Stable determination of polyhedral interfaces from boundary data for the Helmholtz equation
We study an inverse boundary value problem for the Helmholtz equation using the Dirichlet-to-Neumann map. We consider piecewise constant wave speeds on an unknown tetrahedral partition and prove a Lipschitz stability estimate in terms of the Hausdorff distance between partitions.
Keywords. Inverse boundary value problem, Helmholtz equation, Lipschitz stability
MSC: 35R30, 35J08, 35J25
We consider an inverse boundary value problem for the Helmholtz equation
where and is the wavespeed. The data are the Dirichlet-to-Neumann map and the objective is to recover the wavespeed. The uniqueness of this inverse problem was established by Sylvester and Uhlmann  for . Concerning stability, conditional logarithmic continuous dependence of the wavespeed on the Dirichlet-to-Neumann map has been proven in  in the case of wavespeeds in with . We refer to Novikov  for a refinement of this stability estimate. The logarithmic rate of stability is optimal . For the inverse conductivity problem the authors of  proposed restricting the class of unknown coefficients to a finite dimensional set to obtain Lipschitz stability estimates. The result was extended to complex-valued conductivities in . In this finite dimensional setting, in [4, 5], a Lipschitz stability estimate for the recovery of piecewise constant wavespeeds for a given domain partition from boundary data for the Helmholtz equation, and an estimate for the stability constant in terms of the number of domains in the partition, were obtained.
Here, we study the problem of determining the finite partition from boundary data given a (possibly large) finite set of attainable values for the wavespeed. Due to the severe nonlinearity of the problem the derivation of Lipschitz stability estimates is more subtle. For this reason, we consider a partitoning of the domain with a (regular) unstructured tetrahedral mesh. In fact, an unstructured tetrahedral mesh admits a local refinement and, with piecewise constant wavespeeds, can accurately approximate realistic models in applications. In geophysics, we mention as an example the work of Rüger and Hale . Here, knowledge of a set of attainable values for the wavespeed can be motivated by the general knowledge of relevant rock types. The deformation allows one to adjust the mesh and recover structures in the models. In geodynamics, these structures can be an imprint of the local geology and tectonics . Moreover, one can parametrize major discontinuities at (polyhedral) surfaces by connecting boundaries of subdomains in the partition via a segmentation for example.
In this paper, we establish a Lipschitz stability estimate expressed in terms of the Hausdorff distance between partitions using tetrahedra from the Dirichlet-to-Neumann map. Lipschitz stability estimates provide a framework for optimization, specifically, iterative reconstruction of the wavespeed with a convergence radius determined by the stability constant [7, 8]. The recovery of polyhedral interfaces then becomes a shape optimization. The analysis in  makes explicit use of a Landweber iteration. Via successive approximations, and making use of estimates for the corresponding growth of the stability constant, the reconstruction can be cast into a multi-level scheme  effectively enlarging the radius of convergence. As an important application, we mention so-called time-harmonic full waveform inversion (FWI) developed in reflection seismology [14, 15, 19, 21] with the goal to image wavespeed variations in Earth’s interior. The data, here, are essentially the single-layer potential operator. However, stability estimates for the Dirichlet-to-Neumann map directly carry over to stability estimates for this operator.
We give an outline of the paper. We first state the main result and the main assumptions (Section LABEL:sec:2). Then we establish a rough stability estimate for the potentials using complex geometrical optics (CGO) solutions following the outline of an estimate in Beretta et al.  (Section LABEL:sec:3). The CGO solutions were introduced by Sylvester and Uhlmann  in their proof of uniqueness of this inverse boundary value problem. The CGO solutions in our analysis differ slightly from theirs to obtain better constants in the stability estimates as proposed in . We proceed with establishing the recovery of the number of tetrahedra in the mesh from the potential, and with expressing the Hausdorff distance between meshes in terms of the difference of piecewise constant potentials defined on these meshes. Naturally, the information on the Hausdorff distance between meshes can be transformed to information on the vertices of the tetrahedra forming the meshes (Section LABEL:sec:4). The main part of the proof of our result pertains to obtaining a lower bound for the Gateaux derivative of the Dirichlet-to-Neumann map under mesh deformation (Section LABEL:sec:5).
We use the Fourier transform convention,
If the function is defined on a subset of , it is extended to attaining the value zero. We denote by the inverse Fourier transform of ,
We introduce coordinates, , in , where and . We denote the open ball in centered at of radius by , and the open ball in centered at of radius by .
2 Assumptions and main result
We let be a bounded domain in such that is connected,
that is, for any point , there exists a rigid transformation of coordinates under which and
where is a Lipschitz continuous (level set) function in such that
We consider the boundary value problem for the Helmholtz equation,
for , and introduce the Dirichlet-to-Neumann map
The normal derivative is defined in the weak sense as
for every . In the above, is identified with where denotes the wavespeed. The solution of (LABEL:4.1) exists in and is unique if is not in the Dirichlet spectrum of on .
We introduce , such that and
where is the first eigenvalue of on . We recall that . (If we detect the spectrum, we substitute the true first eigenfrequency for .) We then assume that
Unstructured tetrahedral mesh
We let be a regular partition of into tetrahedra, namely a collection of closed tetrahedra such that
|a common edge or a common facet;|
We say that two different tetrahedra of such regular partition are adjacent if they share a common facet.
Assumption (LABEL:1.5.5), together with (LABEL:1.1) implies that the tetrahedra of the partition are not degenerate. In particular, there are two positive numbers and (depending on and only) such that
Indeed, we point out that assumptions (LABEL:1.5.5) and (LABEL:1.1) are equivalent to the following
There exists a positive constant such that
for every , every , and .
We show an illustration of a typical model and the assumptions pertaining to the mesh in Figure LABEL:fig:1.
We introduce a finite set of numbers,
representing the possible values which the wavespeed can attain in the domain ,
The potentials are piecewise constant and of the form
such that is a regular partition of with
for some ,
We denote by the norm in defined by
We refer to the values of , , , , , , , and as to the a priori data. In the sequel we will introduce a number of constants that we will always denote by and, unless otherwise stated, will depend on a priori data only. The values of these constants might differ from one line to the other.
We state the main result
Given a domain satisfying (LABEL:1.1) and (LABEL:1.1.5), a set of values , and , there exist two positive constants and depending on the a priori data and on only such that, for every pair of potentials
satisfying Assumptions LABEL:Ass:1 and LABEL:Ass:2, if
and the order of the tetrahedra can be rearranged so that for every we have
where denotes the Hausdorff distance.
3 A rough stability estimate
We begin with developing a rough stability estimate for the recovery of the potential or wavespeed.
Given , , and as in Theorem LABEL:MainTheorem, there exist two positive constants and depending on , , , , , such that, for ,
Proof. We proceed as in . Alessandrini’s identity states that
for every pair of functions and such that
where we use the shorthand notation, .
We fix and let and be unit vectors in such that is an orthogonal set of vectors. We let be a parameter to be chosen later, and set, for ,
As can be easily checked,
of the form
Inserting (LABEL:e2.3) into (LABEL:e1.2), we get
With (LABEL:e2.3) and (LABEL:e3.1) we find that there exists a constant depending only on such that, for ,
, where . Hence,
where . But then, for ,
To estimate the integral in (LABEL:e4.2) we show that for every
where . Indeed, by  we have
Finally, by inserting (LABEL:e5.2) into (LABEL:e4.2), we get that
where . We then choose
and observe that there is a constant depending only on such that, for ,
We now take
and assume that
so that . Then
where . The claim follows upon choosing .
Next, we establish an estimate for the Haussdorff distance between two domain partitions in terms of the difference of potentials defined on these partitions.
Given , and as in Theorem LABEL:MainTheorem, there exists a positive constant depending on , , and such that, if
and the order of the tetrahedra can be rearranged so that for every
where is given by (LABEL:1.3) and by (LABEL:2.2).
Proof. We write
For every we let
We note that
If and then, by (LABEL:1.3),
hence, by (LABEL:P2.3) and (LABEL:P2.0), we have
By assumption (LABEL:2.1), estimate (LABEL:P3.2) implies that is close to . To make this precise, we introduce
and prove that
Indeed, assume that for some , and that there is a point such that
that is, . Using assumption (LABEL:2.1) and (LABEL:2.2) in Remark LABEL:rem1, it then follows that
if . By (LABEL:P3.2)
Thus (LABEL:P3.3) holds provided that
Now we consider for and for some . Since is a partition of , we can write
and we then obtain
If and , then and cannot be adjacent by assumption (LABEL:3.4). This means that there is a unique such that
and, with (LABEL:P5.1),
Thus we proved that for every there is a unique index such that
In particular, this implies that .
By interchanging the roles of and it follows that , is a permutation on and
that, by (LABEL:P3.4), gives (LABEL:P1.4).
Combining Theorem LABEL:rough and Proposition LABEL:proptetr, we obtain the following logarithmic stability estimate
Under the assumptions of Theorem LABEL:rough, there is a constant depending only on the a priori data such that, if
and the order of tetrahedra can be rearranged so that
4 Geometric estimates, construction of an intermediate partition and augmenting the domain
Here, we map the information on the Haussdorff distance of tetrahedra in information on the distance between vertices of these tetrahedra. It is straightforward to see that if , , are tetrahedra generated by vertices , , that then
where denotes a permutation on the set . Moreover, if and satisfies assumption (LABEL:1.5.5) for , then there exists a positive constant , depending on and only, such that
Using Corollary LABEL:cor1 we then obtain
Under the assumptions of Theorem LABEL:rough, there is a positive constant such that if
then for every vertex of (with ) there is a unique vertex of such that
for as in LABEL:2.1.
Proof. It is sufficient to consider , such that
and the statement follows.
We introduce a deformation of the tetrahedra forming the partition of . To this end, for each , we define tetrahedra by its vertices,
The resulting partition is a regular partition of satisfying condition (LABEL:1.5.5). We point out that, by (LABEL:geo1) and (LABEL:geo2), there is a positive constant such that