshop 6 p. 8 / 3

11 Section 3 /DFG-CNRS Workshop 6 p. 9 / 3

12 Reminder Brenier 83 : the indicatrix function is the minimum of a certain minimization problem I {<ξ<u} = M(u; ) = argmin { g gdξ=u } g(ξ)s (ξ)dξ, S, S >. Generalize : Minimization of weighted L norms, under convex constraints M N (u N ) = argmin g N (ξ, ω)s (ξ)dξdµ(ω), for admissible S, g N K N (u N ) I where K N (u N ) = { g N (, ) P N ω, u N (ω) = } g N (ξ, ω)dξ, g N for ω I. /DFG-CNRS Workshop 6 p. 9 / 3

13 A remark Make the assumption : for all u N there exists a unique M N (we call it a kinetic polynomial) such that M N (u N ) = argmin g N (ξ, ω)s (ξ)dξdµ(ω), for a given S. g N K N (u N ) I Then the solution of tfε N uε N (x, ω, t) = + a(ξ) xfε N + ε f N ε = ε MN (u N ε ; ξ, ω), f N ε (x, ξ, ω, t)dξ, f N ε (t = ) = M N (u init ; ξ, ω), is in bounds, f N, and is conservative tuε N (x, ω, t) + xgε N (x, ω, t)dξ =, uε N = fε N dξ, Gε N (x, ω, t) = a(ξ)fε N dξ. ξ This last property does not hold with the convolution method. ξ /DFG-CNRS Workshop 6 p. / 3

14 Entropy inequality Moreover it satisfies the entropy inequality d f N (x, ξ, ω, t)dxs (ξ)dξdµ(ω) dt ξ ω x = ( ) M N S (ξ)dξdω f N S (ξ)dξdω. ε x ξ ω Note that it passes (at least formally) to the limit ε : one gets the equation tu N (x, ω, t) + xg N (x, ω, t)dξ =, G N (x, ω, t) = a(ξ)m N (u N )dξ which satisfies the entropy inequality t a(ξ)m N (u N )dξ + x a(ξ)m N (u N )S (ξ)dξ. ξ ξ ξ ω ξ /DFG-CNRS Workshop 6 p. / 3

15 Designing M N is the issue For simplicity rewrite the time and space variables as t ξ and x ω. Set S (ξ) = S (t) = t for simplicity. Consider the interval x I = [, ]. write n = N. The problem rewrites as : Given a given non negative polynomial u n P + n (I ), find with the constraint v n(t) U n {w n P n w n(x) for all x I } T v n(t, x)dt = u n(x) x I, such that T v n = argmin w n(t, x) tdtdx. I :( the structure of U n is complex with an infinite number of constraints and the functional is linearly degenerate. :) it can interpreted in the setting of Optimal Control. /DFG-CNRS Workshop 6 p. / 3

16 Reformulation Set y n(t, x) = T v n(t, x)dt. The problem writes : Find a v n(t) U n such that the state y n d yn = vn, yn() =, dt reaches the objective y n(t ) = u n and minimizes the cost function T C(v n) = v n(t, x)tdtdx. I First obvious result : if T u n L then is a (a priori) non optimal solution. non opt. vn (t, x) = un(x) u n L /DFG-CNRS Workshop 6 p. 3 / 3

17 Properties Wish/claim : there exists a unique solution v n(t) to the Optimal problem. Hint of the proof : mix the Bojanic-Devore theorem (one-sided L polynomial approximation) with optimal. Theorem (Pontryagin maximum principle) : for all optimal trajectories, there exists a multiplier λ n P n such that either the trajectory is normal v n(t) = argmax wn Un (λ n(x) t)w n(x)dx or the trajectory is abnormal v n(t) = argmax wn Un λ n(x)w n(x)dx. Next simulations are with the AMPL code, very popular in the Optimal Control community. /DFG-CNRS Workshop 6 p. 4 / 3

18 Feasible solution : constant one layer the other ξ = L+ u ξ =u ξ infinite upper layer M= M= last non zero layer... second layer first layer bottom layer ξ = ω /DFG-CNRS Workshop 6 p. 5 / 3

19 Numerical scheme (with the feasible solution) Let a D mesh j x. Consider u N j u N j t + F N [u N j ] F N [u N j ] x =, unknown uj N P N (ω) is a polynomial in ω of degree N (fixed) in cell j, flux F N [uj N ] P N (ω) in cell j is constructed with the kinetic polynomial formula for the Burgers equation F N [u N ] n = l (F (ξ l+ ) F (ξ l )) hl N (ω)p n(ω)dµ(ω), F (ξ) = ξ. Assume init. is bounded : U m u N j (ω) U M <, j and ω I. Theorem : Assume the CFL condition U M t x. Then U m u N j (ω) U M, j and ω I. Proof. Either use the kinetic formulation, or prove directly. /DFG-CNRS Workshop 6 p. 6 / 3

20 Numerical illustration : Burgers equation Set up : dµ(ω) = π d ω, N =. The moments model is explicit t a b c + x a +b +c ab + bc ac + b =. Compare solutions of the moment model, the kinetic polynomial method with feasible solution, and the standard non intrusive approach (quadrature points, close to MC). /DFG-CNRS Workshop 6 p. 7 / 3

21 N-convergence for Burgers eq. Consider the system of N + conservation laws with dµ(ω) = dω ( n N tu + unϕn(ω)) x ϕ (ω)dω =,... tu n + x ( n N unϕn(ω)) ϕ N (ω)dω =. There is an entropy-entropy flux pair = hyperbolicity n N un t + x Θ ( n N unϕn(ω) ) 3 3 dω. /DFG-CNRS Workshop 6 p. 8 / 3

22 u ini (x, ω) = 3 for x < / and < ω <, 5 for x < / and < ω <, for / < x and < ω <. Shock burgers.dat using :: burgers.dat using :: projection of the initial data exact solution t =.4 burgers.dat using ::3 burgers.dat using :: moment solution t =.4 new method t =.4 /DFG-CNRS Workshop 6 p. 9 / 3

23 u ini (x, ω) = for x ω/5 < /, for x ω/5 < 3/, (x ω/5 /) in between. Compressive solution burgers.dat using ::3 burgers.dat using :: projection of the initial data exact solution t =. burgers.dat using ::3 burgers.dat using :: moment solution t =. new method t =. /DFG-CNRS Workshop 6 p. / 3

24 Non intrusive moments with quadrature points 3 Use ω =, 4 ω = and ω3 = 3 4. Then reconstruct burgers.dat using :: burgers.dat using :: reconstructed initial data reconstructed solution at t =. The test is performed with the compressive initial data. Initialization of the shock problem is ambiguous at t =. /DFG-CNRS Workshop 6 p. / 3

25 Larger N (with a moment method, Poette PHD) /DFG-CNRS Workshop 6 p. / 3

26 The kinetic formulation of conservation laws is a convenient tool for the analysis of conservation laws with intrusive uncertainties. The theory of convolution-based method is OK, but the practice not clear due the spurious damping. The alternative is kinetic (Maxwellian plus ) which are at their infancy. The theory is full of open problems : existence, uniqueness, error estimates,...and connection with optimal. Two issues were not addressed in this talk : interpretation of the results in terms of probability ; the curse of dimension ω R large. /DFG-CNRS Workshop 6 p. 3 / 3