A Carleman estimate and the balancing principle in the Quasi-Reversibility method for solving the Cauchy problem for the Laplace equation

A Carleman estimate and the balancing principle in the Quasi-Reversibility method for solving the Cauchy problem for the Laplace equation Hui Cao joint work with Michael Klibanov and Sergei Pereverzev supported by Austrian funds FWF Project P2235-N18 Conference on Applied Inverse Problems 29 July 2-24, 29 H Cao (RICAM) July 29 1 / 25

Overview 1 Carleman estimate and quasi-reversibility method 2 Parameter choice: Balancing principle 3 Numerical tests H Cao (RICAM) July 29 2 / 25

Overview 1 Carleman estimate and quasi-reversibility method 2 Parameter choice: Balancing principle 3 Numerical tests H Cao (RICAM) July 29 3 / 25

The classical Cauchy problem Find u H 2 (Ω), such that u(x) =, x Ω, u(x) = h(x), x Γ Ω, u(x) = g(x), n x Γ Ω. Here Ω R n is bounded, connected. Γ Ω, Γ C 2, meas(γ) >. H Cao (RICAM) July 29 4 / 25

Problem reformulation for quasi-reversibility method: Suppose there exists a function F H 2 (Ω), such that F Γ = h, F n Γ = g. For F = F, find u G such that u = F, where G = { v H 2 (Ω) : v Γ =, v n Γ = } Then ũ = F u solves the Cauchy problem. We assume that the function F is given with an error: F δ L 2 (Ω) and F δ F L2 (Ω) δ. H Cao (RICAM) July 29 5 / 25

Quasi-reversibility method for noisy data Lattès and Lions (1967): J ε (v) = v F δ 2 L 2 (Ω) + ε v H 2 (Ω). u δ ε = arg min{j ε (v), v G }. Similar discussion can be also found in [L Bourgeois, 26, Inverse Problems]. H Cao (RICAM) July 29 6 / 25

u u δ ε Benefits from Carleman estimates u uε }{{} + uε δ uε }{{} approximation error noise propagation error (convergence rate) (stability bound) Let u ε, u δ ε be the minimizers of J ε (v) for exact data and noisy data respectively. Carleman estimates allow an improvement of the classical stability bound in a subdomain Ω σ Ω. Namely, u δ ε u ε H 2 (Ω) K δ ε, u δ ε u ε H 2 (Ω σ) K δ ε ν, where ν = 1 2 σ 2(m 2σ), (note that < ν < 1 2 ), m = max{ϕ(x), x Ω }, Ω σ = {x Ω : ϕ(x) > σ}, and ϕ is a function appearing in Carleman estimate. H Cao (RICAM) July 29 7 / 25

Carleman estimates ϕ: u H 2(Ω ) and λ λ = λ (Ω ) > 1, Ω = {x Ω : ϕ(x) > } ( u) 2 e 2λϕ dx C λ u 2 e 2λϕ dx Ω Ω + C λ 3 u 2 e 2λϕ dx + C (D α u) 2 e 2λϕ dx. Ω λ Illustration for ϕ and Ω σ in 2-D Ω α =2 ϕ(x 1, x 2 ) = ( x 2 + x 1 2 β 2 + 1 ) µ ( ) 1 µ. 4 2 Ω σ = {x 2 >, ϕ(x 1, x 2 ) > σ}, is a region between a parabola and x 1 -axis.! "!1!.8!.6!.4!.2.2.4.6.8 1 Figure: Ω = [ 1, 1] [, 1] µ = 1, β = 1.2, σ =.1 H Cao (RICAM) July 29 8 / 25

The use of a Carleman estimate in Q-R method Theorem If d > 3, σ > are such that then where ν = 1 2 Ω dσ := {x Ω : ϕ(x) > dσ}, u ε u δ ε H 2 (Ω dσ ) K δ ε ν, u u ε H 2 (Ω dσ ) K 2 u H 2 (Ω) ε1/2 ν, σ 2(m 2σ), m = max{ϕ(x), x Ω }, K = K (σ, Ω ). Remark 1: In R 2 and R 3 the bounds above are valid for C-norms. Remark 2: A convergence rate O(ε 1/2 ν ) has been obtained without imposing source conditions. H Cao (RICAM) July 29 9 / 25

The use of a Carleman estimate in Q-R method Theorem If d > 3, σ > are such that then where ν = 1 2 Ω dσ := {x Ω : ϕ(x) > dσ}, u ε u δ ε H 2 (Ω dσ ) K δ ε ν, u u ε H 2 (Ω dσ ) K 2 u H 2 (Ω) ε1/2 ν, σ 2(m 2σ), m = max{ϕ(x), x Ω }, K = K (σ, Ω ). A priori choice ε = ε ap = O(δ 2 ) leads to K 2 u H 2 (Ω) ε1/2 ν K δ ε ν. Numerical experiments show that such a choice is too rough. H Cao (RICAM) July 29 1 / 25

A priori choice ε opt Let ψ(ε) be the least non-decreasing majorant of u u ε, i.e., u u ε ψ(ε). It is sufficient to choose ε = ε opt balancing ψ(ε) and K δ/ε ν, i.e. this ε opt solves the equation ψ(ε) = K δ ε ν and can be represented as ε opt = θ 1 (K δ), where θ(ε) = ψ(ε)ε ν. Then u u δ ε opt 2ψ(ε opt ) = 2ψ(θ 1 (K δ)). Accuracy estimation u δ ε u ψ(ε) + K δ ε ν. H Cao (RICAM) July 29 11 / 25

Overview 1 Carleman estimate and quasi-reversibility method 2 Parameter choice: Balancing principle 3 Numerical tests H Cao (RICAM) July 29 12 / 25

Balancing principle If K is known... u uε δ u uε }{{} ψ(ε) + u δ ε u ε }{{} K δ/ε ν { ε(k ) = max ε n Υ N : uε δ n uε δ m C 4K δ ε ν, m =, 1,..., n 1 m where Υ N = {ε n = ε q n, n =, 1,..., N }. },! opt $(!) K"/! # (! m! n uε δ n uε δ m u uε δ n + u uε δ m ψ(ε n ) + K δ ε ν n ) + ψ(ε m ) + K δ ε ν m H Cao (RICAM) July 29 13 / 25

Application of the balancing principle to choose ε Assumptions: Two-sided estimation for stability bound ck δ ε ν u ε u δ ε C K δ ε ν, < c < 1 There are two adjacent terms k l, k l+1 K p := {k j = k p j, j =, 1,..., M } such that k l ck < K k l+1. The adaptive strategy selects a regularization parameter from Υ N. Let { } ε(k j ) = max ε n Υ N : uε δ n uε δ δ m C 4k j, m =, 1,..., n 1, ε ν m and ε = ε(k i ) = min { ε(k j ) : ε(k j ) ε ( 3(p 2 + 1) p 1 ) 1/ν }. ε(k l ) ε = ε(k i ) ε(k l+1 ).The final regularization parameter is next to ε, i.e. ε + = ε(k i+1 ). It does not require a priori knowledge of K and ψ. H Cao (RICAM) July 29 14 / 25

Theorem The accuracy of the approximation solution uε δ + is only by the factor 3p 2 q ν worse than the best bound (2ψ(ε opt )), i.e. u u δ ε + C 6p 2 q ν ψ(ε opt ) = 6p 2 q ν ψ(θ 1 (K δ)). Literatures on Balancing principle: [Pereverzev S V & Schock E, 25, SIAM J: Numer. Anal.] [Lazarov R D, Lu S & Pereverzev S V 27, Numer. Math.] [Cao H, Klibanov, & Pereverzev S V, 29, Inverse problems] H Cao (RICAM) July 29 15 / 25

Overview 1 Carleman estimate and quasi-reversibility method 2 Parameter choice: Balancing principle 3 Numerical tests H Cao (RICAM) July 29 16 / 25

Numerical tests for Q-R method and balancing principle Consider { Ω = [ 1, 1] [, 1] and Γ = {(x 1, x 2 ) : x 1 ( 1, 1), x 2 = }. ( Ω σ = (x 1, x 2 ) : ϕ(x 1, x 2 ) = x 2 + x 1 2 β 2 + 1 ) µ ( ) µ 1 > σ} ; 4 2 ( ( ) ) 2 x P = {(x 1, x 2 ) : x 2 >, ϕ 1 (x 1, x 2 ) = x 2 + (1 2δ 2 ) 1 1 1 2δ 1 > }. Parameters involved: µ = 1, β = 1.2, σ =.2; δ 1 =.2, δ 2 =.2. 1.9!! 1.8.7.6.5.4 p.3.2.1 " 3#!1!.8!.6!.4!.2.2.4.6.8 1 Regions described by ϕ and ϕ 1 respectively. H Cao (RICAM) July 29 17 / 25

Numerical tests for Q-R method Examples We test three solutions considered also by Klibanov & Santosa (1991) and by Bourgeois (25): u k (x 1, x 2 ) = cos(γ k x 1 ) cosh(γ k x 2 ), k = 1, 2; γ 1 = π 2, γ 2 = π, and u 3 (x 1.x 2 ) = x 2 1 x 2 + x 3 2 /3. For simulated noise δ = 1 3. H Cao (RICAM) July 29 18 / 25

Numerical tests for Q-R method Parameters set-up for balancing principle: K p = {k j =.4 (1.1) j, j = 1, 2,..., 2}; Υ N = {ε n = 2.5 1 8 (1.3) n, n =, 1,..., 6}. ( 3(p 2 ) 1/ν + 1) Threshold: ε = 3.643 1 4, p 1 since in the considered case m = max{ϕ(x 1, x 2 ), x 2, ϕ(x 1, x 2 ) } = 2, and ν = 1 2 σ 2(m 2σ) = 7, σ =.2, µ = 1, β = 1.2. 16 The sequence of {ε(k j )} 2 j =1 for the case u = u 1: 8.3 1 6, 1.44 1 5, 1.44 1 5, 1.357 1 5, 1.357 1 5, 1.764 1 5, 2.293 1 5, 2.981 1 5, 3.876 1 5, 5.38 1 5, 6.55 1 5, 8.515 1 5, 1.17 1 4, 1.871 1 4, 3.162 1 4, 5.343 1 4, 9.3 1 4,.15,.2,.2. K = k 17 =.1838. H Cao (RICAM) July 29 19 / 25

Numerical tests for Q-R method u ε + ε opt err ab C (ε opt ) err ab C (ε + ) err re C (ε + ) u 1 9.3 1 4 5 1 4.33.34.23 u 2 1.17 1 4 2.5 1 5.2787.338.982 u 3.44.3.85.89.1238 Note that for a priori choice ε = ε ap = δ 2 = 1 6 in case of u = u 1 we have err ab C (ε ap ) =.1451. H Cao (RICAM) July 29 2 / 25

Numerical tests for Q-R method T 1 =T 2 =2,!=9.3*1!4 T 1 =T 2 =2,!=1!6.2 12.15.3 15 1.2.1 1 8.1!.1.5 5 6 4!.2!.5 2!.3!.1!5!.4 1.8.6.4.2!1!.5.5 1!.15!.2!1 1.8.6.4.2!1!.5.5 1!2!4!6 The behaviour of the errors for u = u 1. Left: ε = ε + = 9.3 1 4 (chosen by balancing principle); Right: ε = 1 6 (chosen a priori) H Cao (RICAM) July 29 21 / 25

Numerical tests for Q-R method T 1 =T 2 =2,!=1.17*1!4 T 1 =T 2 =2,!=.44 4.12.1 5 3.15 4.8 3 2 2.1.6 1 1.5.4!1!2.2!3 1.8.6.4.2!1!.5.5 1!1!2!.5 1.8.6.4.2!1!.5.5 1!.2!.4 The behaviour of the errors for u 2 and u 3. Parameters ε = ε + are chosen by balancing principle. H Cao (RICAM) July 29 22 / 25

Numerical tests for Q-R method In fact, to realize the quasi-reversibility method numerically, we approximately solve variational equation using finite elements based on tensor products of univariate cubic splines, which correspond to the partition of the domain Ω = [ 1, 1] [, 1] into T 1 T 2 subdomains [ i T1 T 1, (i+1) T 1 ] Ω T 1,T 2 i,j = T 1 j =, 1,..., T 2 1. In the previous tests, T 1 = T 2 = 2. [ j T 2, j +1 T 2 ], i =, 1,..., 2T 1 1, H Cao (RICAM) July 29 23 / 25

Numerical tests for Q-R method u ε + ε opt err ab C (ε opt ) err ab C (ε + ) err re C (ε + ) u 1.12 5 1 4.462.521.327 u 2 1.439 1 4 5 1 5.3516.4333.1286 u 3.57.22.193.246.3412 Table: The results of experiments with the discretization level T 1 = T 2 = 25. u ε + ε opt err ab C (ε opt ) err ab C (ε + ) err re C (ε + ) u 1 5.343 1 4 3 1 4.337.348.235 u 2 5.38 1 5 3 1 6.2618.311.894 u 3.34.28.38.4.558 Table: The results of experiments with the discretization level T 1 = T 2 = 16. H Cao (RICAM) July 29 24 / 25

Numerical tests for Q-R method Some remarks on discretization: It may happen that an additive noise spoils data up to such an extent that for given noisy data original non-discretized problem cannot be solved with a reasonable accuracy; under some moderate discretization such strong additive noise is not completely involved in a computational process, and it allows an approximate solution with a desirable accuracy; by making a discretization finer, one also involves more noise, and the approximation accuracy decreases. The value of the stability power ν = 1 2 2(m 2σ) obtained due to Carleman estimate is rather accurate, since in all tests performed for a larger domain P Ω 3σ it allows a parameter choice leading to the same accuracy level as the optimal (but not numerically feasible) parameter ε = ε opt. σ H Cao (RICAM) July 29 25 / 25