A Source Reconstruction Formula for the Heat Equation Using a Family of Null Controls

Transcription

1 A Source Reconstruction Formula for the Heat Equation Using a Family of Null Controls Axel Osses (Universidad de Chile) DIM - Departamento de Ingeniería Matemática CMM - Center for Mathematical Modeling (UMI 261 CNRS) Center for Climate and Resilience Research Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Santiago. Séminaire GdT Contrôle - LJLL-Paris 6, le 4 Août 213

2 Outline 1 Motivation 2 Source recovering and null controls 3 Our reconstruction formula 4 Numerical Results 2/35

3 Motivation Source inverse problem Inverse problem: To identify (uniqueness, stability, reconstruction) the source f (x, t) in the heat equation u t u = f (x, t) in Ω (, T ) u() = in Ω u = on Γ (, T ) from local measurements of the solution (boundary or internal values). Counterexample: u = a(t)φ(x), φ Cc (ω), ω strictly included in Ω, then u t u = f = a φ a φ Cc (ω) for each t, so we obtain null data outside ω. 3/35

4 Motivation (uncomplete) Review (focused on reconstruction/control) Yamamoto 1995: f = σ(t)f (x) reconstruction of f (x) for wave equation using exact controls (exact controllability, Volterra eqns). Avdonin-Belishev-Rozhkov-Yu 1997: inverse main coefficient problem for the heat equation (boundary control method). Cannon-DuChateau 1998, Boulakia-Grandmont-O. 29: f = f (u) structural identification from Cauchy data in reaction-diffusion eqns (data-parameters integral formula/carleman). Hettlich-Rundell 21: f = χ D identification of spatially discontinuous sources (domain derivative method). Yamatani-Ohnaka 1997, El Badia-Ha Duong 22: f = N j=1 p jδ xj,t j identify pointwise sources in heat eqn (asymptotic behavior of backward heat solutions/exact controllability). El Badia-Ha Duong-Hamdi 25: 1-d heat-transport from internal up-downstream strategic pointwise data (strategic from control theory). Ikehata 26. f (x, T ) when and where the source fistly appeared (plane solutions for the backward heat equation, indicator fncs). 4/35

5 Motivation Some applications (which motivated us...) Some applications: varying sources): (pointwise, distributed sources, stationnary, time Pollution detection and emission identification in atmospheric chemistry (world, regional, city scales) c.f. Newsam-Enting 1988, Enting 22, Saide, Bocquet, O., Gallardo 29. Water pollution detection in rivers, lakes, c.f. Okubo 198, Linfield Optimal network monitoring design/observation systems c.f. Rodgers 2, O., Faundez, Gallardo 213 DADA: data assimilation/detection and attribution, c.f. Puel 22, Garcia, O., Puel 211, Hannart 212. Other: thermography, heat and phase-transition inverse problems, Homberg, Lu, Sakamoto, Yamamoto, /35

6 Motivation World: estimations of CO2 emissions (28-29) 6/35

7 Motivation World: estimations of CO2 emissions (28-29) Observations: Ibuki-GOSAT satellite (29) 7/35

8 Motivation Regional: nuclear/mining emissions, when and where Observations: OMI satellite (27) 8/35

9 Motivation City scale: air-quality forecasting (Santiago, Chile) f (x) (space variation) σ(t) (time variation) 9/35

10 Motivation City scale: air-quality forecasting (Santiago, Chile) diffusion dominated (morning) transport dominated (afternoon) 1/35

11 Motivation City scale: air-quality forecasting (Santiago, Chile) % correction % correction (morning) (afternoon) 11/35

12 Source recovering and null controls Statement of our inverse problem Our inverse problem: Given an (small) observatory O Ω, T >, σ(t) known, to identify the source f (x) in u t u = σ(t)f (x) in Ω (, T ) (1) u() = in Ω u = on Γ (, T ) from local measurements of the heat solution u O (,T ). We are interested in: uniqueness and stability of f (x) w.r.t. measurements. reconstruction algorithm for f (x) using null controls. 12/35

13 Source recovering and null controls What are null controls? A null control is a locally supported function v that drives the backward heat equation: ϕ t ϕ = v O (,τ) in Ω (, τ) (2) ϕ(τ) = ϕ in Ω ϕ = on Γ (, τ). exactly to zero in finite time τ > from ϕ(τ) = ϕ to ϕ() =. It is possible to show (for instance using a global Carleman inequality [Fursikov-Imanuvilov 1996]) that there exists a minimal norm null control v (τ) such that ( ) v (τ) C1 L2 (O (,τ)) C exp ϕ L2 (Ω). τ 13/35

14 Source recovering and null controls Example: null controlled eigenfrequencies control/observation 14/35

15 Source recovering and null controls Example: null controlled eigenfrequencies 1 L 2 norm initial condition non controled solution at final time controled solution at final time frequency number 15/35

16 Source recovering and null controls A much simpler problem: recover final conditions Problem 1: In order to explain the relationship between null controls and the inverse source problem we consider here, let us start with the simpler problem: u t u = in Ω (, T ) (3) u() = u in Ω u = on Γ (, T ). If we want to recover u L 2 (Ω) from u O (,T ) without any other a priori information is severily ill posed (see example later). Nevertheless, recover u(t ) for any T > from u O (,T ) and without knowing u is well posed. 16/35

17 Source recovering and null controls Null controllability and recovery Multiplying u by the null controlled solution ϕ we obtain T O uv (τ) dxdt = u(t ) ϕ(t ) Ω }{{} ϕ dx } {{ } duality + u ϕ() dx Ω }{{} In this way we derive the following reconstruction formula that has been used in data assimilation problems [Puel 22,García-Osses-Puel 21]: Ω T u(t )ϕ dx = uv (T ) dxdt. } O {{ } C It does not depend on u and only in local observations. From Carleman we obtain Lipschitz stability: u(t ) L2 (Ω) Ce C1/T u L2 (O (,T )). 17/35

18 Source recovering and null controls Another simpler inverse source problem Problem 2: Let us now consider another very simple source inverse problem. To identify f (x) in u t u = f (x) in Ω (, T ) (4) u() = in Ω u = on Γ (, T ) from local observations u O (,T ). We can reduce this problem to the previous one by taking time derivative: w = u t w t w = in Ω (, T ) (5) w() = f (x) in Ω w = on Γ (, T ) Since we cannot recover w() from the local measurements w O (,T ) = u t O (,T ), our problem is to recover f appearing in the final condition w(t ) = u(t ) + f (x) 18/35

19 Source recovering and null controls Another recovery formula w(t ) = u(t ) + f (x) So, by multiplying the equation of w by the controlled solution ϕ as before and using duality, we obtain the recovery formula: Ω T f (x)ϕ dx = u(t )ϕ dx u t v (T ) dxdt. Ω } {{ } } O {{ } L C An extra annoying local in time measurement of u(t ) appears!! The corresponding stability inequality is easily obtained from Carleman: f L2 (Ω) C u(t ) L2 (Ω) + Ce C1/T u t L2 (O (,T )). 19/35

20 Source recovering and null controls The annoying measure u(t ) We cannot drop u(t ) in the previous inequality due to high frequencies. Example: O (, π) = Ω. Take u n (x, t) = 1 n 2 2 π (1 e n2t ) sin(nx) that satisfies 2 ut n u n = π sin(nx) := f n(x) in (, π) (, T ) u n () = in (, π), u(, t) = u(π, t) = on (, T ). Then f n L 2 (,π) = 1 u t L 2 (O (,T )) 1 n 2, n. π but u(t ) L 2 (,π) = (1 e n2t ) 1. 2/35

21 Source recovering and null controls Log estimates This shows that if f is only L 2 (Ω) a log estimate for α (, 1] does not hold. f L 2 (Ω) C log ut L α 2 (O (,T )) If f is more regular, say f D(( )ε ) M for some ɛ (, 1), you still have logarithmic conditional stability: f L2 (Ω) C M,ε log u t L2 ε (,T ;L 2 1 ε (O)) as proved by [Garcia-Takahashi, 211]. The first estimates of this type were obtained by [Li-Yamamoto-Zou 29] for recovering regular u. 21/35

22 Our reconstruction formula Return to the problem Our problem: In order to recover f (x) in u t u = σ(t)f (x) in Ω (, T ) (1) u() = in Ω u = on Γ (, T ) from u O (,T ) we reduce the problem by changing variables: u = t Then if w solves σ(t τ)w(τ)dτ := Kw (notice w = u t if σ = 1) w t w = in Ω (, T ) (6) w() = f (x) in Ω w = on Γ (, T ) u solves (1). So this time we will try to recover f from: σ()w(t ) = u(t ) + σ(t )f (x) T σ (T τ)w(τ)dτ 22/35

23 Our reconstruction formula Volterra equation and duality The definition of K naturally leads to solve a Volterra equation of second kind. Given v L 2 (, T ; L 2 (O)),!θ H 1 (, T ; L 2 (O)) such that θ(t ) = and K θ := σ()θ t + T t (σ(s t)θ(s) + σ (s t)θ t (s)) ds = v(t) with continuous dependence and w L 2 (, T ; L 2 (O)) (w, K θ) L 2 (,T ;L 2 (O)) = (Kw, θ) H 1 (,T,L 2 (O)). This duality was previously used by [Yamamoto 1995] to derive a reconstruction source formula for the wave equation. 23/35

24 Our reconstruction formula Intermediate reconstruction formula T σ()w(t ) = u(t ) + σ(t )f (x) σ (T τ)w(τ)dτ By introducing the family of null controls v (τ) controlling from ϕ(τ) = ϕ to ϕ() = and K θ (τ) = v (τ), Kw = u we have σ(t ) f (x)ϕ dx = = = Ω Ω Ω u(t )ϕ + σ() u(t )ϕ σ() Ω T wv (T ) O } {{ } (w,k θ (T ) ) T w(t )ϕ + T σ (T τ) σ (T τ) T Ω w(τ)ϕ dτ wv (τ) dt dτ O }{{} (w,k θ (τ) ) 24/35

25 Our reconstruction formula Intermediate reconstruction formula Theorem 1 Let σ W 1, (, T ), σ(t ) then ϕ L 2 (Ω) f ϕ = σ(t ) 1 ( u(t ), ϕ ) L 2 (Ω) σ()σ(t ) 1 (u, θ (T ) ) H Ω }{{} 1 (L 2 (O)) }{{} L C 1 T σ(t ) 1 σ (T τ)(u, θ (τ) ) H1 (L 2 (O))dτ } {{ } C 2 where θ (τ) are the Volterra solutions associated to the null controls v (τ) for τ (, T ]. Moreover, if σ (t) = for t (T ε, T ] or σ (t) = e C2/(T t) ρ(t), t (, T ), ρ L (, T ) for large C 2 then f L2 (Ω) C( u(t ) L2 (Ω) + u H1 (,T ;L 2 (O))). with C O(e C1/ε ) or C O(1). 25/35

26 Our reconstruction formula Eliminating u(t )... Fortunately, in the particular case that ϕ = ϕ k the eigenfunctions of the Laplacian f ϕ k = σ(t ) 1 ( u(t ), ϕ k ) L 2 (Ω) σ()σ(t ) 1 (u, θ (T ) k ) H Ω }{{} 1 (L 2 (O)) }{{} L k T σ(t ) 1 σ (T τ)(u, θ (τ) k ) H1 (L 2 (O))dτ }{{} C 2k it is possible to replace the annoying first term by using the alternative expression: f ϕ k = ( u(t ), ϕ k) L2 (Ω) T Ω λ k e λ k (T s) σ(s)ds where λ k > are the corresponding eigenvalues. C 1k 26/35

27 Our reconstruction formula Main result: new reconstruction formula Theorem 2 Let f L 2 (Ω) and let {(λ k, ϕ k )} k be the eigenvalues and L 2 -orthonormal eigenvectors of the in Ω with homogeneous Dirichlet boundary conditions. Given σ W 1, (, T ), σ(t ) such that a k := 1 λ T k e λ k (T s) σ(s)ds, (1) σ(t ) for some k, then we have the reconstruction formula: f ϕ k = a 1 k (C 1k + C 2k ), (2) Ω where C 1k = C 1 (ϕ k ) and C 2k = C 2 (ϕ k ) only depend on the local observations (u, u t ) O (,T ) of the solution of (1). 27/35

28 Our reconstruction formula Examples of functions for which a k k a) σ = σ constant (in this case a k = e γλ k T > ), b) σ a non negative and increasing function (in this case a k e γλ k T ) c) σ = σ +b cos(ωt), σ constant, b, and ω R\D, where D is some discrete set that results from the intersections of tan(ωt ) (solid line) and a sequence of functions associated to each eigenfrequency (dotted lines). There are no intersections in regions where tan(ωt ) and cos(ωt ) have opposite signs (arrows) π/t.5π/t π/t 1.5π/T 2π/T 2.5π/T 3π/T 3.5π/T 4π/T 4.5π/T 5π/T 28/35

29 Numerical Results Choices of σ(t) Numerical results: Three different choices for the time dependency of the source σ(t) σ (t) σ (t) a) σ 1 (t) σ 1 (t) b) c) case a) case b) case c) σ 2 (t) σ 2 (t) 29/35

30 Numerical Results Main features Null controls pre-calculated (independent of the unknown source). This involves hard numerics, even in 1-d! cf. A. Münch, Zuazua, 21, Fernández-Cara, Münch 212, 213, Martin, Rosier, Rouchon, 213. Solution measured in 1/3 of the spatial domain with 5% noise. Source f reconstructed from our proposed formula based on null controls for σ(t) in cases a) constant, b) increasing and c) oscillating. We optimize w.r.t. the cutoff frequency. Next fig. d) is an optimization of case b) starting from f by modifying only the relevant frequencies (p = 2) or sparse (p = 1) using proxy algorithms. min f = K k= f k ϕ k u meas u(f ) 2 H 1 (,T,L 2 (O)) + β f f 2 L p (Ω) 3/35

31 Numerical Results Ω = (, 1) 2, O = (, 1) (.3,.7). 5% noise. 47 eigen. 31/35

32 Numerical Results Contribution of each term coefficients.2.4 C 1 C 2 C 1 +C 2.6 estimated real frequency number 32/35

33 Numerical Results Zoom on crux source: control terms 33/35

34 Numerical Results Zoom on crux source: optimized frequencies coefficients.1.2 C 1 C C 1 +C 2 estim real opt frequency number 34/35

35 Numerical Results Zoom on crux source: optimization 35/35

36 References: ** G. García, A. Osses, M. Tapia, A heat source reconstruction formula from single internal measurements using a family of null controls, J. Inverse Ill-Posed Probl., Ahead of print, Published Online: January 213 **A. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, Lecture Notes, Research Institute of Mathematics, Seoul National University, Korea, **G. García, A. Osses and J.-P. Puel, A null controllability data assimilation methodology applied to a large scale ocean circulation model, M2AN 45(2), , 211. **G. García, T. Takahashi, Inverse problem and null-controllability for parabolic systems. J. Inv. Ill-Posed Problems 19 (211), **J. Li, M. Yamamoto and J. Zou, Conditional stability and numerical reconstruction of initial temperature. Commun. Pure Appl. Anal. 8 (29), no. 1, **A. Münch and E. Zuazua, Numerical approximation of null controls for the heat equation : ill-posedness and remedies. Inverse Problems 26(8) 8518 (21). **J.-P. Puel, Une approche non classique d un problème d assimilation de données, C. R. Math. Acad. Sci. Paris, 335 (22) **J.-P. Puel, A nonstandard approach to a data assimilation problem and Tychonov regularization revisited, SIAM J. Control Optim (29), **M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), /35