Identification of a memory kernel in a nonlinear parabolic integro-differential problem

Transcription

1 FACULTY OF ENGINEERING AND ARCHITECTURE Identification of a memory kernel in a nonlinear parabolic integro-differential problem K. Van Bockstal, M. Slodička and F. Gistelinck Ghent University Department of Mathematical Analysis Numerical Analysis and Mathematical Modelling Research Group 15th International Conference Computational and Mathematical Methods in Science and Engineering, July 6-10, 2015, Rota, Cadiz - Spain.

2 Outline Problem setting Variational formulation Time discretization Numerical experiment Conclusion 2 / 25

3 Ω R d, d 1: bounded domain with Lipschitz continuous boundary Γ = Ω, final time T. Determine the solution u(x, t) and the convolution kernel K(t) such that t u ( β(u)) + K u =?, in Ω [0, T ], β(u) ν = g, on Γ [0, T ], u(x, 0) = u 0 (x), in Ω, when an additional global measurement u(x, t)dx = m(t), t [0, T ] is satisfied. The sign denotes the convolution product Ω (K u(x)) (t) := t 0 K(t s)u(x, s)ds, (x, t) Ω [0, T ]. 3 / 25

4 Such type of integro-differential problems arise in the theory of reactive contaminant transport [Delleur, 1999] and in the modelling of phenomena in viscoelasticity [MacCamy, 1977]. [De Staelen and Slodička, 2015] studied the reconstruction of K based on the same measurement in the semilinear equation t t u u + K(t)h + K(t s)u(x, s) ds = f (u, u). 0 Main idea: measure the equation into space, i.e. m (t) + g(t) + K(t) h(t) + (K m)(t) = f (u(t), u(t)). Γ Ω Ω 4 / 25

5 The measured problem m (t) + g(t) + K(t) h(t) + (K m)(t) = f (u(t), u(t)) Γ Ω Ω (MP) together with the variational formulation for φ H 1 (Ω) ( t u, φ) + ( u, φ) + (g, φ) Γ + K(t)(h, φ) + (K u, φ) = (f (u, u), φ) (P) represent the variational formulation of the inverse problem. The inverse problem is reformulated into a direct problem! Existence and uniqueness of a solution is proved using Rothe s method [Kačur, 1985]. Uniform boundedness of K is crucial into the analysis (to obtain global in time solvability). 5 / 25

6 Uniform boundedness of K follows from (MP) and Grönwall s lemma: K(t) h(t) f (u(t), u(t)) + (K m)(t) + m (t) + g(t) Ω Ω C + t 0 K(s) ds if f : R R is uniformly bounded, g C ( [0, T ], L 2 (Γ) ) and m C 1 ([0, T ]). max K(t) C if min (h(t), 1) ω > 0. t [0,T ] t [0,T ] In this talk, the crucial K(t)h-term is skipped out of the PDE. What are the implications? Γ 6 / 25

7 Measure the problem: m (t) + g(t) + (K m)(t) = Γ Idea: take the time derivative of this equation to obtain K(t) seperately, i.e. m (t) + t g(t) + K(t)m(0) + (K m )(t) = t?. Γ Ω What is possible for?? f (u). We make a safe choice for the right-hand side?, i.e. t 0 f (u(, s)) ds. Ω?. We have made the problem nonlinear by introducing the possible nonlinear function β : R R. 7 / 25

8 Determine the solution u(x, t) and the convolution kernel K(t) such that t t u ( β(u)) + K u = f (u(, s)) ds + F, in Ω [0, T ], 0 β(u) ν = g, on Γ [0, T ], u(x, 0) = u 0 (x), in Ω, u(x, t)dx = m(t). Ω 8 / 25

9 The coupled direct variational problem is given by m (t)+k(t)m(0)+(k m )(t) = (f (u(t)), 1)+(F (t), 1) (g (t), 1) Γ (MP2) and ( t u(t), φ) + ( β(u(t)), φ) + ((K u)(t), φ) ( t ) = f (u(s)) ds, φ + (F (t), φ) (g(t), φ) Γ, (P2) where F (t) := t F (t) and g (t) := t g(t). 0 9 / 25

10 Rothe s method [Kačur, 1985]: divide [0, T ] into n N equidistant subintervals (t i 1, t i ] for t i = iτ, where τ = T /n < 1 and for any function z z i z(t i ), t z(t i ) δz i := z i z i 1 τ. Based on (P2) and (MP2), the following decoupled system for approximating the unknowns (K, u) at time t i, 1 i n, is proposed and ( i ) (δu i, φ) + ( β(u i ), φ) + K k u i k τ, φ m i + K i m(0) + i k=1 = k=1 ( i 1 ) f (u k )τ, φ + (F i, φ) (g i, φ) Γ (DP2i) k=0 K k m i kτ = (f (u i 1 ), 1) + (F i, 1) (g i, 1) Γ. (DMP2i) 10 / 25

11 We assume that f : R R bounded, β : R R is everywhere differentiable and satisfies β(0) = 0, 0 < β 0 β (s) β 1, s R, u 0 L 2 (Ω), g C 1 ( [0, T ], L 2 (Γ) ), F C 1 ( [0, T ], L 2 (Ω) ), m C 2 ([0, T ]) with m(0) 0. and refer to these conditions as ( ). 11 / 25

12 { } m(0) Set τ 0 = min 1, 2 m. Then for any τ < τ 0, we get by the triangle (0) inequality that m(0) + m (0)τ m(0) m (0) τ m(0) 2 > 0. For each i {1,..., n}, the following recursive deduction can be made: Let u 0,..., u i 1 L 2 (Ω) and K 1,..., K i 1 R be given. Then, (DMP2i) implies the existence of a unique K i R such that K i [m(0) + m (0)τ] = (f (u i 1 ), 1) + (F i, 1) (g i, 1) Γ i 1 k=1 K k m i kτ m i. Monotone operator theory gives the existence of a unique solution u i H 1 (Ω) to problem (DP2i) when the assumptions ( ) are fulfilled [Vainberg, 1973]. 12 / 25

13 A priori estimates Lemma Let ( ) be satisfied. Then, there exists a positive constant C such that for any τ < τ 0 holds that max i=1,...,n K i C. Lemma Let ( ) be satisfied. Then there exist positive constants C such that for any τ < τ 0 holds that n max u j 2 + β(u i ) 2 τ C 1 j n and max β(u j) C 0 j n i=1 and n i=1 u i 2 H 1 (Ω) τ C. 13 / 25

14 A priori estimates Lemma Let ( ) be satisfied and u 0 H 1 (Ω). Then there exist positive constants C such that for any τ < τ 0 holds that n i=1 δu i 2 τ + max 1 j n β(u j) 2 + j β(u i ) β(u i 1 ) 2 C i=1 and j max u j 1 j n H 1 (Ω) C and u i u i 1 2 C. i=1 14 / 25

15 Rothe functions Piecewise constant and linear in time spline of the solutions u i, i = 1,..., n. u n u n n 2 n 1 n step (a) n 2 n 1 n step (b) Figure : Rothe s piecewise constant function u n (a) and Rothe s piecewise linear in time function u n (b). Similarly, we define K n, F n, F n, g n, g n, m n and m n. 15 / 25

16 Using these so-called Rothe s functions, (DP2i) and (DMP2i) can be rewritten on the whole time interval as ( t τ = i and t τ = i 1 when t (t i 1, t i ]) and t τ ( t u n (t), φ) + ( β(u n (t)), φ) + K n (t k )u n (t t k )τ, φ k=1 t τ = f (u n (t k ))τ, φ + ( F n (t), φ ) (g n (t), φ) Γ k=1 k=0 t τ m n(t) + K n (t)m(0) + K n (t k )m n(t t k )τ We want to pass to the limit n (term by term). = (f (u n (t τ)), 1) + ( F n(t), 1 ) ( g n(t), 1 ) Γ. 16 / 25

17 Theorem (Existence and uniqueness) Suppose that the conditions ( ) are fulfilled. Moreover, assume that u 0 H 1 (Ω) and let f : R R be global Lipschitz continuous. Then, there exists a unique weak solution K, u to the problem (P2)-(MP2), where K L 2 (0, T ) and u C ( [0, T ], L 2 (Ω) ) L 2 ( (0, T ), H 1 (Ω) ) with t u L 2 ( (0, T ), L 2 (Ω) ). Proof. Uses Lemma of [Kačur, 1985] (compactness argument: H 1 (Ω) L 2 (Ω)). Note that by the a priori estimates holds that for every t [0, T ] t τ k=1 τ t τ K n(t k )u n(t t k )τ = (K n u n)(t) + K n(s)u n(t s) ds = (K n u n)(t) + O (τ) t and t τ t f (u n(t k ))τ = f (u 0)τ + k=0 0 t f (u n(s)) ds f (u n(s)) ds = τ t τ t f (u n(s)) ds + O (τ) / 25

18 We have u n u in C ( [0, T ], L 2 (Ω) ), u n u in L 2 ( (0, T ), L 2 (Ω) ). Only weak convergence of the Rothe functions K n to K is proved up to now (K n K). Extra assumptions are needed for the strong convergence. Lemma Let the assumptions ( ) be fulfilled and u 0 H 1 (Ω). Moreover, assume that β(u 0 ) H(div; Ω), g C 2 ( [0, T ], L 2 (Γ) ), F C 2 ( [0, T ], L 2 (Ω) ), m C 3 ([0, T ]) and f : R R is global Lipschitz continuous. Then, there exist positive constants C and τ 0 such that for all τ < τ 0 holds that n δk i 2 τ C. i=1 By the Arzelà-Ascoli theorem [Rudin, 1987, Theorem 11.28], {K n } converges uniformly on [0, T ] to K, i.e. K C([0, T ]). 18 / 25

19 Error estimates (speed of convergence) Theorem Let the assumptions of the previous lemma be fulfilled. Then there exist positive constants C and τ 0 such that for all τ < τ 0 holds that T 0 K n (t) K(t) T 2 dt + u n (t) u(t) 2 Cτ 2. 0 The convergence of the numerical approximations (K n, u n ) to the exact solution (K, u) is optimal in time. 19 / 25

20 Numerical experiment: setting Ω = [0, 1]. The forward coupled problems in this procedure are discretized in time according to the backward Euler method with timestep 2 j T, j = 2,..., 8. At each time-step, the resulting elliptic problems are solved numerically by the finite element method (FEM) using first order (P1-FEM) Lagrange polynomials for the space discretization. A fixed uniform mesh consisting of 100 intervals is used. At each timestep, the nonlinearity β(u i ) is approximated by β (u i 1 ) u i. The error on K is denoted by E Kex (τ) = T 0 K n (t) K ex (t) 2 dt. Implementation: in FEniCS [Logg et al., 2012]. 20 / 25

21 β linear T = 1, f (s) = β(s) = s + 1, m(t) = 4/3 t 2 + 4/3 t + 4/3, u ex (x, t) = ( 1 + t + t 2) ( 1 + x 2), K ex (t) = exp(t) log 2 E Kex = log 2 τ Kernel K(t) log 2 E Kex t exact solution τ = 2-3 τ = 2-5 τ = 2-7 (a) Kernel reconstruction log 2 τ (b) Error E Kex (τ) 21 / 25

22 β nonlinear T = 1 2, f (s) = s + 5, β(s) = s2 + s, m(t) = π t3 + π t t 3 + π t + 2 t 2 + π + 2 t + 2, π u ex (x, t) = ( 1 + t + t 2 + t 3) (1 + sin (π x)), K ex (t) = sin(2πt) log 2 E Kex = log 2 τ Kernel K(t) log 2 E Kex t exact solution τ = 2-3 τ = 2-5 τ = 2-7 (c) Kernel reconstruction log 2 τ (d) Error E Kex (τ). 22 / 25

23 Conclusion: A nonlinear parabolic problem of second order with an unknown solely time-dependent convolution kernel is considered. A numerical scheme based on Backward Euler s method together with a time-discrete convolution is presented in oder to reconstruct the unknown convolution kernel based on an integral overdetermination. The convergence is of first order in time: Kn (t) K ex (t) L 2 ((0,T )) O(τ). Numerical experiments support the theoretically obtained results. 23 / 25

24 References I De Staelen, R. and Slodička, M. (2015). Reconstruction of a convolution kernel in a semilinear parabolic problem based on a global measurement. Nonlinear Analysis: Theory, Methods & Applications, 112(0): Delleur, J. W. (1999). The Handbook of Groundwater Engineering. CRS Press. Kačur, J. (1985). Method of Rothe in evolution equations, volume 80 of Teubner Texte zur Mathematik. Teubner, Leipzig. Logg, A., Mardal, K.-A., Wells, G., et al. (2012). Automated Solution of Differential Equations by the Finite Element Method. Springer. MacCamy, R. (1977). A model for one-dimensional, nonlinear viscoelasticity. Q. Appl. Math., 35: Rudin, W. (1987). Real and complex analysis. McGraw-Hill. 24 / 25

25 References II Vainberg, M. M. (1973). Variational method and method of monotone operators in the theory of nonlinear equations. translated from russian by a. libin. translation edited by d. louvish. A Halsted Press Book. New York-Toronto: John Wiley & Sons; Jerusalem- London: Israel Program for Scientific Translations. xi, 356 p. (1973). 25 / 25