Since we will also be dealing with weak convergences between Lp spaces the following result will be also of interest
Theorem 3.7.8 ([5, Theorem 3.9]). Let f be a Caratheodory function, and sup-pose that the superposition operator F generated byf acts fromLp(Ω) into Lq(Ω). Then F is weakly continuous if and only if f satises
f(x, u(x)) = c(x) +a(x)u(x).
So superposition operators that come from a truly nonlinear functionf can not be weakly continuous.
Remark 3.7.9. The theorems regarding dierentiability and weak continuity will explain the extended analysis that we have to make for the parameter identication problem compared to the usual analysis of parameter identication problems in Hilbert spaces. We will come back to this when we analyze the parameter-to-state map of our parameter identication problem in Chapter 6.
In particular, we need the following
Theorem 3.8.3 ([42, Theorem 5.30]). LetA:D(A)⊂X →Y be a linear, closed and densely dened operator. IfA−1 exists and is bounded, then(A∗)−1 :Y∗ →X∗ exists and is bounded and it holds
(A∗)−1 = (A−1)∗.
CHAPTER 4
Parameter identication
In a parameter identication problem, the outcome of a system like in (1.3) is at least partially known, that means one can measure the solutions of the dierential equation on a subset Ω0 ⊂ ΩT, and one wants to extract certain parameters from these measurements. For example, in (2.4) we can measure the genetic concentrations at certain time instances and one wants to know the interaction of dierent genes. A problem that almost always comes with this task is the posedness of this inverse problem. We distinguish two dierent kinds of ill-posedness. The rst question one always has to ask if there is a unique dependence of the parameters onto the data or mathematically spoken, if the forward operator is injective. The second question then is, if the parameters depend continuously on the data. Especially in problems involving partial dierential equations this is usually not the case and one has to deal with this.
4.1 Identiability
The rst question one may ask is, if for a given solution of a dierential equation one can obtain an at least locally unique set of parameters. To examine this question further, we introduce the concept of identiability, which we adapt from [7]:
Denition 4.1.1. Let P be the parameter space of a parameter identication problem F : P → Y and d : P ×P → R be a distance function. In a parameter identication problem F(p) = y, p is called globally identiable if F is injective.
The parameter p is called locally identiable if there exists a ε > 0 such that for each p∗ with d(p, p∗)< ε and F(p) = F(p∗) it holds p=p∗. Otherwise, p is called unidentiable.
Local non identiability in fact is much worse than non continuous dependence of the parameters on the data, because even with perfect data, one can not expect to come close to the true solution in general. In this case, one can only hope to characterize the set of parameters S := {p ∈ P | F(p) = F(p†)} and possibly use a priori information to pick the right parameter. However, depending on the structure of the problem, characterizing such a set might be impossible.
4.1.1 Identiability in parabolic systems
In systems of dierential equations, identiability is always a problem and often only holds under strong restrictions on the parameters [60, 16]. In scalar equations, where only one parameter has to be identied, identiability often can be shown, at least locally under mild assumptions on the measurements, see Chapter 7. The more variables are involved in a system and the bigger the system becomes, it seems more and more unlikely that identiability holds and thus the conditions needed to show identiability become more and more restrictive. Especially in parabolic equations or systems with more than one parameter involved, one cannot expect identiability in general if all parameters are space and time dependent.
To show this, we consider our example from Section 2.2.
Theorem 4.1.2. LetN ≥2. LetP andU be Banach spaces, withP =P×...×P and W =U ×...×U and let both spaces be equipped with the product-one-norm.
Assume that for every parameter W ∈ P the equation (2.1) has a unique solution in the space W. Further assume U ↪→P. Then the interaction parameter W ∈ P in equation (2.1) is unidentiable with respect to the P-norm.
Proof. Without loss of generality we assume N = 2. We show that in any ε -neighbourhood of a given parameterW ∈ W there is at least one W∗ with ∥W− W∗∥ ≤ ε but F(W) = F(W∗). Let ε > 0, W ∈ W and let u be the solution corresponding to W. Without loss of generality, we assume u ̸= 0. Note that if ui(x, t) = 0, then the parameter W has no inuence at the point (x, t). Let
∥u∥=∥u1∥P +∥u2∥P and dene W11∗ :=W11− ε
2 u2
∥u∥P and W12∗ :=W12+ ε 2
u1
∥u∥P
as well as
W21∗ :=W21 and W22∗ :=W22.
The parameter W∗ is a well dened element from P because of the continuous embedding U ↪→ P. Further, let u∗ be the solution to (2.1) with the parameter
W∗. For t∈(0, T]we get
− u2
∥u∥u∗1+ u1
∥u∥u∗2
=− u2
∥u∥u∗1+ u1
∥u∥u∗2+ u2
∥u∥u1− u1
∥u∥u2
=− u2
∥u∥(u∗1−u1) + u1
∥u∥(u∗2−u2).
Since both, u and u∗ solve a dierential equation, we get by subtracting the respective equations
(u∗1 −u1)t− ∇ ·D∇(u∗1 −u1) +W11(u∗1−u1) +W12(u∗2−u2)
− ε 2
u2
∥u∥u∗1 +ε 2
u1
∥u∥u∗2
=(u∗1 −u1)t− ∇ ·D∇(u∗1 −u1) +W11(u∗1−u1) +W12(u∗2−u2)
− ε 2
u2
∥u∥(u∗1−u1) + ε 2
u1
∥u∥(u∗2−u2)
=(u∗1 −u1)t− ∇ ·D∇(u∗1 −u1) +W11∗(u∗1−u1) +W12∗(u∗2−u2) = 0 and
(u∗2−u2)t− ∇ ·D∇(u∗2−u2) +W21∗(u∗1−u1) +W22∗(u∗2−u2) = 0.
Hence, v =u∗−u solves the dierential equation vt− ∇ ·D∇v+W∗v = 0 in ΩT
∂
∂νv(0, t) = 0 on ∂Ω×[0, T] v(x,0) = 0 on Ω× {0}.
Clearly, v = 0 is a solution of this dierential equation as well. Thus, by our assumption that the solution is unique, it must hold u = u∗. Hence, F(W) = F(W∗). Further it holds
∥W −W∗∥P =
2
∑
i=1 2
∑
j=1
∥Wij −Wij∗∥P
≤ ε
2∥u∥∥u2∥P + ε
2∥u∥∥u1∥P
≤ε.
This concludes the proof.
Remark 4.1.3. The assumptions we made to show the non-uniqueness are not very strong. For example, if one choosesP =L2(Ω)andU =W21(and restricts the domain ofF in an appropriate way), existence and uniqueness follows by classical weak solution theory (see Chapter 5 or cf. [25]).
Remark 4.1.4.
i) In the case of Theorem 4.1.2,N2 space and time dependent parameters have to be identied, but there is only data for two space and time dependent functions. So the data is highly underspecied.
ii) One can easily construct similar examples for non uniqueness in the case that multiple parameters that have to be identied in a scalar equation (or in a system).
iii) We have used the norm of the parameter space as distance function in The-orem 4.1.2. But even for more general distance functions that can be related to the spaces P and W the parameter stays unidentiable. This is especially interesting for Tikhonov-regularization, because here, usually certain norms are used as a prior to highlight properties of the function.
For now on, we will leave the identiability issue and will return to it in Section 7.