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Stability and Reconstruction in Gel'fand Inverse Boundary Spectral Problem

We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the Laplace-Beltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in stable way when manifold is a priori known to satisfy natural geometrical conditions related to curvature and other invariant quantities.


I. Introduction
In this paper we consider the questions of stability and approximate reconstruction in the inverse boundary spectral problem which is also called the generalized Gelfand inverse problem [13] for Riemannian manifolds. To formulate the problem and the main results, we need to introduce some basic notations. We will denote by (M, g) an unknown, m-dimensional, compact connected Riemannian manifold with a (smooth) metric g and non-empty boundary ∂M . Our goal is to find (M, g) from boundary data. The boundary ∂M is itself an (m − 1)-dimensional compact differentiable manifold. We note that we do not assume the knowledge of the metric i * (g) generated on ∂M by the metric g , where i : ∂M → M is an embedding or even the corresponding area element dS g .
Because in the inverse problems the boundary ∂M is usually known, we will consider a class M = M ∂M of compact, connected Riemannian manifolds which have the same, i.e. diffeomorphic, boundary, ∂M .

Definition 1
The collection {λ j , ϕ j | ∂M ; j = 1, 2, . . .} is called the boundary spectral data of (M, g) and the map D -the boundary spectral map.
It was shown by Belishev-Kurylev [7] who used the boundary control method [5] and the unique continuation result of Tataru [35] that (For earlier uniqueness results in the Gelfand inverse problem for isotropic operators obtained by the boundary control method see, e.g. [6]). The complex geometric optics method [33] was applied to this problem in [26], [27].) In this paper, we will first analyse the question of stability of the inverse problem, i.e. the question of continuity of D −1 . Later, we will also describe a procedure of an approximate reconstruction of (M, g).
Alessandrini [1], [2] considered an inverse boundary problem for a Schrödinger operator, −∆ + q in a bounded domain of R m and obtained a log-type stability estimate in the Gel'fand inverse problem (see also [23]).
Alessandrini and Sylvester [3] and Stefanov and Uhlmann [31] considered inverse boundary value problems for the wave equation which are closely related to the Gel'fand inverse problem. Let where a(x, D) = −∆+q in [3] and a(x, D) = −∂ i a ij ∂ j in [31] and the inverse data is given as a non-stationary Dirichlet-Neumann map. For sufficiently large T and, in [31] a ij close to δ ij , they showed Hölder-type stability. These results raise the question of an optimal stability estimates in the Gel'fand inverse problem and inverse boundary value problem for the wave equation. Although exact stability estimates lie outside the scope of this paper which is devoted to the analysis of geometric conditions upon (M, g) to provide continuity of D −1 , we believe that Hölder-type estimates could be valid only under rather strong requirements upon (M, g), like strong geodesic property (see e.g. [10], [30]). Further results on stability for different types of inverse boundary value problems could be found in [16]. For counterexamples see e.g. [24].
Clearly, to discuss the continuity of D −1 it is necessary to define appropriate topologies in M and B. However, it is well-known (see e.g. [11]) that inverse problems are, in general, ill-posed. Therefore, to gain stability we need to know a priori that the unknown object, which we are going to reconstruct, belongs to some compact class K (see e.g. [16]). To obtain stability results, the topologies of M and B have to satisfy two properties. First, the map D : K → B has to be continuous. Second, D has to be injective. When these two properties are satisfied and K ⊂ M is a compact set then the continuity of the map D −1 : D(K) → K follows from basic results of general topology. In the first part of the paper we carry out the constructions of the appropriate topologies. We would like to note that a proper topology on M is the Gromov-Hausdorff topology and the (pre)-compact subsets we actually use are the manifolds of bounded geometry (see e.g. [9], [14], [28], [29]). The topology on B is defined from the boundary spectral convergence.
In the second part of the paper our attention will be focused on the problem of an approximate reconstruction of (M, g) when we do not know all boundary spectral data. Rather, we know only the first eigenvalues, λ j < δ −1 with some small δ > 0 and boundary values of the corresponding eigenfunctions and, furthermore, we know them with some error. Namely, we have a finite collection of pairs {µ j , ψ j ; j = 1, . . . , n(δ −1 )}, µ j ∈ R, ψ j ∈ L 2 (∂M ) and know that they are close to {λ j , ϕ j | ∂M ; j = 1, . . . , n(δ −1 )}, where n(δ −1 ) is the number of eigenvalues that are smaller than δ −1 . Our definition of topology on B is adjusted to make the intuitive notion of closedness of {µ j , ψ j } to {λ j , ϕ j | ∂M ; j ≤ n(δ −1 )}, rigorous.
In the second part of the paper we describe the stable reconstruction procedure but omit the proof showing that this procedure is stable. These proof are to be published elsewhere. We note that to carry out the reconstruction procedure we will need to work under stronger requirements on (M, g) than those that are needed for stability. Similarly, in the first part we will only formulate necessary results omitting all proofs. The proofs and detailed description of the reconstruction procedure will be given in a forthcoming paper.
II. Main Results. Stability. We start with a proper topology on M.
We remind that Remark 1. In the future we will identify diffeomorphic manifolds assuming implicitly that desired statements are valid after an automorphism of M . Here Rm is the Riemannian curvature tensor on (M, g) considered as a 4-linear form on T M , ∇ is the covariant derivative, and S is the second fundamental form with respect to the inner product, i.e. a quadratic form on T ∂M . Furthermore, diam is the diameter of (M, g) and inj is the injectivity radius of (M, g). Condition iv) consists of three different conditions: (α) Let x ∈ ∂M . Denote, as usual, by B ∂M (x, r) the open metric ball (on ∂M ) of the radius r with center in x. Then, for any r < i 0 , B ∂M (x, r) is a domain of normal coordinates on ∂M centered at x.
Then, for any r < i 0 , the cylinder C(x, r) is a domain of boundary normal coordinates. Here C(x, r) consists of all points y ∈ M such that d(y, ∂M ) < r and the unique boundary point z(y) ∈ ∂M, nearest to y is in the ball B ∂M (z(x), r).
Next we introduce topology on the set B. Definition 4 (Boundary spectral topology.) Let δ > 0. Collections Clearly, condition iv) depends on the choice of the boundary measure used in the definition of L 2 (∂M ). However, due to the compactness of ∂M different smooth measures on ∂M determine equivalent L 2 -norms.
We note that such topology was introduced by Alessandrini [1], [2] who studied stability in the Gel'fand inverse problem for a Schrödinger operator.
We are now able to formulate the principal stability result for inverse boundary spectral problem.
Theorem 2 For any Λ, D, i 0 > 0, the map D −1 exists and is continuous on and their boundary spectral data, are δ-close then M 1 and M 2 are diffeomorphic. Moreover, for any α ∈ (0, 1), Remark 2. The convergence of the metrics in the C 1,α -topology means that in a proper coordinate system on M = M i , i = 1, 2, the metric tensors g ij converge in the C 1,α -sense, see e.g. [28].
Remark 3. It follows from definition 4 that to apply Theorem 2 it is sufficient to know only the parts of the boundary spectral data of (M 1 , g 1 ) and (M 2 , g 2 ) that correspond to the eigenvalues λ i j < δ −1 . Theorem 2 and the fact that in the class M(Λ, D, i 0 ) the Gromov-Hausdorff topology is equivalent to the Lipschitz topology (see ( [21]) implies the following corollary that describes the Lipschitz convergence of Riemannian manifolds.
Then, for any ε > 0 there is δ > 0 such that if the boundary spectral data of these manifolds are δ-close then M 1 and M 2 are diffeomorphic and Although the boundary control method that was instrumental to prove Theorem 1 is, in general, not applicable to C 1,α Riemannian manifolds it was possible to show that In Theorem 2, the condition iii) in M(Λ, D, i 0 ) is automatically satisfied from an estimate of λ 2 in [25]. The other conditions are natural for the stability of the inverse problem because, otherwise, the convergence of the boundary spectral data does not imply the convergence of the corresponding Riemannian manifolds. We illustrate this thesis with the following examples. Example 1. Let (M, g) be a two-dimensional manifold obtained by the following surgeries of the two-sphere S 2 . Add a handle H near the North pole and cut-off a small piece B to make a boundary near the South pole. If the metric size of the handle tend to 0 uniformly then, in the boundary spectral topology, the boundary spectral data of (M, g) tend to the boundary spectral data of the sphere with a hole but without a handle (see e.g. [34]). This shows the necessities of i) and iv) in M(Λ, D, i 0 ). Figure 1: Two representations of the same manifold: Cylinder in R 3 pinched in the middle and its representation as an annulus in R 2 with variable metric. When ε → 0 the arclength of circles in the region A tend to 0 so that the manifold splits into two components.

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Example 2. Let (M, g) be cylinder in R 3 with the boundary made of two components, ∂M = ∂M 1 ∩ ∂M 2 (see Fig.1). We pinch the cylinder in the middle so that the diameter tends to 0. Therefore, the manifold eventually "splits" into two disjoint semispheres. Meanwhile, in the boundary spectral topology on ∂M , the boundary spectral data of (M, g) tend to the boundary spectral data of this disconnected manifold. Again, conclusion of the theorem is no more valid. This second example may be also described in terms of a perturbation of the metric in an annulus A(1, 4) = {(r, θ) : 1 ≤ r ≤ 4} , where (r, θ)) are the polar coordinates in R 2 . We define the length element by Then, for ε → 0 each circle r ∈ (2, 3) shrinks to a point so that the internal part near r = 1 of A becomes separated from the external part near r = 4. We would note that although this example does not precisely satisfies the definition of M ∂M because the limiting manifold is no longer connected. However, this definition can be extended onto the case of, probably disconnected Riemannian manifolds with the same boundary ∂M which have no components without the boundary if we change the condition of boundedness of diam(M ) to the boundedness of vol(M ).
Example 3 Take X and Y to be isospectral manifolds without boundary of different topological types (see e.g. [32]) and Z be a Riemannian manifold with boundary ∂Z. Let M ε be a Riemannian manifold obtained by connecting Z with X by a thin tube of radius ε 2 of length ε. Similarly, let N ε be a Riemannian manifold obtained by connecting Z with Y by a thin tube of radius ε 2 of length ε. When ε → 0, the boundary spectral data of M ε tend to the union of the boundary spectral data of Z and {µ j , 0}, where µ j is the spectrum of X. Clearly, the boundary spectral data of N ε tend to the same limit. Thus, for any δ > 0, there exists ε 0 > 0 such that the boundary spectral data of M ε and N ε are δ-close when ε ≤ ε 0 . However, M ε and N ε are of different topological types.
In these examples Rm → ∞ and inj → 0 when ε → 0 which show the importance of conditions i) and iv) of the theorem. Similar examples can be given to illustrate the importance of condition ii). To construct a metric space approximation to (M, g) we will assume further regularity properties of the class of manifolds.
The algorithm to construct (Y, d) is described in section V.
Remark 5. By definition 4 the δ-closedness does not involve {µ i j , ψ i j } with |µ j | > δ −1 . Similarly, the construction of (Y, d) does not use these data.
IV. Geometric convergence and stability. The proof of Theorem 2 is based upon the following result. ii) for any α ∈ (0, 1), F * n (g n ) converge to g in the C 1,α -topology. Remark 6. Proposition 1 is obtained by applying the ideas developed by M. Anderson ([4], see also [15]) in the interior of the manifold and the fundamental equation of Riemannian geometry (see [28]) near the boundary. The latter equations is also known as the Riccati equation for the second fundamental form along normal geodesics. This equation gives the desired regularity estimates for the metric tensor in the boundary normal coordinates.
The geometric convergence described by Proposition 1 yields the continuity of the direct problem. Remark 7. Proposition 1 was proven by Kodani [21] in the case of weaker regularity. Namely, Kodani showed that the manifolds satisfying i) Rm(M, g) ≤ Λ, iii) diam(M, g) ≤ D, ii) S(M, g) ≤ Λ, iv) inj(M, g) ≥ i 0 . are pre-compact in the Lipschitz topology rather than the C 1,α -topology. This result is sufficient for continuity of D. This can be obtained by means of the perturbation theory of quadratic form (see e.g. [12] in the case of manifolds without boundary). Therefore, the result of Kodani would have been sufficient to prove the continuity of D V. Construction of a finite ε-net. The procedure of constructing an approximation (Y, d) consists of several steps which we will explain separately. We will also point out smoothness requirements which are necessary to carry out different steps of the procedure.
a. Construction of Fourier coefficients of waves. For simplicity, we assume here that the metric on the boundary, i * (g), i : ∂M → M is given. Consider an initial-boundary value problem associated with the Neumann Laplace operator, where C 1 p ([0, D], L 2 (∂M )) consists of piecewise C 1 -functions of t. We introduce the wave operator W t given by where dS g is the boundary area element and This formula makes possible to find approximately the first Fourier coefficients of the waves u f if we know {µ j , ψ j ; j = 1, . . . , n(δ −1 )}.
b. Construction of domains of influence. Let Γ ⊂ ∂M be open.
By the fundamental result of Tataru ([35], see also [18], Ch.2.5), we have Theorem 5 Assume that the metric tensor g is C 1 -smooth. For any τ > 0 and Γ ⊂ ∂M , where L By Theorem 5, for any σ > 0 there is where χ Ω is the characteristic function of a set Ω. It then follows from the general results of control theory that there is a finite linear combination f , such that The coefficients a j of the function (5) which satisfies equation (6) can be found using the boundary spectral data {λ k , ϕ k | ∂M ; k = 1, . . . , K}, K > J by means of some variational procedure.
Theorem 6 For any σ > 0 there are parameters C, N, I, K, δ, N ≤ I ≤ K ≤ n(δ −1 ), such that if {µ j , ψ j } is δ-close to the boundary spectral data {λ j , ϕ j | ∂M } of a Riemannian manifold (M, g), then a minimizer f * ∈ H N of the functional A N,I,K such that satisfies the equation By Proposition 2 the boundary spectral data depend continuously on the Gromov-Hausdorff distance restricted on M(Λ, D, i 0 ). Consider as a function of (M, g) ∈ M(Λ, D, i 0 ). By Proposition 1, the perturbation theory [19] shows that the wave u f (M,g) (·, t) depends continuously (in the c. Construction of the boundary distance map. As Proposition 1 makes it possible to evaluate an approximate volume, vol a (M α ). We will use this observation to construct a finite approximation to the set of the boundary distance functions associated with (M, g). Namely, for any x ∈ M , let r x ∈ L ∞ (∂M ) be given by The boundary distance functions determine the boundary distance map R : It turns out that the map R or, more precisely, its image R(M, g) is sufficient to reconstruct (M, g) (see [22] for the procedure of the reconstruction of (M, g) from R(M )). We will use an approximation R * to R(M, g) to construct the desired metric space (Y, d) in Theorem 3. Note that the metric d is different to the induced metric as a (metric) subspace of L ∞ (∂M ) (cf.
Step 1 in subsection d.
When vol a (M * β ) ≥ σ we associate with this β a function r β ∈ L ∞ (∂M ), Thus, if vol(M * β ) ≥ 2σ so that vol a (M * β ) ≥ σ, there is an approximate function r β such that for any x ∈ M * β , In the construction of the approximate volume of a set M * β = ∅ such that vol(M * β ) < 2σ it may happen that vol a (M * β ) < σ. Thus, according to the procedure, R * do not contain r β corresponding to this multi-index β. Consider next a point x ∈ M * β for such β. By condition (3) there is an upper bound N (η) for the number of all multi-indexes β. Moreover, due to condition iv) of definition 3, vol Summarizing the previous considerations we obtain the following result.
Here d H is the Hausdorff distance between subsets in the metric space L ∞ (∂M ). Moreover, δ can be chosen uniformly on any M(Λ, D, i 0 ).
However, condition (9) is not valid for general manifolds. Therefore, d ∞metric is inappropriate to construct an approximation to (M, g) in the Gromov-Hausdorff topology. In this section we will describe how to equip the finite metric space R * with another metric d * so that (Y, d) = (R * , d * ) becomes close to (M, g) in the Gromov-Hausdorff metric. In our description we will assume that the whole R(M ) rather then R * is known. We will show how to find an approximation to d g (x, y) from R(M ). This procedure is valid also in the case when we know only R * . The corresponding result is given at the end of this section.
1. Assume first that x 1 , x 2 ∈ M int can be connected by a unique shortest geodesic, γ(x 1 , x 2 ). Assume in addition that this geodesic can be continued beyond x 2 to a boundary point z ∈ ∂M so that γ(x 1 , z) is a shortest geodesic between x 1 and z. Then, can be found from R(M, g). It remains to identify those r 1 , r 2 ∈ R(M ) which correspond to x 1 , x 2 with this property. As i * g on ∂M is known, it is possible to extend (M, g) to a larger Riemannian manifold ( M , g) which has a continuous, piecewise C 1 metric g. We can then extend each function r x ∈ C(∂M ) to the function r x ∈ C( M \ M ), r x (y) = d M (x, y). Take a point z ∈ ∂M and consider minimal geodesics γ(x i , z), i = 1, 2. Assume that γ(x i , z) \ {z} ⊂ M int , z is not a cut point along any of γ(x i , z) and γ(x i , z) are transversal to ∂M . Then these geodesics can be extended as minimal geodesics beyond z upto some points y i ∈ M \ M . Denote by γ i (z, y i ) ⊂ M \ M these extensions. Then the geodesic γ(x 1 , x 2 ) connecting x 1 and x 2 extends to a minimal geodesic to z ∈ ∂M if and only if γ 1 (z, y 1 ) = γ 2 (z, y 2 ).

2.
Let now x 1 , x 2 ∈ M int lie on the sides γ(y 3 , y 1 ) and γ(y 3 , y 2 ) of a good geodesic triangle ∆(y 1 , y 2 , y 3 ). A geodesic triangle ∆ is called good if all three geodesics γ(y i , y j ) can be continued to ∂M as the shortest geodesics (see Fig. 2). Then, by part 1 we can find d g (y i , y j ) and also d g (x i , y i ). Let ∆( y 1 , y 2 , y 3 ) be a triangle in R 2 with d e ( y i , y j ) = d g (y i , y j ), where d e stands for the Euclidean distance. Let x i , i = 1, 2 be the points on the sides [ y 3 , , y i ] of ∆ with d e ( x i , y i ) = d g (x i , y i ). Then, by the Alexandrov lemma, where σ = max d g (y i , y j ) and c is uniform on M(Λ, D, i 0 ). Therefore, if for some given x 1 , x 2 , we can find a proper geodesic triangle ∆(y 1 , y 2 , y 3 ) of a small size, we can find d g (x 1 , x 2 ) with an error of order σ 2 .
3. The existence of a good geodesic triangle for any sufficiently close x 1 , x 2 ∈ M int is based on the following result. . Let x ∈ M . Then there is a constant ρ > 0 such that for any (M, g) ∈ M(Λ, D, i 0 ) and x ∈ (M, g) with d g (x, ∂M ) ≥ Cρ there is a vector u ∈ T x M |u| = 1 with the following property: Let (y, v) ∈ B ρ (x, u), |v| = 1. Then the geodesic γ y (tv), t ≥ 0, can be extended as a shortest geodesic to the boundary.
Here B ρ (x, u) is the ball of radius ρ with center at (x, u) in the Sasakian metric on T M . Combining steps 1-3 we can find an approximation to d g (x 1 , x 2 ) when x 1 , x 2 are sufficiently close to each other and not too close to ∂M . (For the definition of the Sasakian metric, see e.g. [29].) There are unique z i ∈ ∂M with d g (x i , z i ) = r i (z i ). Denote d = d(x 1 , x 2 ) = d 2 ∂M (z 1 , z 2 ) + |r 1 (z 1 ) − r 2 (z 2 )| 2 1/2 , where d ∂M (·, ·) is the distance along ∂M . Then, Therefore, d is a good approximation to d when x i are close to ∂M .

5.
Steps 1-4 define approximate distance d(x 1 , x 2 ) for sufficiently close x 1 , x 2 . Using a standard procedure we can extend d(x 1 , x 2 ) onto arbitrary When we know the metric space (R * , d ∞ ) rather then R(M, g) we can carry out the above constructions approximately. This gives rise to the following result.
Lemma 3 Let Y ⊂ L ∞ (∂M ) be a set described in Lemma 1. It is possible to equip Y with a metric structure d(r 1 , r 2 ) so that where ε(η) → 0 when η → 0. The function ε(η) can be found uniformly for M(Λ, D, i 0 ).
Remark 9. All considerations remain valid for the Dirichlet Laplacian. We need just to take into account that ϕ 1 = 0 in M int . Remark 10. It is possible to use the convergence of the heat kernels restricted to ∂M instead of the boundary spectral convergence. Besides, it is possible to modify the procedure to find an approximation to the heat kernel associated with −∆ g rather then R * . This puts our results within the framework of the spectral convergence of Riemannian manifolds (see e.g. [8], [17]). This will be described in the forthcoming paper.
Remark 11. Requirements of C 1 -regularity of Rm and S are of technical nature. We intend to improve reconstruction procedure to be valid for weaker regularity assumptions for Rm and S.
Part of our results also briefly announced in [20].