The objective of quantitative photoacoustic tomography (qPAT) is to reconstruct the diffusion, absorption and Grüneisen thermodynamic coefficients of heterogeneous media from knowledge of the interior absorbed radiation. It has been shown in Bal and Ren (2011 Inverse Problems 27 075003), based on diffusion theory, that with data acquired at one given wavelength, all three coefficients cannot be reconstructed uniquely. In this work, we study the multi-spectral qPAT problem and show that when multiple wavelength data are available, all coefficients can be reconstructed simultaneously under minor prior assumptions. Moreover, the reconstructions are shown to be very stable. We present some numerical simulations that support the theoretical results.

In this paper, we focus on the analysis of discrete versions of the Calderón problem in dimension d ⩾ 3. In particular, our goal is to obtain stability estimates for the discrete Calderón problems that hold uniformly with respect to the discretization parameter. Our approach mimics the one in the continuous setting. Namely, we prove discrete Carleman estimates for the discrete Laplace operator. A main difference with the continuous ones is that in our case, the Carleman parameters cannot be taken arbitrarily large, but should be smaller than some frequency scale depending on the mesh size. Following the by-now classical complex geometric optics (CGO) approach, we can thus derive discrete CGO solutions, but with a limited range of parameters. As in the continuous case, we then use these solutions to obtain uniform stability estimates for the discrete Calderón problems.

We consider a periodic Jacobi operator J with finitely supported perturbations on the half-lattice. We describe all eigenvalues and resonances of J and give their properties. We solve the inverse resonance problem: we prove that the mapping from finitely supported perturbations to the Jost functions is one-to-one and onto, we show how the Jost functions can be reconstructed from all eigenvalues, resonances and from the set of zeros of S(λ) − 1, where S(λ) is the scattering matrix.

We present a novel approach to nonlinear constrained Tikhonov regularization from the viewpoint of optimization theory. A second-order sufficient optimality condition is suggested as a nonlinearity condition to handle the nonlinearity of the forward operator. The approach is exploited to derive convergence rate results for a priori as well as a posteriori choice rules, e.g., discrepancy principle and balancing principle, for selecting the regularization parameter. The idea is further illustrated on a general class of parameter identification problems, for which (new) source and nonlinearity conditions are derived and the structural property of the nonlinearity term is revealed. A number of examples including identifying distributed parameters in elliptic differential equations are presented.

We discuss the inverse boundary value problem of determining the conductivity in two dimensions from the pair of all input Dirichlet data supported on an open subset Γ ₊ and all the corresponding Neumann data measured on an open subset Γ ₋. We prove the global uniqueness under some additional geometric condition, in the case where , and we prove also the uniqueness for a similar inverse problem for the stationary Schrödinger equation. The key of the proof is the construction of appropriate complex geometrical optics solutions using Carleman estimates with a singular weight.

We propose and test stable algorithms for the reconstruction of the internal conductivity of a biological object using acousto-electric measurements. Namely, the conventional impedance tomography scheme is supplemented by scanning the object with acoustic waves that slightly perturb the conductivity and cause the change in the electric potential measured on the boundary of the object. These perturbations of the potential are then used as the data for the reconstruction of the conductivity. The present method does not rely on 'perfectly focused' acoustic beams. Instead, more realistic propagating spherical fronts are utilized, and then the measurements that would correspond to perfect focusing are synthesized. In other words, we use synthetic focusing. Numerical experiments with simulated data show that our techniques produce high-quality images, both in 2D and 3D, and that they remain accurate in the presence of high-level noise in the data. Local uniqueness and stability ...

An inverse scattering transform method corresponding to a Riemann–Hilbert problem is formulated for CH2, the two-component generalization of the Camassa–Holm (CH) equation. As an illustration of the method, the multi-soliton solutions corresponding to the reflectionless potentials are constructed in terms of the scattering data for CH2.

The authors explain how the major results which were obtained recently in Eggermont et al ( 2009 Inverse Problems 25 115018) can be derived from a more general perspective of recent regularization theory. By pursuing this further, the authors provide a general view on regularization under general noise assumptions, including weakly and strongly controlled noise. The prospect is not to generalize previous work in this direction, but rather to envision the intrinsic structure present in regularization under general noise assumptions. In particular, the authors find variants of the discrepancy and the Lepski principle to choose the regularization parameter, albeit within different context and under different assumptions.

We solve direct and inverse spectral problems for Sturm–Liouville operators with singular nonlocal potentials and nonlocal boundary conditions.

The validity of the synthesis of a globally convergent numerical method with the adaptive FEM technique for a coefficient inverse problem is verified on time-resolved experimental data. The refractive indices, locations and shapes of dielectric abnormalities are accurately imaged.