Volume 31, Issue 1

We present novel reconstruction and stability analysis methodologies for two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show that the 2-D, limited-data MRI inverse problem, whereby the missing parts of 𝐤 {\mathbf{k}} -space (Fourier space) are lines parallel to either k 1 {k_{1}} or k 2 {k_{2}} (i.e., the 𝐤 {\mathbf{k}} -space axis), can be reduced to a set of 1-D Fredholm type inverse problems. The Fredholm equations are then solved to recover the 2-D image on 1-D line profiles (“slice-by-slice” imaging). The technique is tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and phantom data. Our method is shown to offer optimal performance, in terms of structural similarity, when compared against similar methods from the literature, and when the 𝐤 {\mathbf{k}} -space data is sub-sampled at random so as to simulate motion corruption. In addition, we present a Singular Value Decomposition (SVD) and stability analysis of the Fredholm operators, and compare the stability properties of different 𝐤 {\mathbf{k}} -space sub-sampling schemes (e.g., random vs uniform accelerated sampling).

This paper is concerned with the inverse scattering problem of time-harmonic acoustic waves from a penetrable cavity bounded by a layered structure and seeks to determine the shape and location of the cavity from interior near-field measurements. Of particular interest is that the near-field operator does not satisfy the main theorem of the factorization method, so we introduce a modified near-field operator and prove that it can be used to reconstruct the cavity. Numerical examples demonstrate the feasibility and effectiveness of our algorithm.

A new method for solving inverse spectral problems on quantum star graphs is proposed. The method is based on Neumann series of Bessel function representations for solutions of Sturm–Liouville equations. The representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse spectral problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the representation is sufficient for the recovery of the potential. The method for solving the inverse spectral problem on the graph consists in reducing the problem to a two-spectra inverse Sturm–Liouville problem on each edge. Then a system of linear algebraic equations is derived for computing the first coefficient of the series representation for the solution on each edge and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.

The geodesic ray transform, the mixed ray transform and the transverse ray transform of a tensor field on a surface can all be seen as what we call mixing ray transforms, compositions of the geodesic ray transform and an invertible linear map on tensor fields. We provide an approach that uses a unifying concept of symmetry to merge various earlier transforms (including mixed, transverse, and light ray transforms) into a single family of integral transforms with similar kernels.

In this paper, we study the robust recovery of signals for general noise by ℓ 1 - 2 / ℓ p \ell_{1-2}/\ell_{p} ( 2 ≤ p < + ∞ 2\leq p<+\infty ) minimization with partially known support (PKS). A recovery condition for ℓ 1 - 2 / ℓ p \ell_{1-2}/\ell_{p} minimization by incorporating prior support information is established, and an error estimate is obtained. In particular, the obtained results not only provide a new theoretical guarantee to robustly recover signals for general noise, but also improve and generalize the state-of-the-art ones. In addition, a series of numerical experiments are carried out to confirm the validity of the proposed method, which show that incorporating prior support information for ℓ 1 - 2 / ℓ p \ell_{1-2}/\ell_{p} minimization exhibits better recovery performance than ℓ 1 - 2 / ℓ p \ell_{1-2}/\ell_{p} minimization.

Local reconstruction from localized projections attains importance in Computed Tomography (CT). Several researchers addressed the local recovery (or interior) problem in different frameworks. The recent sparsity based optimization techniques in Compressed Sensing (CS) are shown to be useful for CT reconstruction. The CS based methods provide hardware-friendly algorithms, while using lesser data compared to other methods. The interior reconstruction in CT, being ill-posed, in general admits several solutions. Consequently, a question arises pertaining to the presence of target (or interior-centric) pixels in the recovered solution. In this paper, we address this problem by posing the local CT problem in the prior support constrained CS framework. In particular, we provide certain analytical guarantees for the presence of intended pixels in the recovered solution, while demonstrating the efficacy of our method empirically.

In this paper, we discuss the reconstruction of a local volatility surface from American option prices. First of all, the American option prices are calculated by an accurate and fast finite difference scheme. Then the local volatility is obtained by minimizing the distance between theoretical prices and market option prices, which yields an optimization problem. The Bicubic spline regularization technique is used to overcome the ill-posedness of the reconstruction problem. We solve the nonlinear optimization problem by using a gradient-based optimization algorithm. Finally, we test our model with numerical examples and real market American put option data. The results show the good performance of our method.

This paper is concerned with the stability of the inverse source problem for Maxwell’s equations in an inhomogeneous background medium. We show that the stability estimate consists of the Lipschitz-type data discrepancy and the high frequency tail of the source function, where the latter decreases as the upper bound of the frequency increases. The analysis employs scattering theory to obtain the holomorphic domain and an upper bound for the resolvent of the elliptic operator.

Motivated by the application to helioseismology, we demonstrate uniqueness and stability for a class of inverse problems of the weighted transport equation. Using 𝐴-analytic functions, this inverse problem is expressed as a Cauchy problem. In this form, we show that, for a finite even trigonometric polynomial weight function, the resulting system is well-conditioned numerically and permits a Carleman-like estimate with boundary terms.

In this paper, we study secant-type iteration for nonlinear ill-posed equations involving 𝑚-accretive mappings in Banach spaces. We prove that the proposed iterative scheme has a convergence order at least 2.20557 using assumptions only on the first Fréchet derivative of the operator. Further, using a general Hölder-type source condition, we obtain an optimal error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter.