A mollifier approach to regularize a Cauchy problem for the inhomogeneous Helmholtz equation Pierre MARECHAL´ ,∗ Walter Cedric SIMO TAO LEE,† Faouzi TRIKI‡ May 7, 2021 Abstract The Cauchy problem for the inhomogeneous Helmholtz equation with non-uniform refraction index is considered. The ill-posedness of this problem is tackled by means of the variational form of mollification. This approach is proved to be consistent, and the proposed numerical simulations are quite promising. 1 Introduction Let V be a C3;1 bounded domain of R3 with boundary @V . For x0 @V , we denote by ν(x0) the unit normal vector to @V pointing outward V . Let Γ be a nonempty2 open subset of @V . We consider the Cauchy problem for the inhomogeneous Helmholtz equation ∆u(x) + k2η(x)u(x) = S(x); x V; (1) 0 0 0 2 @νu(x ) = f(x ); x Γ; (2) 2 u(x0) = g(x0); x0 Γ: (3) 2 1 arXiv:2105.02665v1 [math.AP] 6 May 2021 Here, u = u(x) is the unknown amplitude of the incident field, η L (Ω) is the refraction index, k is a positive wave number, S L2(Ω) is the source function,2 and f L2(Γ) and 2 2 ∗Institut de Math´ematiques universit´e Paul Sabatier, 31062 Toulouse, France. Email: [email protected] †Institut de Math´ematiques universit´e Paul Sabatier, 31062 Toulouse, France. Email: wsimo- [email protected] ‡Laboratoire Jean Kuntzmann, UMR CNRS 5224, Universit´eGrenoble-Alpes, 700 Avenue Centrale, 38401 Saint-Martin-d'H`eres,France. Email: [email protected]. This work is supported in part by the grant ANR-17-CE40-0029 of the French National Research Agency ANR (project MultiOnde). 1 g L2(Γ) are empirically known boundary conditions. 2 The Helmholtz equation arises in a large range of applications related to the propagation of acoustic and electromagnetic waves in the time-harmonic regime. In this paper, we con- sider the inverse problem of reconstructing an acoustic or electromagnetic field from partial data given on an open part of the boundary of a given domain. This problem called the Cauchy problem for the Helmholtz equation is known to be ill-posed if Γ does not occupy the whole boundary @V [13,9, 18,1]. In [17], the above system was considered in the particular case where the refraction index is constant. However, in practice, the emitted wave travels through an environment in which the refraction index fails to be constant, and we need to investigate the corresponding problem. We are facing a linear inverse problem. Our aim is to derive a stable approximation method for this problem, which yields stable and amenable computational scheme. Our main focus will be on mollification, in the variational sense of the term, which turns out to be both flexible and numerically efficient. Mollifiers were introduced in partial differential equations by K.O. Friedrichs [27, 11]. The term mollification has been used in the field of inverse problems since the eighties. In the original works on the subject, mollifiers were used to smooth the data prior to inversion. In his book, D.A. Murio [23] provides an overview of this approach and its application to some classical inverse problems. Let us also mention the paper by D.N. Hao [14], which provides a wide framework for the mollification approach in this initial meaning. In [21], A.K. Louis and P. Maass proposed another approach, based on inner product duality. This approach has been subsequently referred to as the method of approximate inverses [24]. The approximate inverses are particularly well adapted to problems in which the adjoint equation has explicit solutions. A third approach, based on a variational formulation, also appeared in the same period of time. In [20], A. Lannes et al. gave such a formulation while studying the problems of Fourier extrapolation and deconvolution. This variational formulation was not studied further until the papers by N. Alibaud et al. [2] and by X. Bonnefond and P. Mar´echal [7], where convergence properties of the variational formulation was considered. A definite advantage of the variational approach to mollification lies in the fact that it offers a quite flexible framework, just like the Tikhonov regularization, while being more respectful of the initial model equation than the latter. The paper is organized as follows. In Section2, we introduce the linear operator associated to the Cauchy problem for the inhomogeneous Helmholtz equation. We propose a regularized variational formulation of the ill-posed problem based on mollification. Under an additional smoothness assumption on the targeted solution we show in Theorem7 that the unique min- imizer converges strongly to the minimum-norm least square solution. Section3 is devoted to numerical experiments. We consider two numerical examples in order to illustrate the efficiency of our regularization approach. 2 2 Functional setting and regularization We shall work in a functional space which enables us to interpret the ideal (exact) data as the image of u by a bounded linear operator. We observe that: (I) the Laplacian ∆ is a bounded operator from H2(V ) to L2(V ), so that, since η L1(Ω), 2 2 2 2 the operator T1u = (∆ + k η)u is also bounded from H (V ) to L (V )[12]; 1 3 1=2 (II) u (H (V )) , so that u @νu H (Γ) is a continuous linear operator, which r 2 7! 2 2 2 implies in turn that the operator T2u = @νujΓ is compact from H (V ) to L (Γ); 2 3=2 (III) the trace operator u uj@V maps H (V ) to H (Γ) continuously, so that the operator 7! 2 2 T3u = ujΓ is compact from H (V ) to L (Γ). Therefore, a natural choice for our workspace is H2(V ). We can then write our system in the form T u = v; (4) in which T : H2(V ) L2(V ) L2(Γ) L2(Γ) −! × × 2 u ∆u + k ηu; @νu; u 7−! and v = (S; f; g) G := L2(V ) L2(Γ) L2(Γ): 2 × × We first show that T is injective. Proposition 1. The linear bounded map T defined by (4) is injective. Proof. For a proof of this classical result, we refer to the Fritz John's book [19]. More recent proofs based on Carleman estimates can be found in [18,9,1, 26]. The principal idea is to show that a part of Γ is non-characteristic with respect to the Helmholtz operator, which in turns leads to the existence of a small neighborhood of that part of the boundary in V where 1 the solution is identically zero. Since η L (Ω), the Helmholtz operator T1 possesses the unique continuation property in V , and2 hence the solution is identically zero in the whole domain V which completes the proof. We are now going to set up our approach to the regularization of the problem. We consider the mollifier 1 x 3 'α(x) = ' ; α (0; 1]; x R ; α3 α 2 2 in which ' is an integrable function such that 3 (1) supp ' Br := x R x r for some positive r, ⊂ f 2 j k k ≤ g (2) R '(x) dx = 1. 3 Desirable additional properties of ' are, as usual, nonnegativity, isotropy, smoothness, radial decrease. 2 3 We denote by Cα the convolution operator by 'α: for every u L (R ), Cαu := 'α u. Our 2 ∗ regularization principle will control smoothness by means of Cα, and α will play the role of the regularization parameter. One difficulty lies in the fact that convolving u by 'α entails extrapolating u from V to the larger set V + αBr. Zero padding is obviously forbidden here since we wish to preserve H2 regularity. It is then necessary to introduce an extension operator that preserves the properties of the solution on V . 2+s R3 2+s R3 For s [0; 1), we denote by H0 ( ) the set of functions in H ( ) having a compact 2 2+s 2+s R3 support. Our objective is to derive an extension operator E : H (V ) H0 ( ) that in 2+s ! addition of satisfying Eu V = u for all u H (V ), is bounded and invertible. Note that there are many extensionj operators satisfying2 these properties. We next provide a complete characterization of a useful extension operator suited for C3;1 smooth bounded domains. For other extension operators on Sobolev spaces under weaker regularity assumptions on the domain V; see, for example, [10, 22,8, 25]. For " (0; 1) small enough define the tubular domains 2 ± 0 0 0 + V := x tν(x ):(x ; t) @V (0;") ; and V" := V @V V : (5) " f ± 2 × g [ [ " − We first notice that V" V V" for all " (0; 1). Due to the regularity of @V , the function defined by ⊂ ⊂ 2 φ(x0; t) = x0 tν(x0); (6) − is a C2;1-diffeomorphism from @V (0;") onto V − for " small enough. × " 2 2 Let u be fixed in H (V ). The first step is to construct an extension u" of u to H (V"=2). For α; β R, set 2 u~(x0 + tν(x0)) = αu(x0 tν(x0)) + βu(x0 2tν(x0)); for (x0; t) @V (0; "=2): (7) − − 2 × 2 2 + Since ν is a C vector field,u ~ lies in H (V" ), and verifies 0 0 0 0 0 u~(x ) = (α + β)u(x ); and @νu~(x ) = ( α 2β)@νu(x ); x @V: (8) − − 2 Let u" be defined by + u~(x); x V"=2; u"(x) = 2 (9) u(x); x V: 2 2 2 + By construction, we have u" H (V ) H (V ).