Sparsity regularization of the diffusion coefficient identification problem: well-posedness and convergence rates Pham Quy Muoi† March 13, 2014 †
Department of Mathematics, Danang University of Education - University of Danang 459 Ton Duc Thang, Danang, Vietnam Email:
[email protected] Abstract In this paper, we investigate sparsity regularization for the diffusion coefficient identification problem. Here, the regularization method is incorporated with the energy functional approach. The advantages of our approach are to deal with convex minimization problems. Therefore, the well-posedness of the problem is obtained without requiring regularity property of the parameter. The convexity of regularized problems also allows to use the fast algorithms developed recently. Furthermore, the convergence rates of the method are obtained under a simple source condition. The main results of the paper are the well-posedness and convergence rates of sparsity regularization. We also obtain some new results of the continuity and the differentiability of related operators.
2010 Mathematics Subject Classification: 47A52, 49N30, 49N45 Keywords: Sparsity regularization, Diffusion coefficient identification problem, Nonlinear inverse problems, Well-posedness, Convergence rate.
1
Introduction
The diffusion coefficient identification problem is to identify the coefficient σ in the equation − div (σ∇φ) = y in Ω, φ = 0 on ∂Ω from noisy data φδ ∈ H01 (Ω) of φ such that
∗
φ − φδ 1 ≤ δ. H (Ω)
(1)
(δ > 0)
This problem has attracted great attention of many researchers. For surveys on this problem, we refer to [18, 40, 14, 27, 34, 28, 11, 38, 1, 9] and the references therein. It is well-known that the problem is ill-posed and thus need to be regularized. There have been several regularization methods proposed. Among of them, Tikhonov regularization [18, 13] and the total variational regularization [40, 8] are most popular. In some applications, the coefficient σ ∗ , which needs to be recovered, has a sparse presentation, i.e. the number of nonzero components of σ ∗ − σ 0 are finite in an orthonormal basis (or frame) of L2 (Ω) . The sparsity of σ ∗ − σ 0 promotes to use sparsity regularization. Sparsity regularization has been of interest by many researchers for the last years. The well-posedness and some convergence rates of the method have been analyzed for linear inverse problems [12] as well as for nonlinear inverse problems [17, 6, 37]. Some numerical algorithms have also been proposed [29, 12, 4, 3, 33, 2]. It is shown that sparsity regularization is simple for use and very efficient for inverse problems with sparse solutions. This method has been investigated and applied very successfully to some fields such as for compressive imaging [36, 39] and electrical impedance tomography [23, 16, 24]. Note that it is possible to apply the least squares approach in [17] or more general misfit term in [15] for our problem. However, it is not clear that the solution operator FD (·) y, which maps each parameter σ into the solution of (1), is weakly sequentially closed in L2 (Ω) without additional conditions. Therefore, if the such approaches in [17, 15] are applied, then it needs further conditions. Moreover, the approaches lead
1
to a non-convex minimization problem and the source conditions are difficult to be checked for the problem, see e.g. [18]. To overcome this shortcoming, we use the energy functional approach incorporating with sparsity regularization, i.e. considering the minimization problem min Fφδ (σ) + αΦ σ − σ 0 , (2) σ∈A
where • A is an admissible set defined by A = σ ∈ L∞ (Ω) : λ ≤ σ ≤ λ−1 a.e. on Ω and supp σ − σ 0 ⊂ Ω0 ⊂⊂ Ω .
(3)
Here λ ∈ (0, 1) is a given constant ,σ 0 is the background value of σ, and Ω0 is an open set with the smooth boundary that contained compactly in Ω. • α > 0 is a regularization parameter. • Fφδ (σ) and Φ (ϑ) are defined by Z
2 σ ∇ FD (σ) y − φδ dx, Ω X p Φ (ϑ) := ωk |hϑ, ϕk i| , (1 ≤ p ≤ 2)
Fφδ (σ) :=
(4) (5)
where {ϕk } is an orthonormal basis (or frame) of L2 (Ω) , h·, ·i is the scalar product in L2 (Ω), and ωk ≥ ωmin > 0 for all k. Let us give a short discussion on the admissible set A. The first condition σ ∈ [λ, λ−1 ] is required to obtain the unique existence of the solution of (1). The second condition, supp σ − σ 0 ⊂ Ω0 ⊂⊂ Ω, means that σ − σ 0 has a sparse expansion in a suitable basis {ϕk } [31]. Under this assumption, sparsity regularization should be applied [12, 17]. Note that our assumption is satisfied if the values of the parameter (we need to recover) are known on the boundary ∂Ω. In such a situation any σ 0 ∈ L∞ (Ω) that has the same values of the parameter on the boundary is satisfied. This assumption is often used in many papers such as [40, 26, 18]. Note that the energy functional approach was first introduced by Zou [40] and then was used by Knowles in [26]. However, the authors in those papers did not consider the well-posedness and convergence rates of Tikhonov-type regularization, they have used the energy functional for finding the numerical solution using some numerical algorithms. Recently, Hao and Quyen have used this approach incorporating with either Tikhonov regularization or the total variation regularization for some problems [18, 19, 20, 21] and the well-posedness and convergence rates are obtained under some conditions. This paper is motivated by the numerical results in [29, 31]. In those papers, we have applied some numerical algorithms for problem (2). The algorithms have worked very well and very effective, but the well-posedness of problem (2) has not been analyzed. Here, we will investigate the differentiability of Fφδ (·) and the well-posedness of problem (2) as well as convergence rates. We will prove that problem (2) is convex ∗ 0 and well-posed, and if there exists w∗ such that ξ = (FD (σ + ) y) w∗ ∈ ∂Φ σ + − σ 0 , then the convergence rates
√
p
p δ (1 < p ≤ 2) , Dξ σα,δ , σ + = O (δ) and σα,δ − σ+ =O L2 (Ω)
p are obtained as δ → 0 and α ∼ δ. Note that σα,δ is a minimizer of (2), σ + is a Φ-minimizing solution of the p p diffusion coefficient identification problem and Dξ σα,δ , σ + is the Bregman distance of two elements σα,δ
and σ + [20, 21]. Comparing the standard conditions in [17] and the references therein, our source condition is very simple and does not require the smallness. Furthermore, the objective functional in (2) is now convex and thus its global minimizers are easy to find by available efficient algorithms [29, 31].
2
2
Auxiliary Results
We recall that a function φ in H01 (Ω) is a weak solution of (1) if the identity Z Z σ∇φ · ∇vdx = yvdx Ω
(6)
Ω
holds for all v ∈ H01 (Ω) . If σ ∈ A and y ∈ L2 (Ω) , then there is a unique weak solution φ ∈ H01 (Ω) of (1) [18], which satisfies the inequality 1 (7) kφkH 1 (Ω) ≤ kykL2 (Ω) , C where C > 0 is a constant depending only on Ω and λ. In the next sections, two following inequalities are used: • For any η ∈ H01 (Ω) and σ ∈ A, in virtue of the Poincar´e-Friedrichs inequality we have Z 2 2 σ |∇η| dx ≥ C kηkH 1 (Ω)
(8)
Ω
with C > 0 defined by (7). • For any y ∈ Lr (Ω) , r ≥ 2 with a bounded set Ω ⊂ Rd , we have 1
kykL2 (Ω) ≤ |Ω| 2
− r1
kykLr (Ω) .
(9)
We shall endow the set A with the Lq (Ω) −norm, q ∈ [1, ∞) and define the nonlinear coefficient-tosolution mapping FD (·) y : A ⊂ Lq (Ω) → H01 (Ω) which maps the coefficient σ ∈ A to the solution u = FD (σ) y of problem (1). Before considering sparsity regularization for the problem, we analyze some properties of FD (·) y and Fφδ (·) with respect to the Lq −norm. These properties are needed for investigating the well-posedness and convergence rates of the method as well as numerical algorithms. They are derived by exploiting Meyers’ gradient estimate [30], which has recently been employed by [35, 24]. Theorem 1 (Meyers’ theorem) Let Ω be a bounded Lipschitz domain in Rd (d ≥ 2) . Assume that σ ∈ d L∞ (Ω) satisfies λ < σ < λ−1 for some fixed λ ∈ (0, 1) . For z ∈ (Lr (Ω)) and y ∈ Lr (Ω) , let φ ∈ H 1 (Ω) be a weak solution of the equation − div (σ∇φ) = − div (z) + y in Ω. Then, there exists a constant Q ∈ (2, +∞) depending on λ and d only, Q → 2 as λ → 0 and Q → ∞ as 1,r λ → 1, such that for any 2 < r < Q, φ ∈ Wloc (Ω) and for any Ω0 ⊂⊂ Ω k∇φkLr (Ω0 ) ≤ C 0 kφkH 1 (Ω) + kzkLr (Ω) + kykLr (Ω) , where the constant C 0 depends on λ, d, r, Ω0 and Ω. Using this result, we can show that the mappings FD (·) y and Fφδ (·) are continuous and continuous Fr´echet differentiable on the set A with respect to the Lq -norm. These results are shown in the following lemmas. i 2Q Lemma 2 Let q ∈ Q−2 , ∞ , 1q + 1r = 12 and y ∈ Lr (Ω) . For σ, σ + ϑ ∈ A, we have k∇FD (σ + ϑ) y − ∇FD (σ) ykL2 (Ω) ≤ C kϑkLq (Ω0 ) kykLr (Ω) , where C is a positive constant depending on λ, d, r, Ω0 and Ω. Proof. The weak solution formulas of FD (σ) y and FD (σ + ϑ) y give Z Z σ∇FD (σ) y · ∇vdx = (σ + ϑ) ∇FD (σ + ϑ) y · ∇vdx, ∀v ∈ H01 (Ω) , Ω
Ω
3
i.e.
Z
Z
ϑ∇FD (σ + ϑ) y · ∇vdx, ∀v ∈ H01 (Ω) .
σ∇ (FD (σ + ϑ) y − FD (σ) y) · ∇vdx = − Ω
Ω
Taking v = FD (σ + ϑ) y − FD (σ) y ∈ H01 (Ω) in the last equation, we obtain Z Z 2 σ |∇ (FD (σ + ϑ) y − FD (σ) y)| dx = − ϑ∇FD (σ + ϑ) y · ∇ (FD (σ + ϑ) y − FD (σ) y) dx Ω ZΩ =− ϑ∇FD (σ + ϑ) y · ∇ (FD (σ + ϑ) y − FD (σ) y) dx Ω0
≤ kϑkLq (Ω0 ) k∇FD (σ + ϑ) ykLr (Ω0 ) k∇ (FD (σ + ϑ) y − FD (σ) y)kL2 (Ω) , where 1q + 1r = 12 . The assumption q ∈
i
2Q Q−2 , ∞
implies that r ∈ (2, Q). By Theorem 1, there exist constants
0
C and C such that (7),(9) ≤ C kykLr (Ω) . k∇FD (σ + ϑ) ykLr (Ω0 ) ≤ C 0 kFD (σ + ϑ) ykH 1 (Ω) + kykLr (Ω) It follows that there exists a constant C such that k∇FD (σ + ϑ) y − ∇FD (σ) ykL2 (Ω) ≤ C kϑkLq (Ω0 ) kykLr (Ω) .
Remark 3 1) Note that for σ, σ + ϑ ∈ A and 1 ≤ q1 ≤ q2 , we have |Ω|
−1/q1
−1/q2
kϑkLq1 (Ω) ≤ |Ω|
and q
kϑkL2q2 (Ω) ≤ 2λ−1
q2 −q1
kϑkLq2 (Ω) , q
kϑkL1q1 (Ω) .
This means that the convergence of ϑ to zero with respect to the Lq1 (Ω) −norm and the Lq2 (Ω) −norm are equivalent. q 2) By thei above lemma, FD (·) y is Lipschitz continuous on A with respect to the L (Ω) −norm for q∈
2Q Q−2 , ∞ q
. Furthermore, by the above remark, it implies that FD (·) y is continuous on A with respect
to the L (Ω)-norm for any q ≥ 1. Next, we consider the differentiability of FD (·) y and Fφδ (·) . Here the continuous Fr´echet differentiability is understood as in [24]. i 2Q Lemma 4 Let q ∈ Q−2 , ∞ , 1q + 1r = 12 and y ∈ Lr+ (Ω) with some > 0. Then, the mapping FD (·) y : A ⊂ Lq (Ω) → H01 (Ω) is continuously Fr´echet differentiable on A and for each σ ∈ A, the Fr´echet derivative 0 0 FD (σ) y of FD (·) y has the property that the differential η := FD (σ) y (ϑ) , with any ϑ ∈ L∞ (Ω0 ) extended by zero outside Ω0 , is the (unique) weak solution of the Dirichlet problem − div (σ∇η) = div (ϑ∇FD (σ) y) in Ω, η = 0 on ∂Ω in the sense that it satisfies the equation Z Z 0 σ∇FD (σ) y (ϑ) · ∇vdx = − ϑ∇FD (σ) y · ∇vdx Ω
(10)
Ω
for all v ∈ H01 (Ω) . Moreover, 0 kFD (σ) y (ϑ)kH 1 (Ω) ≤ C1 kykLr (Ω) kϑkLq (Ω0 ) , ∀ϑ ∈ L∞ (Ω0 ) ,
where C1 is a positive constant depending on λ, d, r, Ω0 and Ω.
4
(11)
0 Proof. Note that variational equation (10) has the unique solution η := η (ϑ) = FD (σ) y (ϑ) ∈ H01 (Ω) q 0 with σ ∈ A. We first show that ifor a fixed σ in A, η = η (ϑ) defines a bounded linear operator from L (Ω )
2Q to H01 (Ω) for any q ∈ Q−2 , ∞ . From (10), η is a linear operator of ϑ. By the weak solution formula of η and the generalized H¨ older inequality, we have Z Z σ∇η · ∇ηdx = − ϑ∇FD (σ) y · ∇ηdx Ω ZΩ =− ϑ∇FD (σ) y · ∇ηdx Ω0
≤ kϑkLq (Ω0 ) k∇FD (σ) ykLr (Ω0 ) k∇ηkL2 (Ω) . From the last inequality and (8), there exists a constant C such that kηkH 1 (Ω) ≤ C kϑkLq (Ω0 ) k∇FD (σ) ykLr (Ω0 ) . Besides, the assumption q ∈
2Q Q−2 , ∞
i
(12)
implies r ∈ (2, Q) . By Theorem 1, (7) and (9), there exist positive
constants C, C 0 , C 00 such that k∇FD (σ) ykLr (Ω0 ) ≤ C 0 kFD (σ) ykH 1 (Ω) + kykLr (Ω) 1 0 ≤C kykL2 (Ω) + kykLr (Ω) C ≤ C 00 kykLr (Ω) .
(13)
Thus, due to two last inequalities, η is a bounded linear operator from Lq (Ω0 ) → H01 (Ω) and there exists a positive constant C1 such that 0 kFD (σ) y (ϑ)kH 1 (Ω) ≤ C1 kykLr (Ω) kϑkLq (Ω0 ) , ∀ϑ ∈ L∞ (Ω0 ) .
We now show that FD (·) y is Fr´echet differentiable. Note that the function R := FD (σ + ϑ) y−FD (σ) y− η ∈ H01 (Ω) is the weak solution of the equation − div ((σ + ϑ) ∇R) = div (ϑ∇η) in Ω. Taking R as the test function in the weak solution formula of R gives Z Z Z 2 (σ + ϑ) |∇R| dx = − ϑ∇η · ∇Rdx = − Ω
ϑ∇η · ∇Rdx
Ω0
Ω
≤ kϑkLq (Ω0 ) k∇ηkLr (Ω0 ) k∇RkL2 (Ω) . This implies that kRkH 1 (Ω) kϑkLq (Ω0 ) 0 FD
≤ C k∇ηkLr (Ω0 ) .
(14)
To show that FD (·) y : A ⊂ Lq (Ω) → H01 (Ω) is continuously Fr´echet differentiable and its differential (σ) y (ϑ) is η, we need to prove that k∇ηkLr (Ω0 ) converges to zero as kϑkLq (Ω0 ) converges to zero. By Theorem 1, there exists a positive constant C such that k∇ηkLr (Ω0 ) ≤ C kηkH 1 (Ω) + kϑ∇FD (σ) ykLr (Ω0 ) .
Since kηkH 1 (Ω) converges to zero as kϑkLq (Ω0 ) converges to zero by (12), we need to prove that kϑ∇FD (σ) ykLr (Ω0 ) → 0.
5
Take any small 1 ∈ (0, ) such that r0 = r + 1 ∈ (r, Q) . Using H¨older’s inequality, we deduce Z Z r r r |ϑ∇FD (σ) y| dx = |ϑ| |∇FD (σ) y| dx Ω0
Ω0
Z ≤
|ϑ|
rr 0 r 0 −r
1− r0 Z r
dx
Ω0
r0 r |∇FD (σ) y| dx . r0
(15)
Ω0
Z
r
≤ C2 kykLr0 (Ω)
rr 0
1− r0 r
|ϑ| r0 −r dx
,
Ω0
where we have applied Theorem 1 to the term k∇FD (σ) ykLr0 (Ω0 ) , see (13). By Remark 3, the convergence of ϑ to zero with respect to the Lq1 (Ω) −norm and the Lq2 (Ω) −norm (q1 , q2 ∈ [1, ∞)) are equivalent. Therefore, kϑ∇FD (σ) ykLr (Ω0 ) converges to zero as kϑkLq (Ω0 ) converges to zero. Remark 5 1) If y ∈ Lr (Ω) , then from the proof above we conclude that FD (·) y : A ⊂ Lq (Ω) → H01 (Ω) is Gˆ auteaux differentiable. 2) This lemma improves the known results on the differentiability of FD (·) y with respect to the L∞ −norm in [26, 18]. There, the authors have shown that FD (·) y : A ⊂ L∞ (Ω) → H01 (Ω) is the Fr´echet differentiable under the condition y ∈ L∞ (Ω) [26] or y ∈ L2 (Ω) [18]. Lemma 6 For φ ∈ H01 (Ω) , the functional Fφ (·) : A ⊂ Lq (Ω) → R defined by Z 2 Fφ (σ) = σ |∇ (FD (σ) y − φ)| dx Ω
has the following properties r q 1) For q ≥ 1 and y ∈ i L (Ω) , Fφ (·) is continuous with respect to the L −norm.
2Q , ∞ , 1q + 1r = 12 , and y ∈ Lr+ (Ω) with > 0, if k∇φkLr (Ω0 ) < ∞, then Fφ (·) is Fr´echet 2) For q ∈ Q−2 differentiable with respect to the Lq -norm and Z 2 2 Fφ0 (σ) ϑ = − ϑ |∇FD (σ) y| − |∇φ| dx. Ω
Furthermore, Fφ (·) is convex on the convex set A and the norm of operators kFφ00 (σ) k are uniformly bounded, i.e. there exists a constant M such that kFφ00 (σ) k ≤ M, ∀σ ∈ A. i 2Q Proof. 1) We first prove for q ∈ Q−2 , ∞ . For σ, σ + ϑ ∈ A, we have Fφ (σ + ϑ) − Fφ (σ) Z 2 2 = (σ + ϑ) |∇ (FD (σ + ϑ) y − φ)| − σ |∇ (FD (σ) y − φ)| dx Ω Z Z 2 2 2 = σ |∇ (FD (σ + ϑ) y − φ)| − |∇ (FD (σ) y − φ)| dx + ϑ |∇ (FD (σ + ϑ) y − φ)| dx. Ω
Ω
Using the triangle inequality, generalized H¨older inequality and Theorem 1, the second term is estimated by Z
2
Z
2
ϑ |∇ (FD (σ + ϑ) y − φ)| dx =
ϑ |∇ (FD (σ + ϑ) y − φ)| dx ≤ kϑkLq (Ω0 ) k∇ (FD (σ + ϑ) y − φ)kL2 (Ω) k∇FD (σ + ϑ) ykLr (Ω0 ) + k∇φkLr (Ω0 ) Ω0
Ω
≤ C kϑkLq (Ω0 ) . On the other hand, by Lemma 2 the first term is estimated by Z 2 2 σ |∇ (FD (σ + ϑ) y − φ)| − |∇ (FD (σ) y − φ)| dx Ω Z −1 ≤λ ∇ (FD (σ + ϑ) y − FD (σ) y) · ∇ (FD (σ + ϑ) y + FD (σ) y − 2φ) dx Ω
≤ C k∇ (FD (σ + ϑ) y − FD (σ) y)kL2 (Ω) ≤ C 0 kϑkLq (Ω0 ) . 6
Therefore, Fφ (·) is Lipschitz continuous on A with respect to the Lq (Ω0 )-norm for q ∈ q
2Q Q−2 , ∞
i
.
0
Finally, by Remark 3 Fφ is continuous on A with respect to the L (Ω ) −norm for q ≥ 1. 2) From Lemma 4, it implies that Fφ (·) is Fr´echet differentiable and Z Z 2 0 Fφ0 (σ) ϑ = ϑ |∇ (FD (σ) y − φ)| dx + 2 σ∇ (FD (σ) y − φ) · ∇FD (σ) ϑdx. Ω
Since FD (σ) y − φ ∈
Ω
H01
(Ω) and (10), the last equation yields Z Z 2 0 Fφ (σ) ϑ = ϑ |∇ (FD (σ) y − φ)| dx − 2 ϑ∇FD (σ) y · ∇ (FD (σ) y − φ) dx Ω Ω Z 2 2 =− ϑ |∇FD (σ) y| − |∇φ| dx. Ω
(Ω ) and extended by zero outside Ω0 , the second derivative of Fφ (·) is given by Z Z 2 0 0 Fφ00 (σ) (ϑ, ϑ) = −2 ϑ∇FD (σ) y · ∇FD (σ) y (ϑ) dx = 2 σ |∇FD (σ) y (ϑ)| dx ≥ 0.
For ϑ ∈ L
∞
0
Ω
Ω
Therefore, Fφ (·) is convex. Furthermore, by Lemma 4, it implies that kFφ00 (σ) k is uniformly bounded on A. Remark 7 The uniform boundedness of kFφ00 (σ) k on A implies that Fφ0 (·) is Lipschitz continuous with i 2Q ,∞ . respect to the Lq −norms with q ∈ Q−2
3
The Well-posedness
We now assume that there exists some σ ∗ ∈ A such that φ∗ = FD (σ ∗ ) y and only noisy data φδ ∈ H01 (Ω) of φ∗ such that
∗
φ − φδ 1 ≤δ H (Ω) with δ > 0 are given. Our problem is to reconstruct σ ∗ from φδ . Because of the ill-posedness of the problem and the assumption of sparsity of σ ∗ − σ 0 , using sparsity regularization incorporated with the energy functional approach leads to considering the minimization problem (2). Note that noisy data is normally assumed to be in L2 (Ω) and thus the assumption φδ ∈ H01 (Ω) seem to be not a real situation. In fact this assumption is not too tight since if noisy data is in L2 (Ω), we can obtain new noisy data in H01 (Ω) by using some methods of data processing. The assumption φδ ∈ H01 (Ω) is also used in some papers, e.g. in [18]. We now analyze the well-posedness of problem (2), which consists of the existence, stability and convergence. Note that the well-posedness of problem (2) is still valid if the sparsity promoting penalty (5) is replaced by any convex, lower semicontinuous functional in Lq . Because the set A is weakly compact. Before proving the main results, we introduce some properties of the functional (5) and the notion of Φ-minimizing solution. Lemma 8 The functional Φ defined by (5) has the following properties 1) Φ is non-negative, convex and weakly lower semi-continuous. 2) There exists a positive constant C such that for any σ ∈ L2 (Ω), p
Φ (σ) ≥ ωmin C p/2 kσk . This implies that Φ is weakly coercive, i.e. Φ (σ) → ∞ as kσk → ∞. 3) If {σ n }n∈N ⊂ L2 (Ω) weakly converges to σ ∈ L2 (Ω) and Φ (σ n ) converges to Φ (σ) , then Φ (σ n − σ) converges to zero. Proof. Φ is non-negative, convex and weakly lower semi-continuous because it is the sum of non-negative, convex and weakly continuous functionals. The proofs of 2) and 3) can be found in [17, Remark 3.] and [17, Lemma 2.], respectively. 7
Lemma 9 The set Π (φ∗ ) := {σ ∈ A : FD (σ) y = φ∗ } is nonempty, convex, bounded and closed with respect to the L2 (Ω)-norm. Thus, there exists a solution σ + of the problem min∗ Φ σ − σ 0 σ∈Π(φ )
which is called a Φ-minimizing solution of the diffusion coefficient identification problem. The Φ-minimizing solution is unique if p > 1. Proof. It is trivial that the set Π (φ∗ ) is nonempty, convex and bounded. The closeness of Π (φ∗ ) in the L2 (Ω) −norm is proven similarly as that of [18, Lemma 2.1]. We now prove that there exists at least a Φ-minimizing solution. Suppose that there does not exist a Φ-minimizing solution in Π (φ∗ ) . There exists a sequence {σ k } ⊂ Π (φ∗ ) such that Φ σ k − σ 0 → c and c < Φ σ − σ0
for all σ ∈ Π (φ∗ ) .
(16)
Since Π (φ∗ ) is weakly compact, there exists a subsequence of {σ k }, denoted by {σ k } again, which weakly converges to σ ˜ ∈ Π (φ∗ ) . From the weakly lower semi-continuity of Φ, it follows that Φ σ ˜ − σ 0 ≤ lim inf Φ σ k − σ 0 = c. k→∞
This gives a contradiction to (16). For p > 1, Φ (·) is strictly convex and thus the Φ-minimizing solution is unique. Theorem 10 (Existence) Problem (2) has at least one solution. Proof. Since the functional Fφδ (·) is convex and continuous with respect to the L2 (Ω)-norm, it is weakly lower semi-continuous. Besides, Φ (·) is also convex and weakly lower semi-continuous with respect to the L2 (Ω)-norm (see Lemma 8). Therefore, the objective functional of problem (2) is convex and weakly lower semi-continuous on A. On the other hand, since A is nonempty, convex, bounded and closed with respect to the L2 (Ω)-norm, it is weakly compact. Therefore, there exists at least one solution of (2). Theorem 11 (Stability) For a fixed regularization α > 0, let the sequence {φn } converge to φδ in H01 (Ω) and σ n ∈ argmin Fφn (σ) + αΦ σ − σ 0 . σ∈A
Then, there exist a subsequence {σ
nk
p } of {σ n } and a minimizer σα,δ of (2) such that
nk p
σ − σα,δ
L2 (Ω)
→ 0.
p p In addition, if the minimizer σα,δ is unique, then the sequence {σ n } converges to σα,δ with respect to the L2 (Ω)-norm.
Proof. By the definition of σ n , we have Fφn (σ n ) + αΦ σ n − σ 0 ≤ Fφn (σ) + αΦ σ − σ 0 2 ≤ λ−1 kFD (σ) ykH 1 (Ω) + C + αΦ σ − σ 0
(17)
2
for any σ ∈ A, where the constant C is independent of n such that kφn kH 1 (Ω) ≤ C for all n. This follows that {Φ σ n − σ 0 } is bounded. Since Φ is weakly coercive in L2 (Ω) (see Lemma 8), the sequence {σ n } is also bounded in L2 (Ω). Therefore, there exist a subsequence of {σ n } denoted by {σ nk } and an element p p σα,δ ∈ L2 (Ω) such that {σ nk } weakly converges to σα,δ in L2 (Ω) . Since A is a convex closed set in L2 (Ω) , p σα,δ ∈ A. On the other hand, since Fφδ (·) and Φ (·) are weakly lower semi-continuous, we have p Fφδ σα,δ ≤ lim inf Fφδ (σ nk ) k
8
(18)
and
p Φ σα,δ − σ 0 ≤ lim inf Φ σ nk − σ 0 .
(19)
Z Fφδ (σ nk ) = Fφnk (σ nk ) + 2 σ nk ∇FD (σ nk ) y · ∇ φnk − φδ dx Ω Z 2 − σ nk ∇ φnk − φδ dx .
(20)
k
Furthermore, we have
Ω
Since φnk → φδ in H 1 (Ω) , the term in brackets on the right-hand side of (20) converges to zero as k → ∞. Therefore, (21) lim inf Fφδ (σ nk ) = lim inf Fφnk (σ nk ) , lim sup Fφδ (σ nk ) = lim sup Fφnk (σ nk ) . k
k
k
k
From (21), (17), (18) and (19), we obtain (18),(19) p p Fφδ σα,δ + αΦ σα,δ − σ0 ≤ lim inf Fφδ (σ nk ) + α lim inf Φ σ nk − σ 0 k
k
(21)
≤ lim inf Fφnk (σ nk ) + αΦ σ nk − σ 0 k ≤ lim sup Fφnk (σ nk ) + αΦ σ nk − σ 0 k
(17)
≤ lim sup Fφnk (σ) + αΦ σ − σ 0 k = Fφδ (σ) + αΦ σ − σ 0
(22)
p for all σ ∈ A. It means that σα,δ is a minimizer of (2). p From (22), setting σ = σα,δ , we get
lim Fφδ (σ nk ) + αΦ σ nk − σ 0 k
p p = Fφδ σα,δ + αΦ σα,δ − σ0 .
p Together with (18) and (19), we deduce that Φ σ nk − σ 0 → Φ σα,δ − σ 0 . Finally, since {σ nk } weakly p p p converges to σα,δ and Φ σ nk − σ 0 → Φ σα,δ − σ 0 as k → ∞, we conclude that Φ σ nk − σα,δ → 0 as
p k → 0, and thus σ nk − σα,δ → 0 as k → ∞ by Lemma 8.
L2 (Ω)
p p In the case the minimizer σα,δ is unique, the convergence of the original sequence {σ n } to σα,δ follows by a subsequence argument.
Theorem 12 (Convergence) Assume that the operator equation FD (σ) y = φ∗ attains a solution in A and that α : R>0 → R>0 satisfies α (δ) → 0 and
δ2 → 0 as δ → 0. α (δ)
Let δn → 0 and kφn − φ∗ kH 1 (Ω) ≤ δn . Moreover, let αn = α (δn ) and σ n ∈ argmin Fφn (σ) + αn Φ σ − σ 0 . σ∈A
Then, there exist a Φ-minimizing solution σ + of FD (σ) y = φ∗ and a subsequence of {σ n } converging to σ + on A with respect to the L2 (Ω) −norm. Proof. Let σ ˜ ∈ A be a solution of FD (σ) y = φ∗ . The definition of σ n implies that Fφn (σ n ) + αn Φ σ n − σ 0 ≤ Fφn (˜ σ ) + αn Φ σ ˜ − σ0 Z 1 2 ≤ |∇ (FD (˜ σ ) y − φn )| + αn Φ σ ˜ − σ0 λ Ω 1 2 ˜ − σ0 ≤ kφ∗ − φn kH 1 (Ω) + αn Φ σ λ 1 ≤ δn2 + αn Φ σ ˜ − σ0 . λ 9
(23)
In particular, when δ → 0 and α ∼ δ 2 , it follows that Fφn (σ n ) → 0 and lim sup Φ σ n − σ 0 ≤ Φ σ ˜ − σ0 .
(24)
n
This implies that {Φ σ n − σ 0 } is bounded. Since Φ (·) is weakly coercive, {σ n } is bounded, too. Therefore, there exist a subsequence {σ nk } of {σ n } and σ + ∈ A such that σ nk weakly converges to σ + . From (24), we deduce Z 2 Fφ∗ (σ nk ) = σ nk |∇ (FD (σ nk ) y − φ∗ )| ZΩ Z 2 2 ≤ σ nk |∇ (FD (σ nk ) y − φnk )| + σ nk |∇ (φnk − φ∗ )| Ω
Ω
≤ Fφ
nk
(σ
nk
−1
)+λ
kφ
nk
−
2 φ∗ kH 1 (Ω)
→ 0 (k → ∞) .
Since Fφ∗ (·) is weakly lower semi-continuous, 0 ≤ Fφ∗ σ + ≤ lim inf Fφ∗ (σ nk ) = 0. k
Thus, Fφ∗ (σ + ) = 0. It implies that kFD (σ + ) y − φ∗ kH 1 (Ω) = 0. Hence σ + is a solution of the equation FD (σ) y = φ∗ . Moreover, since Φ (·) is weakly lower semi-continuous in L2 (Ω) , by using (24) we get Φ σ + − σ 0 ≤ lim inf Φ σ nk − σ 0 ≤ lim sup Φ σ nk − σ 0 ≤ Φ σ ˜ − σ0 . (25) k
k
It implies that σ + is a Φ-minimizing solution. Finally, choosing σ ˜ = σ + in (25), we have Φ σ nk − σ 0 → Φ σ + − σ 0 as k → ∞. Since {σ nk − σ 0 } weakly converges to σ + − σ 0 in L2 (Ω) and Φ σ nk − σ 0 → Φ σ + − σ 0 as k → ∞, Φ (σ nk − σ + ) → 0 as k → 0 and thus kσ nk − σ + kL2 (Ω) → 0. In the case the minimizer σ + is unique, the convergence of the original sequence {σ n − σ 0 } to σ + − σ 0 follows by a subsequence argument.
4
Convergence Rates
As shown before, for σ ∈ A, the operator 0 FD
q
0
(σ) y (·) : L (Ω ) →
H01
(Ω) with q ∈
2Q ,∞ Q−2
is continuous and linear. Denote by ∗
0 (FD (σ) y) (·) : H −1 (Ω) = H01 (Ω)
∗
→ Lq1 (Ω0 ) with
1 1 + = 1, q q1
0 the dual operator of FD (σ) y. Then,
0 ∗ 0 (FD (σ) y) (w∗ ) , ϑ (Lq1 (Ω0 ),Lq (Ω0 )) = hw∗ , FD (σ) y (ϑ)i(H −1 (Ω),H 1 (Ω)) .
(26)
0
To obtain convergence rates of sparsity regularization, an important tool is the Bregman distance relating to a proper convex functional. We briefly introduce this notion here. For a detail discussion on the Bregman distance, we refer to [5, 10, 25, 7]. Let X be a Banach space with its dual space X ∗ and R : X → (−∞, +∞] be a proper convex functional with dom (R) := {x ∈ X : R (x) < +∞} = 6 ∅. The subdifferential of R at x ∈ dom (R) is defined by ∂R (x) := {x∗ ∈ X ∗ : R (y) ≥ R (x) + hx∗ , y − xi(X ∗ ,X) for all y ∈ X}. Then, for a fixed element x∗ ∈ ∂R (x) , the expression DxR∗ (y, x) := R (y) − R (x) − hx∗ , y − xi(X ∗ ,X) 10
is called the Bregman distance of two elements y, x ∈ X with respect to R and x∗ . In the following, we denote Dx∗ (y, x) instead of DxR∗ (y, x) for simplicity. Since ∂R (x) might be empty or multi-valued, Bregman distance might be not defined or multi-valued. However, for a continuously differentiable functional, there is a unique element in the subdifferential and consequently, a unique Bregman distance. In this case, the distance is just the difference at the point y between R (·) and the first order Taylor series approximation to R (·) at x. Furthermore, if R (y) is strictly convex, Dx∗ (y, x) is also strictly convex in y for each fixed x, and therefore Dx∗ (y, x)=0 if and only if y = x. Note that Dx∗ (y, x) is not a distance in the usual metric sense since, in general, D (y, x) 6= D (x, y) and the triangle inequality does not hold. However, it is a measurement of closeness in the sense that Dx∗ (y, x) ≥ 0 and Dx∗ (y, x) = 0 if y = x. Convergence rates of sparsity regularization are given in the following theorem. i
p 2Q , ∞ , 1q + 1r = 12 and y ∈ Lr (Ω) . Assume that φδ − φ∗ H 1 (Ω) ≤ δ and σα,δ is Theorem 13 For q ∈ Q−2 a solution of (2). Moreover, assume that there exists a function w∗ ∈ H −1 (Ω) such that ∗ 0 ξ := FD σ + y (w∗ ) ∈ ∂Φ σ + − σ 0 .
(27)
Then,
p p y − φδ Dξ σα,δ , σ + = O (δ) and FD σα,δ
H 1 (Ω)
= O (δ)
as δ → 0 and α ∼ δ. In particular, for p ∈ (1, 2] , we have
√
p
=O δ .
σα,δ − σ + 2 L (Ω)
p Proof. The proof follows the ideas of Hao and Quyen in [18, 19]. By the definition of σα,δ , we get
p p Fφδ σα,δ + αΦ σα,δ − σ 0 ≤ Fφδ σ + + αΦ σ + − σ 0 .
(28)
Then, we have p p Fφδ σα,δ + αDξ σα,δ , σ+ E D p p p 0 + 0 + = Fφδ σα,δ + α Φ σα,δ − σ − Φ σ − σ − ξ, σα,δ − σ (Lq1 (Ω0 ),Lq (Ω0 )) D E p ≤ Fφδ σ + − α ξ, σα,δ − σ+ (Lq1 (Ω0 ),Lq (Ω0 )) D E 1 p − σ+ . ≤ δ 2 − α ξ, σα,δ λ (Lq1 (Ω0 ),Lq (Ω0 )) From (26) and (27), we get D E p ξ, σα,δ − σ+
(Lq1 (Ω0 ),Lq (Ω0 ))
D E p 0 = w∗ , FD σ + y σα,δ − σ+
(H −1 (Ω),H01 (Ω))
.
By Riesz’s representation theorem, there exists an element w ∈ H01 (Ω) such that D E D E p p 0 0 + + w∗ , FD σ + y σα,δ − σ+ = w, F σ y σ − σ . D α,δ H01 (Ω) (H −1 (Ω),H01 (Ω))
(29)
(30)
(31)
Since σ + ≥ λ > 0, the scalar product Z [φ, v]H01 (Ω) :=
σ + ∇φ · ∇vdx, for all φ, v ∈ H01 (Ω)
Ω
p is equivalent to hφ, viH 1 (Ω) on H01 (Ω) . Therefore, there exists an element w ˆ ∈ H01 (Ω) independent of σα,δ 0 such that Z D E p p 0 + 0 + + w, FD σ + y σα,δ − σ+ = σ ∇ w ˆ · ∇F σ y σ − σ dx. (32) D α,δ 1 H0 (Ω)
Ω
11
From (30), (31) and (32), we have D
p ξ, σα,δ − σ+
Z
E (Lq1 (Ω0 ),Lq (Ω0 ))
= Ω
p 0 σ + ∇w ˆ · ∇FD σ + y σα,δ − σ + dx =: Λ.
p 0 From the weak solution formulas of FD (σ + ) y and FD (σ + ) y σα,δ − σ + (see (6) and (10)), we deduce Z αΛ = α Ω
p 0 σ + ∇w ˆ · ∇FD σ + y σα,δ − σ + dx
Z p ˆ · ∇FD σ + ydx = −α σα,δ − σ + ∇w Z Ω Z p =α σ + ∇w ˆ · ∇FD σ + ydx − α σα,δ ∇w ˆ · ∇FD σ + ydx Ω ZΩ Z p p p =α σα,δ ∇w ˆ · ∇FD σα,δ ydx − α σα,δ ∇w ˆ · ∇FD σ + ydx Ω ZΩ p p =α σα,δ ∇w ˆ · ∇ FD σα,δ y − FD σ + y dx ZΩ Z p p p δ =α σα,δ ∇w ˆ · ∇ FD σα,δ y − φ dx + α σα,δ ∇w ˆ · ∇ φδ − φ∗ dx. Ω
Ω
Using the Cauchy-Schwartz inequality, we obtain Z α |Λ| ≤ α Ω
+α
p σα,δ
Z Ω
1/2 Z
2
|∇w| ˆ dx Ω
p σα,δ
2
2
2 1/2 p δ ∇ FD σα,δ y − φ dx
p σα,δ
1/2 ∇ φδ − φ∗ 2 dx
1/2 Z
|∇w| ˆ dx Ω
1/2 1/2 Z 1/2 α Z
δ
1 2 2 p
φ − φ∗ 1 |∇w| ˆ dx Jφδ σα,δ + |∇w| ˆ dx ≤α H (Ω) λ Ω λ Ω Z Z 1/2 αδ α2 1 2 2 p ≤ |∇w| ˆ dx + Fφδ σα,δ + |∇w| ˆ dx . 2λ Ω 2 λ Ω Here, we used the inequality ab ≤
αa2 2
+
b2 2α
(33)
for the first term. Together with (29), we deduce
1 1 α2 2 αδ p p Fφδ σα,δ + αDξ σα,δ C + C1 , , σ+ ≤ δ2 + 2 λ 2λ 1 λ with C1 =
R Ω
(34)
1/2 2 |∇w| ˆ dx . This inequality implies that p Dξ σα,δ , σ + = O (δ) as α → 0 and α ∼ δ.
By (8) and (34), we have
2
p
FD σα,δ y − φδ
1 p 2 F σ = O δ as δ → 0 and α ∼ δ. δ φ α,δ C H 1 (Ω)
2
p
p In particular, for p ∈ (1, 2] there exists a constant Cp > 0 such that Dξ σα,δ , σ + ≥ Cp σα,δ − σ+ ≤
L2 (Ω)
see [17, Lemma 10]. Therefore, we have
p
σα,δ − σ +
L2 (Ω)
=O
,
√ δ .
Remark 14 Our source condition is very simple if we compare with the source conditions in [22, 17, 13, 32]. Especially, we do not need the smallness requirement in the source condition. 12
5
Conclusion
In this paper, sparsity regularization incorporated with the energy functional approach was analyzed for the diffusion coefficient identification problem. The regularized problem was proven to be well-posed and convergence rates of the method was obtained under a simple source condition. An advantage of the new approach is to work with a convex energy functional. Another advantage is that the source condition of obtaining convergence rates are very simple.
6
Acknowledgement
The author would like to thank all three implicit referees for their very valuable comments. I have learned a lots from these comments and made the paper clearer.
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