Volume 33, Issue 2

Fourier phasing is the problem of retrieving Fourier phase information from Fourier intensity data. The error-reduction (ER) algorithm consists of two projections on the subspaces generated by the Fourier magnitude constraint and the object-domain constraint. The random phase illumination (RPI) and the real-valued constraint on the object significantly reduce the complexity of the intersection of the two subspaces. In this paper, we study how to approximate the projection of the starting point onto the subspace generated by the Fourier magnitude constraint by its projection on the tangent plane and estimate the approximation error by orthogonal decompositions. Moreover, we prove that the local geometric convergence rate of the ER algorithm is less than one almost surely and can be characterized as the cosine of the angle between the two projection spaces. A theoretical estimate of the convergence rate is derived and validated by some numerical experiments.

Mathematically, full waveform inversion is a nonlinear and ill-posed inverse problem, requiring a regularization method to obtain a reasonable result. Total variation regularization is an effective regularization method which can preserve the sharp edges of the solution. It is well-known that the standard total variation regularization usually leads to stair-casing artifacts in slanted structures. Thus, the standard total variation regularization may not be effective in solving the full waveform inverse problem, due to the slanted properties of the subsurface structures. In this paper, we propose a rotational total variation regularization method based on the weighting rotational transform operator and the standard total variation regularization for the full waveform inverse problem. To further improve the resolution of the inverted results for different directional structures, a hybrid regularization method combining the rotational and standard total variation regularization is proposed. An efficient version of the conjugate gradient method, i.e., CGOPT, is used to efficiently solve the proposed methods. Numerical experiments based on Sigsbee and Marmousi2 models are carried out to demonstrate the effectiveness of our methods.

In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand–Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems. Our method incorporates the utilization of training data, using a bounded linear operator, which guides the iterative procedure. At each iteration step, the method randomly chooses one equation from the nonlinear system with data-driven term. When dealing with the precise or exact data, it has been established that mean square iteration error converges to zero. However, when confronted with the noisy data, we employ our approach in conjunction with a predefined stopping criterion, which we refer to as an a priori stopping rule. We provide a comprehensive theoretical foundation, establishing convergence and stability for this scheme within the realm of infinite-dimensional Hilbert spaces. These theoretical underpinnings are further bolstered by a numerical experiment on a system of linearly ill-posed problems and by discussing an example that fulfills the assumptions of the paper.

Determining the unknown order of the fractional derivative in differential equations simulating various processes is an important task of modern applied mathematics. In the last decade, this problem has been actively studied by specialists. A number of interesting results with a certain applied significance were obtained. This paper provides a short overview of the most interesting works in this direction. Next, we consider the problem of determining the order of the fractional derivative in the subdiffusion equation, provided that the elliptic operator included in this equation has at least one negative eigenvalue. An asymptotic formula is obtained according to which, knowing the solution at least at one point of the domain under consideration, the required order can be calculated.

In this article, for a time-degenerate fractional parabolic equation, we study an inverse problem for restoration of the initial condition from the information of the final temperature profile. We show that the considered problem is ill-posed in the sense of Hadamard, i.e., small errors in the measurement data may lead to indefinitely large errors in the solutions. This ill-posed problem is regularized using a modified quasi-boundary value method, and some convergence estimates for the regularized solution are obtained using a priori and posteriori parameter choice rules. Finally, several numerical experiments are presented to demonstrate the accuracy and efficiency of the regularization method.

In this short note, we investigate an inverse (source) problem associated with a nonlocal elliptic equation ( - ∇ ⋅ σ ⁢ ∇ ) s ⁢ u = F {(-\nabla\cdot\sigma\nabla)^{s}u=F} that is given in a bounded open set Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , for n ≥ 3 {n\geq 3} and 0 < s < 1 {0<s<1} . We demonstrate both the leading coefficient σ and the source F can be determined uniquely by using the exterior Dirichlet-to-Neumann (DN) map in Ω e := ℝ n ∖ Ω ¯ {\Omega_{e}:=\mathbb{R}^{n}\setminus\overline{\Omega}} . The result is intriguing in that analogous theory cannot be true for the local case generally, that is, s = 1 {s=1} . The key ingredients to prove the uniqueness are based on the unique continuation principle for nonlocal elliptic operators and the reduction from the nonlocal to the local via the Stinga–Torrea extension problem.

In this review, we extend the boundary control method – an approach to inverse problems based on control theory for dynamical systems – to inverse problems for discrete dynamical systems. We apply our results to classical moment problems, Toda lattices, Weyl functions, de Branges spaces, Krein–Stieltjes strings, and also to problems of numerical simulations.

In this paper, stability estimates for an inverse problem of determining time-dependent coefficients in a system of parabolic equations are established using the Carleman estimate technique.

In this paper, we apply numerical methods to study inverse nodal problems for the Sturm–Liouville operator. At first, we find an approximate function of the potential from the nodal points of the ( n + 1 ) {(n+1)} -th eigenfunction and three constants via the second kind Chebyshev wavelet method (SCW). We have a sharp condition for uniform convergence of Chebyshev series and an error estimate between the approximate solution and exact solution of the potential. Then, a numerical example is provided to show that the approximate solutions become more accurate and the errors decrease as the values of n increase. Also, we show the comparison of SCW with Bernstein method. Finally, we present an application of numerical solutions to reconstruct the potential from parts of nodal set and its mean value. Compared with some well-known results, the nodal data used here is least.

In this paper, we revisit Linear Neural Networks (LNNs) with single-output neurons performing linear operations. The study focuses on constructing an optimal regularized weight matrix Q from training pairs { G , H } {\{G,H\}} , reformulating the LNNs framework as matrix equations, and addressing it as a linear inverse problem. The ill-posedness of linear machine learning problems is analyzed through the lens of inverse problems. Furthermore, classical and modern regularization techniques from both the machine learning and inverse problems communities are reviewed. The effectiveness of LNNs is demonstrated through a real-world application in blood test classification, highlighting their practical value in solving real-life problems.