Volume 27, Issue 2

An integro-differential Dirac system with an integral term in the form of convolution is considered. We suppose that the convolution kernel is known a priori on a part of the interval, and recover it on the remaining part, using a part of the spectrum. We prove the uniqueness theorem, provide an algorithm for the solution of the inverse problem together with necessary and sufficient conditions for its solvability.

The inverse problem of magnetoencephalography is ill-posed and difficult for both analytical and numerical solutions. Additional complications arise from the volume (passive) currents and the associated magnetic fields, which strongly depend on the brain geometry. In this paper, we find approximate analytical solutions for the forward and the inverse problems in the spheroid geometry. We compare the obtained results with the exact solution of the forward problem and deduce that for a wide range of parameters our approximation is valid. The analysis sheds new light on the role of the volume magnetic fields for solving the inverse problem of magnetoencephalography.

In this paper we prove semigroup properties for the mixed Riesz hyperbolic B-potential, find its analytic continuation and describe a space of images of mixed hyperbolic Riesz B-potentials. This problem is closely related to the problem of inversion of the weighted Radon transform on Lorentzian manifolds.

This paper is devoted to the inverse problem of identifying an unknown spacewise-dependent zeroth-order coefficient p⁢(x){p(x)} in the 1D diffusion equation ut=(k⁢(x)⁢ux)x-p⁢(x)⁢u{u_{t}=(k(x)u_{x})_{x}-p(x)u} from boundary Dirichlet measured output f⁢(t):-u⁢(0,t){f(t)\coloneq u(0,t)}, t∈[0,T]{t\in[0,T]}. Compactness and Lipschitz continuity of the input-output operator Φ⁢[p]:-u⁢(x,t;p)|x=0+{\Phi[p]\coloneq u(x,t;p)|_{x=0^{+}}}, Φ⁢[⋅]:𝒫⊂H1⁢(0,l)↦L2⁢(0,T){\Phi[\,\cdot\,]\colon\mathcal{P}\subset H^{1}(0,l)\mapsto L^{2}(0,T)} are proved. Then an existence of a quasi-solution of the inverse problem is obtained. We prove Fréchet differentiability of the Tikhonov functional and derive an explicit gradient formula for the Fréchet gradient through the solutions of the direct and corresponding adjoint problems solutions. This allows to use gradient-type algorithms for the numerical solution of the considered inverse problem.

The convergence of Levenberg–Marquardt method is discussed for the inverse problem to reconstruct the storage modulus and loss modulus for the so-called scalar model by a single interior measurement. The scalar model is the most simplest model for data analysis used as the modeling partial differential equation in the diagnosing modality called the magnetic resonance elastography which is used to diagnose for instance lever cancer. The convergence of the method is proved by showing that the measurement map which maps the above unknown moduli to the measured data satisfies the so-called the tangential cone condition. The argument of the proof is quite general and in principle can be applied to any similar inverse problem to reconstruct the unknown coefficients of the model equation given as a partial differential equation of divergence form by one single interior measurement. The performance of the method is numerically tested for the two-layered piecewise homogeneous scalar models in a rectangular domain and a circular domain.

We prove a global uniqueness theorem of reconstruction of a matrix-potential a⁢(x,t){a(x,t)} of one-dimensional wave equation □⁢u+a⁢u=0{\square u+au=0}, x>0,t>0{x>0,t>0}, □=∂t2-∂x2{\square=\partial_{t}^{2}-\partial_{x}^{2}} with zero Cauchy data for t=0{t=0} and given Cauchy data for x=0{x=0}, u⁢(0,t)=0{u(0,t)=0}, ux⁢(0,t)=g⁢(t){u_{x}(0,t)=g(t)}. Here u,a,f{u,a,f}, and g are n×n{n\times n} smooth real matrices, det⁡(f⁢(0))≠0{\det(f(0))\neq 0}, and the matrix ∂t⁡a{\partial_{t}a} is known.

We consider inverse problems in which the unknown target includes sharp edges, for example interfaces between different materials. Such problems are typical in image reconstruction, tomography, and other inverse problems algorithms. A common solution for edge-preserving inversion is to use total variation (TV) priors. However, as shown by Lassas and Siltanen 2004, TV-prior is not discretization-invariant: the edge-preserving property is lost when the computational mesh is made denser and denser. In this paper we propose another class of priors for edge-preserving Bayesian inversion, the Cauchy difference priors. We construct Cauchy priors starting from continuous one-dimensional Cauchy motion, and show that its discretized version, Cauchy random walk, can be used as a non-Gaussian prior for edge-preserving Bayesian inversion. We generalize the methodology to two-dimensional Cauchy fields, and briefly consider a generalization of the Cauchy priors to Lévy α-stable random field priors. We develop a suitable posterior distribution sampling algorithm for conditional mean estimates with single-component Metropolis–Hastings. We apply the methodology to one-dimensional deconvolution and two-dimensional X-ray tomography problems.

We consider the problem of reconstruction of the Cauchy data for the wave equation in ℝ3{\mathbb{R}^{3}} and ℝ2{\mathbb{R}^{2}} by the measurements of its solution on the boundary of the unit ball.

In this paper we present some aspects of an inverse source problem in an elliptic equation with varying coefficients, using partial Dirichlet boundary measurements. A uniqueness result is established for dipolar sources including their number. Additionally, assuming the number of dipoles known, a stability result is obtained and an efficient numerical identification method is developed. Finally, numerical experiments illustrate the effectiveness of the approach and a discussion is given on electroencephalography (EEG) in neonates.

The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization in Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.