Exercise 2.1.2 A Nonhyperbolic Cauchy Problem for the Wave Equation)
9.2 Lateral overdetermination: single measurements
While stable and computationally feasible, inverse problems with final overdeter- mination do not very often reflect interesting applied situations when one is given only additional lateral data. In many cases, the results of all possible lateral bound- ary measurements are available. However, even in this case, we have a severely ill-posed inverse problem that is challenging both theoretically and numerically.
We start our discussion of lateral overdetermination with single boundary mea- surements; i.e., we are given one set of lateral boundary data{g0,g1}on the lateral boundary∂×(0,T) or on a part of it.
We will make use of the known transform (9.2.1) u(x,t)=(πt)−1/2
∞
0
exp(−τ2/(4t))u∗(x, τ)dτ.
Observe that
(9.2.2) u(,t)=v(,0,t),
9.2. Lateral overdetermination: single measurements 265 wherev=v(τ,t) is the solution to the following standard parabolic problem:
∂tv−∂τ2v =0 inR+×R+, v =u∗onR+× {0},
∂τv =0 on{0} ×R+. (9.2.3)
This follows from the known basic representation of the solution of the one- dimensional Cauchy problem, provided that we extend the initial datau∗ontoR as an even function:
(9.2.4) v(τ,t)=(4πt)−1/2 ∞
0
(e−(τ−θ)2/(4t)+e−(τ+θ)2/(4t))u∗(θ)dθ.
We observe that when∂τ2u∗∈ L2(0,R) for any positive Rand∂τu∗(0)=0, one can differentiate the equation and the initial conditions of the extended Cauchy problem (9.2.3). Since from the heat equation∂tvsolves the Cauchy problem with the initial data∂τ2u∗, we conclude that the operator (9.2.1) transforms∂τ2u∗into∂tu. The transformationu∗ →uis very stable, while the inverse one is quite unstable (as a solution of the lateral Cauchy problem).
Theorem 9.2.1. Let
(9.2.5) u∗∞(0,T)≤T2M2eM T, ∂τu∗∞(0,T)≤T M2(1+T2M2)eM T andε= u(,0)2(T/8,T).
Then there is C depending only on T,T∗, δsuch that
(9.2.6) |u∗(τ)| ≤CC(δ,M)((−logε)−1+M(−logε)−1+δ) where C(δ,M)=σ1/2eσ2M2T+4M T∗.
We outline a proof based on the following auxiliary results.
Lemma 9.2.2. Under the conditions (9.2.5) for a solution v to the problem (9.2.3) we have
|v(τ,t)| ≤2e4M2t+2Mτ, |∂τv(τ,t)| ≤4.2Me4M2t+2Mτ. PROOF. First, we observe the elementary inequality
(9.2.7) (4πat)−1/2 ∞
0
skeMθ−(τ−θ)2/(4at)dθ≤(k/(Me))ke4M2at+2Mτ. Indeed, the functionθke−Mθ attains its maximum atθ =k/M, henceθke−Mθ≤ (k/(Me))k. So the integral in the left side of (9.2.7) is less than
(k/(Me))k(4πat)−1/2
Re2Mθe−(τ−θ)2/(4at)dθ=(k/(Me))ke4M2at+2Mτ because due to the known integral formula for the Cauchy problem the both sides solve the same Cauchy problem for the heat equation∂tv−a∂τ2v=0.
Using the formula (9.2.4) and the first condition (9.2.6) we yield
|v(τ,t)| ≤(πt)−1/2 ∞
0
θ2M2eMθ−(τ−θ)2/(4t)dθ≤2M2(2/(Me))2e4M2t+2Mτ where we used the inequality (9.2.7) witha=1,k=2. So we have the first bound of Lemma 9.2.2. The second bound follows similarly when we use the integral formula (9.2.4) for∂τv(with difference of exponents instead of the sum and with
∂θu(θ) intstead ofu∗(θ)), second condition (9.2.5), the inequality (9.2.7) with a=1 andk=1,3, as well as the elementary inequality 2/e+(3/e)3<2.1.
From Theorem 3.3.10 and from Lemma 9.2.2 we have
(9.2.8) |v| ≤CC1(M)εκ onT,T∗ = {0< τ <T∗,T/4<t <3T/4}
whereC1(M)=e4M(M T+T∗),κ ∈(0,1) and depends only onT,T∗. Now we will obtain the bound of Theorem 9.2.1 by using stability estimates of the analytic continuation ofv(,t) onto (0,3T/4).
Our first claim is that for allτ ∈(0,T∗) the functionv(τ,t) has the complex- analytic continuation onto the sectorS = {t =t1+i t2 :|t|< σt1}ofCand more- over
(9.2.9) |v| ≤C2(M)eσM2|t| where C2(M)=0.55√
σe2Mτ. This complex-analytic continuation for 0<t2
is given by the formula (9.2.4). Using in addition the bound |e−(τ−θ)2/(4t)| ≤ e−(τ−θ)2/(4σ|t|)fort∈ Sand conditions (9.2.6) we yield
|v(τ,t)| ≤M2√
σ(σπ|t|)−1/2 ∞
0
θ2eMθ−(τ−θ)2/(4σ|t|)dθ
√σM2(2/(Me))2e4σM2|t|+2Mτ
by Lemma 9.2.2 withk=2,a=σ. Using that (2/e)2<0.55 we complete the proof of the bound (9.2.9).
The second claim is the following stability estimate for analytic continuation:
(9.2.10) |v(τ,t)| ≤CC3(M)εtπ/(2β)/C whereC3(M)=√
σeσ2M2T+4M T∗,β=cos−1σ−1.
To prove (9.2.10) we introduce the function V(τ)=v(τ)e−(σ2M2+δ1)t with a positive parameterδ1. This function is complex-analytic inSand from the bound (9.2.9) it follows that|V(τ,t)| ≤C2(M)e−δ1/σ|t|whent∈ S. Hencelog|V(τ,)|is a subharmonic function onSwhich tends to zero at infinity.
Let à(t,I) be the harmonic measure of the interval [T/4,3T/4] in S with respect tot which is defined in section 3.3. As in section 6.3 or in the proof of Theorem 9.4.3, by using the conformal mappingz=tπ/(2β),β =cos−1σ−1 and the maximum principles one can show thattπ/(2β) <C(σ)à(t,I) when 0<t <T.
9.2. Lateral overdetermination: single measurements 267 Since the subharmonic functionlog|V| ≤logC2(M) on∂Sand due to (9.2.8)
log|V(,t)| ≤log(Ce4M(2Mt+T∗)εκe−σ2M2t)≤log(Ce4M T∗εκ) onI when 3< σ, from maximum principles we derive that
log|V(,t)| ≤(1−à(t))logC2(M)+à(t)(log(Ce4M T∗)+κlogε) providedt ∈S. Taking exponents of the both sides, replacingàin the first term by 0, using the lower bound onàand lettingδ1→0 we complete the proof of (9.2.10).
To conclude the proof of Theorem 9.2.1 we will utilize the Mean Value Theorem to getv(τ,0)=v(τ,t)−∂tv(τ, θ(τ,t))tfor someθ∈(0,t). Using Lemma 9.2.2 and (9.2.10) we obtain
|v(τ,0)| ≤CC4(σ,M)(εtπ/(2β)/C+Mt).
Givenδ >0 one can choose largeσ(henceβclose toπ/2) so that (2−δ)β/π = 1−δ. Lettingt =(−logε)−β/π(2−δ)in the bound forv(τ,0) we conclude that
|v(τ,0)| ≤CC4(δ,M)(e−(−logε)δ/2/C+M(−logε)−1+δ).
Finally, sincee−wδ/C ≤C(δ)w−1we complete the proof of Theorem 9.2.1.
Returning to inverse problems, we first give a complete solution of the unique- ness question in the one-dimensional case for some special lateral boundary data.
Theorem 9.2.3. Let=∂, where =(0,1) inR. Let g0(t,j),j =0,1, be the transform(9.2.1)of a function g0∗(τ,j)∈Ck([0,∞))whose absolute value is bounded by Cexp(Cτ)with some C and that satisfies the condition g0∗(k−1)(0)=0.
Then (i) the coefficient c=c(x)∈L∞()of equation(9.0.1)with a0=a = 1,b=0 or (ii) the coefficient a=a(x)∈C2() of the same equation with a0=1,b=0,c=0is uniquely determined by the Neumann data(9.0.6)of the parabolic problem(9.0.1)–(9.0.3)with f =0,u0=0, and g=g(,j)at x = j . PROOF. Case (i). Let us consider the hyperbolic problem
∂τ2u∗−∂x2u∗+c(x)u∗=0 in×(0,T∗), u∗=∂τu∗=0 on× {0}, u∗=g0∗on (0,T∗)×∂.
(9.2.11)
Using (9.2.3) and the remarks after it concerning the transformation of∂τ2u∗, we conclude thatuobtained via (9.2.1) solves the parabolic equation (9.0.1), (where a0=1,a =1,b=0). The additional data
(9.2.12) ∂xu∗=g1∗on ∂×(0,T∗)
necessary to find the coefficientc(x) in (9.2.11) can be obtained by inverting the relation (9.2.1), whereu is replaced byg1. As follows from Theorem 9.2.1, the correspondence betweeng1∗andg1is unique, so we can uniquely determineg1∗.
According to known results about one-dimensional inverse hyperbolic problems (Corollary 8.1.2), a solutionc(x) to the inverse problem (9.2.11), (9.2.12) is unique
(and stable) on (0,12) if we choose T∗=1. With this choice ofT∗ we can use Corollary 8.1.2, because due to finite speed of propagation,g∗1on (0,T∗) coincides with the Neumann data for the hyperbolic problem when=(0,∞). To prove uniqueness ofcon (12,1), one can similarly use the datag∗1atx=1.
Part (ii) follows from Corollary 8.1.7, after a similar reduction to the inverse hyperbolic problem with variable speed of propagation.
The proof is complete.
As an explicit example of the Dirichlet data satisfying the conditions of Theorem 9.2.3 we can useg0(t)=1. The corresponding functiong∗0(τ)=1/2τ2. This can be seen from the parabolic intial value problem (9.2.3) withv(t, τ)=t−1/2τ2. The unstable part of the reconstruction procedure described in this proof is the stepg1→g∗1. On the other hand, this logarithmic instability is isolated and is due to the solution of the simplest, standard ill-posed lateral Cauchy problem for the one-dimensional heat equation (9.2.3), which is relatively well understood.
In the multidimensional case the best uniqueness results are available in the case of nonzero initial datau. LetPbe a half-space inRn,ebe the exterior unit normal to∂P,γequal to∂∩P, andx0a point ofPsuch thatx0ãe≥xãewhenx∈γ.
Theorem 9.2.4. Let a=1,b=0,c=0, f =0, and g∗0,u0∈Cl(), ∂∈Cl where(n+7)/2≤l. Let us assume that
(9.2.13) 0< ε0< u0 on0=∩P.
Then the coefficient a0=a0(x)∈Cl(0)of the parabolic equation (9.0.1) with the given initial data (9.0.3) and satisfying condition
(9.2.14) ∇a0ãe≤0,0≤a0+1
2∇a0ã(x−x0)on0
is uniquely determined on0by the Cauchy data u =g0, ∂νu=g1onγ×(0,T).
PROOF. We will make use of the transform (9.2.1) again, this time reducing our parabolic problem to the following hyperbolic one:
a0∂t2u∗−u∗=0 on×(0,T∗), u∗=u0, ∂τu∗=0 on× {0},
u∗=g∗0on∂×(0,T∗).
By uniqueness of inversion g1→g1∗ due to Theorem 9.2.1, we conclude that the data of the inverse parabolic problem uniquely identify the data of the inverse hyperbolic problem for anyT∗>0. Now uniqueness ofa0follows from Corollary 8.2.3.
The proof is complete.
Exercise 9.2.5. Show that in the situation of Theorem 9.2.4 the coefficientc= c(x)∈ L∞() of the equation∂tu−u+cu=0 is uniquely determined by the
9.2. Lateral overdetermination: single measurements 269 additional lateral Neumann data (9.0.6), provided that condition (9.2.14) is replaced by the condition 0< ε0<u0on0.
A disadvantage of Theorem 9.2.4 and of similar results is the condition that the initial condition is not zero. This condition is not satisfies in many applications when the physical field is intiated from the lateral boundary. Another seemingly excessive restriction of Theorem 9.2.4 is the condition (9.2.14) which guarantees absence of trapped rays/validity of appropriate Carleman estimates in the associ- ated hyperbolic problem.
One of few available theoretical results in case of zero initial data and few boundary measurements requires simulteneous overdetermination at∂and inside at some fixed moment of timeθ∈(0,T).
Theorem 9.2.6. Let us assume that a0,a,b,c∈C1(Q)andα, ∂tα∈C(Q). Let γ be any open subset of∂and=γ×(0,T). Letθ∈(0,T)and
(9.2.15) ε0< αon× {θ}
for some positive numberε0.
Then any pair(u,F)∈ H2,1;2()×L2()satisfying the equation (9.0.1) with f =αF ,∂tF =0in Q is uniquely determined by the lateral Cauchy data
u =g0, ∂νu=g1onγ×(0,T) and the intermediate time data
u(, θ)=uθ on.
This result follows from Theorem 8.2.2 because as shown in the proof of The- orem 3.3.10 for any subdomain 0 of which is a diffeomorphic image of a halfsphere, so thatγ0=∂0∪∂is inγ, there is a Carleman estimate (3.2.3) for the equation (9.0.1) with the weight functionϕwithϕ >0 onγ0×(θ−ε, θ+ε) andϕ <0 onQ\(0×(θ−ε, θ+ε)). We would like to emphasize locality of this uniqueness statement: indeed we do not assume that the lateral boundary data are known outside, socan be an arbitrary subdomain of a larger domain. We only need condition (9.2.15).
By using some new Carleman estimates and assuming additional regularity of the coefficients of the parabolic boundary value problem Imanuvilov and Ya- mamoto [IY1] showed that under the additional assumption thatusatisfies a lateral boundary conditionu=0 or∂ν(a)u+b0u=0 on∂×(0,T) one has Lipschitz stability estimate
F2()≤C(uθ(2)()+
w∂tl∂mj2())
where the sum is over j =1, . . . ,n,l,m=0,1,w is some nonnegativeC2(Q)- function which goes to infinity as t goes to 0 or T. In this stability estimateC depends on the coefficients of the parabolic initial boundary value problem, on ,T, γ,α∞(Q)+ ∂α∞(Q) and onε0 from condition (9.2.15). This a best
possible estimate showing a possibility for a very efficient numerical solution of this inverse problem.