We shall give a priori conditions on the illuminations ϕ i such that the solutions to the Helmholtz equationon ∂Ω, and their gradients satisfy certain non-zero and linear independence properties inside the domain Ω, provided that a finite number of frequencies k are chosen in a fixed range. These conditions are independent of the coefficients, in contrast to the illuminations classically constructed by means of complex geometric optics solutions. This theory finds applications in several hybrid problems, where unknown parameters have to be imaged from internal power density measurements. As an example, we discuss the microwave imaging by ultrasound deformation technique, for which we prove new reconstruction formulae.

We prove that an L ∞ potential in the Schrödinger equation in three and higher dimensions can be uniquely determined from a finite number of boundary measurements, provided it belongs to a known finite dimensional subspace W. As a corollary, we obtain a similar result for Calderón's inverse conductivity problem. Lipschitz stability estimates and a globally convergent nonlinear reconstruction algorithm for both inverse problems are also presented. These are the first results on global uniqueness, stability and reconstruction for nonlinear inverse boundary value problems with finitely many measurements. We also discuss a few relevant examples of finite dimensional subspaces W, including bandlimited and piecewise constant potentials, and explicitly compute the number of required measurements as a function of dim W.

This paper provides an analysis of the linearized inverse problem in multifrequency electrical impedance tomography. We consider an isotropic conductivity distribution with a finite number of unknown inclusions with different frequency dependence, as is often seen in biological tissues. We discuss reconstruction methods for both fully known and partially known spectral profiles, and demonstrate in the latter case the successful employment of difference imaging. We also study the reconstruction with an imperfectly known boundary, and show that the multifrequency approach can eliminate modeling errors and recover almost all inclusions. In addition, we develop an efficient group sparse recovery algorithm for the robust solution of related linear inverse problems. Several numerical simulations are presented to illustrate and validate the approach.

We consider the dynamical super-resolution problem consisting in the recovery of positions and velocities of moving particles from low-frequency static measurements taken over multiple time steps. The standard approach to this issue is a two-step process: first, at each time step some static reconstruction method is applied to locate the positions of the particles with super-resolution and, second, some tracking technique is applied to obtain the velocities. In this paper we propose a fully dynamical method based on a phase-space lifting of the positions and the velocities of the particles, which are simultaneously reconstructed with superresolution. We provide a rigorous mathematical analysis of the recovery problem, both for the noiseless case and in presence of noise (in the discrete setting). Several numerical simulations illustrate and validate our method, which shows some advantage over existing techniques. We then discuss the application of this approach to the dynamical super-resolution problem in ultrafast ultrasound imaging: blood vessels' locations and blood flow velocities are recovered with super-resolution.

The focus of this paper is the study of the regularity properties of the time harmonic Maxwell's equations with anisotropic complex coefficients, in a bounded domain with C 1,1 boundary. We assume that at least one of the material parameters is W 1,p for some p > 3. Using regularity theory for second order elliptic partial differential equations, we derive W 1,p estimates and Hölder estimates for electric and magnetic fields up to the boundary, together with their higher regularity counterparts. We also derive interior estimates in bi-anisotropic media.and R k (ϕ 2 ) = −ˆΩE k div(ζ T ϕ 2 ∇χ) + ζ∇χ · ∇ϕ 2 + H k div(µ T ϕ 2 ∇χ) + µ∇χ · ∇ϕ 2 dx.This last system can be reformulated in the form (41), with A given by (25). We then apply Lemma 18 and obtain χE k , χH k ∈ W 1,r (Ω; C), namely E, H ∈ W 1,r (Ω 0 ; C 3 ). Finally, (45) follows from (42).

Abstract. We study the boundary control of solutions of the Helmholtz and Maxwell equations to enforce local non-zero constraints. These constraints may represent the local absence of nodal or critical points, or that certain functionals depending on the solutions of the PDE do not vanish locally inside the domain. Suitable boundary conditions are classically determined by using complex geometric optics solutions. This work focuses on an alternative approach to this issue based on the use of multiple frequencies. Simple boundary conditions and a finite number of frequencies are explicitly constructed independently of the coefficients of the PDE so that the corresponding solutions satisfy the required constraints. This theory finds applications in several hybrid imaging modalities: some examples are discussed.

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms.

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