We deduce formulas for the Fréchet derivatives of cost functionals of several inverse problems for a parabolic integrodifferential equation in a weak formulation. The method consists in the application of an integrated convolutional form of the weak problem and all computations are implemented in regular Sobolev spaces.

1. Introduction

Many methods to solve inverse problems (e.g., the Landweber iteration, conjugate gradient method) use the Fréchet derivatives of the cost functionals of these problems [1]. The explicit formula for the Fréchet derivative in terms of the variation of the unknowns of the inverse problem contains the solution of an adjoint problem.

The derivation of the explicit formula for such a Fréchet derivative includes testing the direct problem with the solution of the adjoint problem and vice versa: testing the adjoint problem with the solution of the direct problem. In the case of the parabolic weak problem, such a procedure is cumbersome, because of the asymmetry of the properties of the solution and the test function. In the classical formulation of the parabolic weak problem (see, e.g., [2] and also (19) below), the test function must have higher time regularity than the weak solution. This means that in case of nonsmooth coefficients neither the solution of the direct problem nor the solution of the adjoint problem can be used as a test function. Another formulation of the weak parabolic problem consists in reducing the problem to an abstract Cauchy problem over the time variable (see, e.g., [3]). In such a case, a partial integration over the time has to be implemented within singular distributions in the derivation procedure.

In this paper, we present a new method that enables the deduction of the formulas for the Fréchet derivatives for cost functionals of inverse problems related to weak solutions of parabolic problems. The method is based on an integrated convolutional form of the weak direct problem. The requirements to the test function are weaker than in the classical case and coincide with the properties of the solution of the direct problem. All computations in the deduction procedure can be implemented within usual regular Sobolev spaces.

More precisely, we will consider inverse problems related to a parabolic integrodifferential equation that occur in heat flow with memory [4–6]. This equation contains a time convolution. Therefore, the convolutional form of the weak problem is especially suitable. Supposedly, the proposed method can be generalised to parabolic systems, as well.

2. Formal Direct Problem: Notation

Let Ω be an n-dimensional domain, where n ≥ 1, and Γ be the boundary of Ω. Let Γ = Γ₁ ∪ Γ₂ where either Γ₁ or Γ₂ is allowed to be an empty set. In case n ≥ 2, we assume that Γ is sufficiently smooth, meas Γ₁ ∩ Γ₂ = 0, and for any j ∈ {1; 2} it holds either Γ_j = ∅ or meas Γ_j > 0. Denote

()

for t ≥ 0. Consider the problem (direct problem) to find u(x, t) : Ω_T → ℝ such that

()

where T > 0 is a fixed number,

()

a_ij, a, u₀ : Ω → ℝ, f : Ω_T → ℝ, ϕ : Ω_T → ℝⁿ, g : Γ_1,T → ℝ, ϑ : Γ₂ → ℝ, h : Γ_2,T → ℝ, m : (0, T) → ℝ are given functions, the subscripts t, x_j, x_i denote the partial derivatives and

()

denotes the time convolution. In case Γ₁ = ∅ (Γ₂ = ∅), the boundary condition (4) and (5) is dropped.

The problem (2)–(5) describes the heat flow in the body Ω with the thermal memory. Concerning the physical background, we refer the reader to [4, 6, 7]. The solution u is the temperature of the body and m is the heat flux relaxation (or memory) kernel. The boundary condition (5) is of the third kind where the term −ν_A · ∇u + m*ν_A · ∇u equals the heat flux in the direction of the conormal vector.

Let us introduce some additional notations. Let t > 0. We use the Sobolev spaces

()

Here, l = 0,1, 2, …, α = (α₁, …, α_n) is the multiindex, |α| = α₁ + ⋯+α_n and

. Further, let X be a Banach space. We denote by C([0, t]; X) the space of abstract continuous functions from [0, t] to X endowed with the usual maximum norm ∥v∥_C([0,t];X) : = max _s∈[0,t]∥v(s)∥. Moreover, let

()

By means of these spaces, we define the following important functional spaces:

()

Convention. In case n = 1, the integrals , j = 1,2 are equal to , where x_k ∈ Γ_j and K is the number of points in Γ_j, and L_p(Γ_j) is simply ℝ^K.

3. Weak Direct Problem and Its Convolutional Form

Let us return to the direct problem (2)–(5). Throughout the paper we assume the following basic regularity conditions on the coefficients, the kernel, and the initial and boundary functions:

()

and the ellipticity condition

()

(for the sake of simplicity we introduced an assumption for the extension of g onto Ω_T).

The first aim is to reformulate the problem (2)–(5) in a weak form. Let us suppose that

, (∂/∂x_i)ϕ_i ∈ L²(Ω_T), i = 1, …, n and (2)–(5) has a classical solution

. Then, we multiply (2) with a test function η from the space

()

and integrate by parts with respect to time and space variables. We obtain the following relation:

()

This relation makes sense also in a more general case when a_ij, ϕ satisfies only (11) and (15) and u does not have regular first-order time and second-order spatial derivatives.

We call a weak solution of the problem (2)–(5) a function from the space 𝒰(Ω_T) that satisfies the relation (19) for any η ∈ 𝒯(Ω_T) and in case Γ₁ ≠ ∅ fulfills the boundary condition (4).

Lemma 1. The following assertions are valid:

(i)
where q₃ = ∞ if n = 1, q₃ ∈ (q₁q₂/(q₁ − q₂), ∞) if n = 2 and q₃ = 2n/(n − 2) if n > 2, where q₁ and q₂ are given in (12) and (14), respectively;
(ii)
for any it holds and , where C is a constant.

Proof. Since , the assertion (i) follows from the continuous embedding of in . The assertion (ii) can be proved by means of Hölder′s inequality.

Theorem 2. The problem (2)–(5) has a unique weak solution. This solution satisfies the estimate

()

where θ₁ = 0 in case Γ₁ = ∅, θ₂ = 0 in case Γ₂ = ∅ and C₀ is a constant depending on Ω, Γ_j, a_ij, a, ϑ and m.

Proof. The assertion of the theorem in case m = 0 is well known from the theory of parabolic equations (see, e.g., [2]). Let 𝒵 be the operator that assigns to the data vector d : = (u₀, f, ϕ, g, h) the weak solution of the problem (2)–(5) in case m = 0. Then it holds

()

where RHS is the right-hand side of (20).

Further, let us formulate the problem for the difference v = u − 𝒵d. Introducing the linear operator 𝒜 by the formula

()

the weak problem (2)–(5) for the solution u ∈ 𝒰(Ω_T) equivalent to the following operator equation for the quantity v:

()

We have to estimate 𝒜. For this purpose, we firstly prove the following auxiliary inequality:

()

for any p ≥ 1 and y ∈ L²((0, t); L^p(Ω)).

Denoting , , making use of the following property of the Bochner integral: for functions w ∈ L¹((0, T); L^p(Ω)) and the Cauchy′s inequality, the relation (24) can be deduced by means of the following computations:

()

Next, let t ∈ [0, T] and introduce the operator

()

Due to the causality we have 𝒵(0, P_tf, P_tϕ, 0,0)(x, t) = 𝒵(0, f, ϕ, 0,0)(x, t) for any (x, t) ∈ Ω_t. Using these relations, the continuity of the linear operator 𝒵, the inequality (24), and the boundedness of a_ij, we compute the following:

()

with some constants

and

depending on Ω, Γ_j, a_ij, a, ϑ. Using Lemma 1, we obtain

()

Using this relation in (27), we arrive at the following basic estimate for 𝒜:

()

where C₂ is a constant depending on Ω, Γ_j, a_ij, a, ϑ. Let us define the weighted norms in 𝒰(Ω_T):

where σ ≥ 0. The estimate (29) implies the further estimate

()

Since

as σ → ∞, there exists σ₀, depending on C₂ and m, such that

. Thus,

. The operator 𝒜 is a contraction in 𝒰(Ω_T). This implies the existence and uniqueness assertions of the theorem.

To prove the estimate (20), we firstly deduce from (23) the inequality . This implies . Using the equivalence relations , where and (21), we reach (20).

We note the upper integration bound T in (19) can be released to be any number t from the interval [0, T]. Indeed, (19) is equivalent to the following problem:

()

for any η ∈ 𝒯(Ω_T). This assertion can be proved using the standard technique defining the test function as follows:

()

and letting the parameter ϵ to approach 0.

Next we transform the weak direct problem (31) to a form that does not contain a time derivative of the test function η. This form enables the extension of the test space. This is useful for treatment of problems for adjoint states of quasisolutions of inverse problems in next sections.

Theorem 3. The function u ∈ 𝒰(Ω_T) satisfies the relation (19) for any η ∈ 𝒯(Ω_T) if and only if it satisfies the following relation:

()

for any η ∈ 𝒰₀(Ω_T).

Here, according to the definition of the time convolution in the previous section, .

Proof. It is sufficient to prove that u ∈ 𝒰(Ω_T) satisfies (31) for any η ∈ 𝒯(Ω_T) if and only if it satisfies (33) for any η ∈ 𝒰₀(Ω_T). Suppose that u ∈ 𝒰(Ω_T) satisfies (31) and choose an arbitrary η ∈ 𝒯(Ω_T). Let t₁ be an arbitrary number on the interval [0, T] and choose some function such that the relation

()

is valid. For instance, it is possible to define

as a periodic function with respect to t, that is,

for t ∈ [0, t₁],

for t ∈ [t₁, 2t₁],

for t ∈ [2t₁, 3t₁] and so on. Using the relation (31) with η replaced by

and setting there t = t₁ we obtain the equality

()

where

()

Note that the time derivative of η can be removed from K₁ by integration. Indeed, let t₂ ∈ [0, T]. Then

()

Changing the order of the integrals over τ and t₁ in the last term, we easily obtain

()

Integrating now the whole equality (35) over t₁ from 0 to t₂, observing (37) and (39), and finally redenoting t₂ by t, we reach the desired relation (33). Summing up, we have proved that (33) holds for any η ∈ 𝒯(Ω_T). But all terms in the right-hand side of (33) are well defined for η ∈ 𝒰₀(Ω_T), too. Since 𝒯(Ω_T) is densely embedded in 𝒰₀(Ω_T), we conclude that (33) holds for any η ∈ 𝒰₀(Ω_T).

It remains to show that (33) implies (31). Suppose that u ∈ 𝒰(Ω_T) satisfies (33) and choose an arbitrary η ∈ 𝒯(Ω_T) and t₁ ∈ [0, T]. Again, let be a function from 𝒯(Ω_T) such (34) is valid. Inserting instead of η into (33), differentiating with respect to t and setting t = t₁ we come to the relation (31). Theorem is proved.

Corollary 4. A function u ∈ 𝒰(Ω_T) is a weak solution of (2)–(5) if and only if it satisfies the relation (33) for any η ∈ 𝒰₀(Ω_T) and in case Γ₁ ≠ ∅ fulfills the boundary condition (4).

4. Inverse Problems and Quasisolutions

In the sequel, let us pose some inverse problems for the weak solution of (2)–(5). These problems are selected in order to demonstrate the wide possibilities of the method that we will introduce in Section 5.

Firstly, we suppose that (2)–(5) has the following specific form:

()

where ω = (ω₁, …, ω_N) is unknown. The coefficients and other given functions f₀, ϕ, u₀, g, h are assumed to satisfy (11)–(17). Moreover,

is prescribed.

IP1. Find the vector

such that the weak solution of (40) satisfies the following instant additional conditions:

()

where 0 < T₁ < T₂ < ⋯<T_N ≤ T and

, i = 1, …, N are given functions (observations of u).

Since for , the weak solution u of (40) exists in 𝒰(Ω_T); hence, it has traces u(·, T_i) ∈ L²(Ω), i = 1, …, N. In practice, the term may represent an approximation of a more general function F(x, t) ∈ L²(Ω_T), where γ_j, j = 1,2, … form a basis in L²(0, T).

Further, let u₀ also be unknown.

IP2. Find the vector

and u₀ ∈ L²(Ω) such that the weak solution of (40) satisfies the following integral additional conditions:

()

where v_i ∈ L²(Ω), i = 1, …, N + 1 are given observation functions and κ_i, i = 1, …, N are given weights that satisfy the following condition:

()

Note that the integral

in (42) belongs to L²(Ω) for any

and u₀ ∈ L²(Ω). Indeed, for such ω and u₀ it holds u ∈ 𝒰(Ω_T) ⊂ L²(Ω_T), which implies

()

In practice, the weights κ_i are usually concentrated in neighborhoods of some fixed values of time t = T_i.

Finally, let us pose a nonlinear inverse problem for the coefficient a and the kernel m. Assume that n ∈ {1; 2; 3}. Then any coefficient a that belongs to L²(Ω) satisfies (12). Moreover, let us set q₁ = 2 if n = 2 and Γ₂≢∅. The other coefficients and the given functions u₀, f, ϕ, g, h are assumed to satisfy (11)–(17).

IP3. Find a ∈ L²(Ω) and m ∈ L¹(0, T) such that the weak solution of (2)–(5) satisfies the following integral additional conditions:

()

where u_T ∈ L²(Ω), v ∈ L²(0, T) are given observation functions and κ is a given weight function such that κ ∈ L^∞((0, T); L²(Γ₂)).

As in IP1, we can show that the trace u(·, T) belongs to L²(Ω). Moreover, using the property u ∈ 𝒰(Ω_T), the embedding of in L²(Γ₂) and Hölder’s inequality, one can immediately check that the term in (45) belongs to L²(0, T).

Available existence, uniqueness, and stability results for IP1–IP3 require stronger smoothness of the data than imposed in the present paper. Let us cite some of these results.

In case N = 1, the well posedness of IP1 was proved in [8]. Partial results were deduced earlier in [9]. A more general problem involving both IP1 and IP2 without the unknown u₀ in case N = 1 was studied in [10] by means of different techniques. IP1 and IP2 in case m = 0 and N = 1 were treated in many papers, for example, [11–14]. The case N > 1 is open even if m = 0. Inverse problems to determine m with given a were studied in a number of papers, for example, [7, 15–23]. The problem for a with given m was treated in [8].

In the present paper, we will deal with quasisolutions of IP1–IP3 and related cost functionals. Denote 𝒵₁ = (L²(Ω)) ^N. Let M⊆𝒵₁. The quasi-solution of IP1 in the set M is an element ω^* ∈ arg min_ω∈MJ₁(ω), where J₁ is the following cost functional

()

and u(x, t; ω) is the solution of (40) that corresponds to a fixed element ω.

Similarly, let

. The quasi-solution of IP2 in the set M is z^* ∈ arg min_z∈MJ₂(z), where J₂ is the cost functional

()

and u(x, t; z) is the weak solution of (40) that corresponds to a given vector z = (ω, u₀).

Finally, defining M⊆𝒵₃ : = L²(Ω) × L²(0, T), the quasi-solution of IP3 in M is an element z^* ∈ arg min_z∈MJ₃(z), where J₃ is the cost functional

()

and u(x, t; z) is the weak solution of the direct problem (2)–(5) corresponding to given z = (a, m). Here, we restricted the space for the unknown m to L²(0, T), because we will seek for the Fréchet derivative of J₃ in a Hilbert space. Moreover, the kernel of the second addend corresponding to m in the representation formula of

(90) is an element of L²(0, T) and in general does not belong to the adjoint space L^∞(0, T).

According to the above-mentioned arguments, the functionals J_k, k = 1,2, 3, are well-defined in 𝒵₁, 𝒵₂, and 𝒵₃, respectively.

5. The Fréchet Derivatives of Cost Functionals of Inverse Problems

5.1. General Procedure

Suppose that the solution u of the direct problem depends on a vector of parameters p that has to be determined in an inverse problem making use of certain measurements of u. Let the quasi-solution of the inverse problem be sought by a method involving the Fréchet derivative of the cost functional (i.e., some gradient-type method). Usually in practice, a solution of a proper adjoint problem is used to represent the Fréchet derivative.

We introduce a general procedure to deduce such adjoint problems. Assume that Δu is the difference of solutions of the direct problem corresponding to a difference of the vector of the parameters Δp. More precisely, we suppose that Δu is a solution of the following problem:

()

with some data f^†, ϕ^†, Δu₀, h^† depending on Δp. We restrict ourselves to the case when the Dirichlet boundary condition of the state u is independent of p. Therefore, the condition (51) for Δu is homogeneous.

In practice, the adjoint parabolic problems are usually formulated as backward problems. In our context, it is better to pose adjoint problems in the forward form. The involved memory term with m is defined via a forward convolution and from the practical viewpoint, it is preferable to have the direct and adjoint problems in a similar form (e.g., to simplify parallelisation of computations).

More precisely, let the adjoint state ψ be a solution of the following problem:

()

where f^∘, ϕ^∘, u^∘, and h^∘ are some data depending on Δu and the cost functional under consideration.

Assume that the quadruplets f^†, ϕ^†, Δu₀, h^†, and f^∘, ϕ^∘, u^∘, h^∘ satisfy the conditions (14)–(16). Then, due to Theorem 2, the problems (49)–(52) and (53) have unique weak solutions in the space 𝒰(Ω_T). Actually, we have Δu, ψ ∈ 𝒰₀(Ω_T) because of the homogeneous boundary conditions on Γ_1,T.

Let us write the relation (33) for Δu and use the test function η = ψ. Then we obtain for any t ∈ [0, T]

()

Secondly, let us write this relation for ψ and use the test function η = Δu. Then we have for any t ∈ [0, T]

()

Subtracting (54) from (55), using the commutativity of the convolution, the symmetricity relations a_ij = a_ji and differentiating with respect to t, we arrive at the following basic equality that can be used in various inverse problems:

()

5.2. Derivative of J₁

Theorem 5. The functional J₁ is the Fréchet differentiable in and

()

where

, i = 1, …, N, are the unique ω-dependent weak solutions of the following problems:

()

i = 1, …, N.

Proof. Let us fix some . One can immediately check that it holds

()

where Δu(x, t; ω) = u(x, t; ω + Δω) − u(x, t; ω) ∈ 𝒰₀(Ω_T) is the weak solution of the following problem:

()

Applying the estimate (20) to the solution of this problem we deduce the following estimate for the second term in the right-hand side of (59):

()

with some constant C₄. This implies that J₁ is the Fréchet differentiable and the first term in the right-hand side of (59) represents the Fréchet derivative, that is,

()

Further, let us use the method presented in Section 5.1 to deduce the proper adjoint problems. Comparing (60) with (49)–(52) we see that , ϕ^† = Δu₀ = h^† = 0. Therefore, the relation (56) has the form

()

In order to deduce a formula for the component σ_i in the quantity

, we set

, h^∘ = f^∘ = ϕ^∘ = 0 and t = T_i in (63). Then we immediately have

()

where according to (53) and the definition of

, the function ψ_i is the weak solution of the problem (58) in the domain Ω_T instead of

. Due to Theorem 2, this problem has a unique solution. From (62) and (64) we obtain (57). The latter formula contains the values of ψ_i in

. Therefore, we can restrict the problem (58) from Ω_T to

To use the formula (57) one has to solve N weak problems for the functions ψ_i in domains . In the following theorem, we will show that computational work related to the evaluation of the Fréchet derivative can be considerably reduced. Actually, it is sufficient to solve N weak problems in the smaller domains , i = 1, …, N. Here, T₀ = 0.

Theorem 6. The Fréchet derivative of the functional J₁ can also be written in the form

()

where

are the unique ω-dependent weak solutions of the following sequence of recursive problems in the domains

()

where l = N, N − 1, …, 2,1. Here,

()

and the function f^l and the vector Φ^l are defined via β_N, β_N−1, …, β_l+1 as follows:

()

and Θ_N = 0, Θ_l = 1 for l < N.

Proof. Firstly, let us check that (66) indeed have unique weak solutions β_l in . To this end we can use Theorem 2. For the problem β_N this is immediate, because the initial condition of the problem for β_N belongs to L²(Ω) and other equations in this problem are homogeneous. Further, we use the induction. Choose some l in the range N > l ≥ 1 and suppose that for all k such that N − 1 ≥ k ≥ l. The aim is to us to show that then the problem for β_l has a unique weak solution in . Let us represent the kth addend in (68) in the form

()

For any k in the range N − 1 ≥ k ≥ l we have

()

where

for t ∈ [0, T_k+1 − T_k], z_k,α(t) = 0 for t ∉ [0, T_k+1 − T_k] and m_k(t) = |m(T_k − T_l + t)|. Since m ∈ L¹(0, T) and

, we have m_k ∈ L¹(0, T_l − T_l−1 + T_k+1 − T_k) and z_k,α ∈ L²(0, T_l − T_l−1 + T_k+1 − T_k). Due to the Young’s theorem for convolutions, we get m_k*z_k,α ∈ L²(0, T_l − T_l−1 + T_k+1 − T_k). Therefore,

. This implies that f_l belongs to

. From the latter relation and a_ij ∈ L^∞(Ω) we immediately have

. Using the embedding theorem and Lemma 1 we see that af^l satisfies the property (14). Finally, the initial condition

belongs to L²(Ω), because

, β_l+1 ∈ C([0, T_l+1 − T_l], L²(Ω)). All assumptions of Theorem 2 are satisfied for the problem for β_l. Consequently, it possesses a unique weak solution in

Secondly, let us define the functions

()

where l = 1 … , N and ψ_i are the solutions of (58). We are going to show that

, l = 1, …, N. From the definition of

using the value of ψ_l(x, 0) and simple computations, we immediately get

()

Let us fix l = 1, …, N and choose some

. We continue η by the formulae η(x, t) = η(x, T_l − T_l−1) for t > T_l − T_l−1 and η(x, t) = η(x, 0) for t < 0. Further, let us define η_i(x, t) = η(x, T_l − T_i + t) where i = l, …, N. By the definition, it holds

Let us write down the weak form (31) for the problem for ψ_i (58) with the test function η_i. We fix some t ∈ [0, T_l − T_l−1] and compute the difference of this weak problem with t replaced by T_i − T_l + t and t replaced by T_i − T_l and take the sum over i = l, …, N. This results in the following expression:

()

where

()

Using the definitions of η and

and the formula (72), we have

()

Similarly, using the definitions of η and

and changing the variable of integration in Z₂, we deduce

()

By the change of variable, the quantity Z₃ is transformed to

()

Let us consider the term

in the latter formula. We compute

()

Thus, (77) reads

()

Using similar computations, we obtain

()

Plugging (75), (76), (79), and (80) into (73), we arrive at a certain weak problem for

that coincides with the weak problem for β_l. Moreover, since

, from (71) we see that

. But we have shown the uniqueness of the weak solutions of the problems for β_l in

. This implies

Finally, from (57), we have

()

Changing here the order of sums over i and l and observing (71) with

replaced by β_l, we obtain (65). The proof is complete.

5.3. Derivative of J₂

Theorem 7. The functional J₂ is the Fréchet differentiable in and

()

where ψ ∈ 𝒰(Ω_T) is the unique z-dependent weak solution of the following problem:

()

Proof. Let us fix some . It holds

()

where Δu(x, t; z) = u(x, t; z + Δz) − u(x, t; z) ∈ 𝒰₀(Ω_T) is the weak solution of the following problem:

()

Using (43), the Cauchy inequality and estimate (20) from Theorem 2 for the problem of Δu(x, t; z), we come to the estimate

()

with some constants C₅ and C₆. Therefore, J₂ is the Fréchet differentiable and the first term in the right-hand side of (84) represents the Fréchet derivative, that is,

()

Comparing (85) with (49)–(52), we see that

, ϕ^† = h^† = 0. Consequently, the relation (56) has the form

()

To deduce a formula for

, we define

()

u^∘ = h^∘ = ϕ^∘ = 0 and t = T in (88). Then from (87) and (88), we obtain (82), where due to (53), ψ_i is the weak solution of the problem (83). In view of Theorem 2, this problem has a unique solution in 𝒰(Ω_T).

5.4. Derivative of J₃

Theorem 8. The functional J₃ is the Fréchet differentiable in L²(Ω) × L²(0, T) and

()

where ψ ∈ 𝒰₀(Ω_T) is the unique z-dependent weak solution of the problem

()

Proof. Due to u(x, t; z) ∈ 𝒰(Ω_T), κ ∈ L^∞((0, T); L²(Γ₂)), v ∈ L²(0, T), and u_T ∈ L²(Ω), the problem (91) satisfies the assumptions of Theorem 2. Therefore, it has a unique weak solution in 𝒰₀(Ω_T).

Let Δz = (Δa, Δm) ∈ L²(Ω) × L²(0, T) and define . We split as follows: , where Δu is the weak solution of the following problem:

()

In view of Lemma 1(i), u ∈ 𝒰(Ω_T), m ∈ L¹(0, T), and the Young’s theorem, it holds

. Therefore, Lemma 1(ii) implies

()

where C₈ and C₉ are some constants depending on u, m. Moreover, since

, by Young’s inequality we have also

()

with some constants C₁₀ and C₁₁ depending on u. The obtained estimates show that assumptions of Theorem 2 are satisfied for the problem (92) and it indeed has a unique weak solution Δu ∈ 𝒰(Ω_T). Moreover, applying the relation (20) from Theorem 2, we get

()

where C₁₂(m, u) is a constant depending on m, u.

Further, writing the problem for and subtracting the problem for Δu, we obtain the following problem for :

()

where

()

Using again Lemma 1 and the Young’s inequality, we deduce the estimates

()

with some constants C₁₃ ⋯ C₁₇. Therefore, applying the relation (20) to the solution of the problem (96) we obtain

()

with some constant C₁₈. In case ∥Δz∥ is small enough, that is,

()

we have

()

In view of (95), this implies

()

with a constant C₁₉.

Similarly, for the solution of the problem (92), we deduce the estimate

()

with a constant C₂₀.

Now, we write the difference of J₃ in the following form:

()

where

()

Using (102), (103), and the property κ ∈ L^∞((0, T); L²(Γ₂)), we obtain the estimate

in case (100). This shows that J₃ is the Fréchet differentiable and

()

Finally, let us prove (90) and (91). Comparing (92) with (49)–(52), we see that f^† = Δa[u − m*u] − Δm*au, and Δu₀ = h^† = 0. Thus, (56) reads

()

In order to obtain a formula for the right-hand side in (106), we set u^∘ = 2[u(x, T; z) − u_T(x)],

()

f^∘ = ϕ^∘ = 0 and t = T. Then, we obtain (90), where in view of (53) the function ψ is the weak solution of (91).

6. Further Aspects of Minimisation

6.1. Existence of Quasisolutions

For the convenience, we will use also the symbol z to denote the argument ω of J₁.

Theorem 9. (i) Let k ∈ {1; 2} and M ⊂ 𝒵_k be bounded, closed, and convex. Then, IPk has a quasi-solution in M. The set of quasisolutions is closed and convex.

(ii) Let k ∈ {1; 2; 3} and M ⊂ 𝒵₃ be compact. Then IPk has a quasi-solution in M.

Proof. Let us prove (i). The existence assertion follows from Weierstrass existence theorem (see [24, Section 2.5, Theorem 2D]) once we have proved that J_k is weakly sequentially lower semicontinuous in ℱ, that is,

()

But (109) follows from the continuity and convexity of J_k [24]. The convexity of J_k can be immediately deduced making use of the linearity of the ingredient u(x, t; z) with respect to z inside the quadratic functional J_k (for similar computations see [25, Theorem 2]). The closedness of the set of quasisolutions is again a direct consequence of the continuity of J_k. The convexity of the set of solutions follows from the convexity of J_k.

Next, we prove (ii). Let m = inf_z∈MJ_k(z) and z_l ∈ M be the minimising sequence, that is, . By the compactness, there exists a subsequence such that . Due to the continuity of J_k we have . Thus, J_k(z^*) = m. This proves (ii).

In practice, the compact set M may be a bounded and closed finite-dimensional subset of 𝒵_k. The proof of weak lower semicontinuity of J₃ may be harder because this functional is not convex.

6.2. Discretisation and Minimisation

Let us consider the penalised discrete problems

()

where k ∈ {1; 2; 3}, 𝒵_k,L is an L-dimensional subspace of Z_k (L ∈ {1,2, …}) and Π_L is a penalty function related to the set M_L = P_LM with P_L being the orthogonal projection onto 𝒵_k,L. The general assumptions for Π_L are

()

Theorem 10. The problem (110) has a solution.

Proof. The proof repeats the proof of the statement (ii) of Theorem 9, because in view of the accretivity of Φ_k,L, a minimizing sequence is bounded and in a finite-dimensional space any bounded sequence is compact.

The Fréchet derivative of Φ_k,L, that is,

can be identified by a certain element in 𝒵_k,L, that is,

()

where

is the inner product of 𝒵_k. In particular, the addend

is identical to the element P_Lw_k(z) where w_k(z) is the kernel of the functional

. Thus, by virtue of (57), (65), (82), and (90), it holds

()

In w₁, the functions ψ_i and β_l are the z- (or, equivalently, ω-) dependent weak solutions of the problems (58) and (66), respectively. In w₂ the function ψ is the weak solution of (83) and in w₃ the functions u and ψ are the z-dependent weak solutions of (2)–(5) and (91), respectively.

Example 11. Consider the case k = 1. Let M = {z ∈ 𝒵₁ : ∥z∥ ≤ ρ}, where ρ > 0. Further, let ξ_j, j = 1,2, …, be an orthonormal basis in L²(Ω) and 𝒵_1,L = (span(ξ₁, …, ξ_L)) ^N. Then is in 𝒵_1,L identical to the element

()

Moreover, it holds M_L = {z ∈ 𝒵_1,L : ∥z∥ ≤ ρ}. Define a convex penalty function Π_L ∈ C^∞[0, ∞) such that Π_L(z) = 0 for ∥z∥ ≤ ρ and Π_L(z) = d(∥z∥² − ρ²) for ∥z∥ ≥ ρ + ε with some d, ε > 0. Then Π_L satisfies (111).

Let k ∈ {1; 2; 3}. Choose some initial guess z₀ ∈ 𝒵_k,L. Compute the approximate solutions by the gradient method

()

where s = 0,1, 2, … and c_s > 0.

Theorem 12. Let k ∈ {1; 2} and c_s be chosen by the rule

()

where δ_s ≥ 0,

. Then it holds dist(z_s, S) → 0 as s → ∞, where S is the set of solutions of (110).

Proof. The assertion follows from Theorem 5.1.2 of [26] once we have proved that is uniformly Lipschitz continuous, Φ_k,L is convex, and the set M(z₀) = {z ∈ 𝒵_k,L : Φ_k,L(z) ≤ Φ_k,L(z₀) + δ} is bounded. The convexity of Φ_k,L follows from the convexity of its addends Π_L and J_k. The boundedness of M(z₀) is a direct consequence of the accretivity of Φ_k,L following from the accretivity of the addend Π_L.

It remains to show the uniform Lipschitz continuity of in 𝒵_k,L (such a property for is assumed in (111)). Let k = 1. Then by (113) and for any , we have

()

where C₂₂ is a constant independent of z and

. Further, observing (58) and (40), the estimate (20) of Theorem 2 and z = ω, we deduce

()

where C₂₃, C₂₄ are independent of z and

. This proves the uniform Lipschitz continuity of

. Such a property of

can be proved in a similar manner.

The convergence of z_s in case k = 3 is an open issue. This case is more complex because IP3 is nonlinear and the Fréchet derivative of J₃ is not uniformly Lipschitz continuous.

The quasisolutions of IP1–IP3 are not expected to be stable with respect to the noise of the data, that is, the problems under consideration may be ill posed. Nevertheless, from the intuitive viewpoint, a discretisation should regularise an ill-posed problem. Such a property of the discretisation has been proved in many cases [27, 28]. Alternatively, the index s of the gradient method could be used as a regularization parameter (see [29, 30]). Moreover, the addend Π_L can be defined to be the stabilizing term of the Tikhonov′s method instead of the penalty function, that is, Π_L = α∥z∥², where α > 0 is the regularisation parameter. Such a Π_L satisfies (111).

Acknowledgments

The paper was supported by the Estonian Science Foundation (Grant 7728), Estonian Ministry of Education and Science target financed theme SF0140011s09, and the Estonian state programme Smart Composites-Design and Manufacturing.

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All articles

Inverse Problems for a Parabolic Integrodifferential Equation in a Convolutional Weak Form

Abstract

1. Introduction

2. Formal Direct Problem: Notation

3. Weak Direct Problem and Its Convolutional Form

4. Inverse Problems and Quasisolutions

5. The Fréchet Derivatives of Cost Functionals of Inverse Problems

5.1. General Procedure

5.2. Derivative of J₁

5.3. Derivative of J₂

5.4. Derivative of J₃

6. Further Aspects of Minimisation

6.1. Existence of Quasisolutions

6.2. Discretisation and Minimisation

Acknowledgments

References

References

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Inverse Problems for a Parabolic Integrodifferential Equation in a Convolutional Weak Form

Abstract

1. Introduction

2. Formal Direct Problem: Notation

3. Weak Direct Problem and Its Convolutional Form

4. Inverse Problems and Quasisolutions

5. The Fréchet Derivatives of Cost Functionals of Inverse Problems

5.1. General Procedure

5.2. Derivative of J1

5.3. Derivative of J2

5.4. Derivative of J3

6. Further Aspects of Minimisation

6.1. Existence of Quasisolutions

6.2. Discretisation and Minimisation

Acknowledgments

References

References

Related

Information

5.2. Derivative of J₁

5.3. Derivative of J₂

5.4. Derivative of J₃