We consider an one-dimensional Schrodinger operator with an even periodic potential perturbed by a small PT-symmetric potential. The unperturbed spectrum can have points associated with two periodic or antiperiodic eigenfunctions. We study how such points bifurcate under the perturbation. We obtain a three-terms asymptotic expansion for the corresponding perturbed band functions with rigorous estimates for the error terms. These asymptotics allow us to establish sufficient conditions for the emergence and absence of a non-real spectrum. The emerging non-real spectrum is a complex curve, the shape of which is described by our asymptotic expansions. We also describe how the Dirichlet eigenavules, which satisfy the Dubrovin equations for finite-gap potentials, depend on the perturbation.
We consider an inverse Sturm-Liouville problem on a star-shaped metric graph, where the aim is to reconstruct edge potentials from spectral data. The inverse map from spectral data to the potentials is highly ill-conditioned, so classical spectral-mapping iterations suffer from severe noise amplification and spurious oscillations. To stabilize the reconstruction, we propose a regularized spectral-mapping method that combines spectral filtering, damping of the nonlinear fixed-point iteration, spatial Tikhonov smoothing with Neumann boundary conditions, and projection onto an affine constraint set encoding endpoint and integral information. The method admits a unified operator formulation that allows a transparent analysis of stability and convergence. Under mild assumptions on the filtered update, we prove that the regularized iteration is locally contractive in the L 2 {L<^>{2}} -norm, which guarantees existence and uniqueness of a fixed point and linear convergence of the iteration. Furthermore, with noise-dependent parameter choice and a discrepancy principle, we establish that the method defines a convergent regularization. The reconstructed potentials converge to the exact solution as the noise level tends to zero. Numerical experiments on star graphs with multiple edges confirm the theoretical results and demonstrate that the proposed regularization significantly improves robustness and accuracy compared with the unregularized spectral-mapping approach.
We study high-frequency stability estimates for the determination of the zeroth order perturbation of the polyharmonic operator with constant attenuation from the partial Dirichlet-to-Neumann map when part of the boundary is inaccessible and flat. Our results extend the recent results obtained in [A. P. Choudhury and A. Kumar T., High-frequency stability estimates for the inverse boundary value problems for the Schr & ouml;dinger and the biharmonic operator with constant attenuation on certain bounded domains, J. Math. Anal. Appl. 556 2026, 1, Article ID 130094] for the Schr & ouml;dinger equation and the biharmonic operator to the polyharmonic case.
Projection-based neurodynamic systems have recently been used for sparse reconstruction, but most existing designs focus on convex L(1 )models and maybe suboptimal for highly sparse signals. This paper develops an inertial projection-based neurodynamic system (PBNS) for the nonconvex L-q-regularized (0 < q < 1) sparse recovery problem. By applying a standard variable splitting and introducing a smooth absolute-value regularization, we transform the original model into the minimization of a continuously differentiable objective over a closed convex set, which enables a rigorous dynamical-system formulation. We design a two-layer inertial PBNS driven by the projected gradient mapping and obtain an implementable reconstruction algorithm via time discretization. The proposed dynamics is shown to be globally well-posed (existence and uniqueness of trajectories), and its trajectory converges asymptotically to the equilibrium set under a cocoercivity-type condition, with an additional local convergence result under local cocoercivity. Extensive experiments on synthetic compressed sensing data and real fetal ECG signals demonstrate that the proposed PBNS-L-q method is stable and achieves improved reconstruction accuracy and efficiency compared with representative convex and non-convex baselines.
This article provides an overview of Jan Boman’s illustrious seventy year career as an approximation theorist, microlocal analyst, and integral geometer. We will include his main mathematical themes and some personal observations.
We study ill-posed continuation problems for partial differential equations, with an emphasis on the mechanisms of ill-posedness and their mitigation via conditional stability and regularization. Three canonical examples of elliptic, parabolic, and hyperbolic type are used to illustrate the underlying ill-posedness. For a second-order elliptic continuation problem, we summarize well-posedness results for the associated direct and adjoint problems, establish conditional stability estimates, and develop an adjoint-based iterative reconstruction method with convergence-rate guarantees. For a parabolic continuation problem, we present corresponding well-posedness results and an adjoint-based iterative scheme. For a hyperbolic continuation problem, we derive a conditional stability result. We further analyze the singular numbers of the continuation operator for a complex-valued Helmholtz equation, thereby characterizing the frequency dependence of the ill-posedness. Finally, we compare Tikhonov regularization with linear neural networks for ill-posed Helmholtz inverse problems, highlighting their complementary strengths.
The paper deals with a nonlinear inverse source problem in a system of two coupled elliptic 2D advection-dispersion-reaction partial differential equations. In such system, we address the identification of multiple unknown mixed point and distributed sources defining the right-hand side of its first equation using some local observations related to the state solution of its second coupled equation. We develop appropriate adjoint functions leading to establish reciprocity gaps fulfilled by the unknown elements defining the sought sources. These adjoint functions are defined by scalar potentials derived from fields collinear to the orthogonal directions pointed by the eigenvectors of the symmetric dispersion tensor. From some interior measuring interfaces suitably set up within the monitored domain, we establish an identifiability result and develop a detection-identification method. Some numerical experiments on the coupled surface water BOD-OD model are presented.
In this paper, we propose a discretized Predictor-Corrector iterative Tikhonov regularization (DPC-ITR) method, which integrates multiscale Galerkin projection for discretization, iterative Tikhonov regularization for inversion, and the Modified Euler Method for time stepping. The proposed DPC-ITR method significantly accelerates computation for ill-posed inverse problems compared to standard iterative Tikhonov methods, while preserving the same order of numerical accuracy. Under specific regularity conditions, we rigorously derive a priori error estimates for the approximate solutions generated by the DPC-ITR method. Furthermore, we propose a novel heuristic parameter choice rule for the DPC-ITR method when applied to linear ill-posed integral equations. The proposed parameter choice rule, under certain conditions, enables the DPC-ITR method to generate approximate solutions that converge at order-optimal rates, as rigorously proven in our analysis. Our numerical experiments confirm that the DPC-ITR method equipped with the proposed heuristic parameter choice rule achieves the theoretically predicted convergence rates while demonstrating improved computational efficiency compared to conventional regularization approaches.
A study is presented of a time-dependent source identification problem for a Schr & ouml;dinger-type involutory differential equation. The problem is formulated in an abstract Hilbert space with a self-adjoint positive definite operator. The unique solvability of the abstract problem is established, and stability estimates for its solution are obtained. These results are further applied to four specific time-dependent source identification problems: A one-dimensional problem with nonlocal conditions, a one-dimensional problem with spatial involution, and multi-dimensional problems with Dirichlet and Neumann boundary conditions. This demonstrates the importance of the presented operator approach for studying various classes of problems for Schr & ouml;dinger-type involutory differential equations with unknown time-dependent source terms.
In this paper, we consider the Cauchy problem for the heat equation. The inverse problem involves reconstructing the missing data on an inaccessible boundary from the measured data on an accessible boundary, which is severely ill-posed. By Green's representation theorem, we transform the Cauchy problem into a system of two boundary integral equations. Noticing the ill-posedness of this system, a stable numerical method via a quasi Tikhonov regularization technique is proposed. Then the well-posedness of the proposed regularizing problem and convergence property of the regularizing solution to the exact one are proven. The advantages of our proposed method are as follows: on one hand, we can obtain simultaneously the Dirichlet data and the Neumann data by solving a regularized system; on the other hand, compared with the method based on the classical Tikhonov regularization technique for solving the equations, our proposed method is simpler from the numerical point of view, and needs relatively small amount of computations in numerical implementations. Based on the Nystr & ouml;m method and trigonometric approximation for singular integrals, we give an effective way of discretizing the boundary integral equations. Some numerical results, with exact and noisy measurement data, are also presented to show the efficiency and accuracy of our scheme. Furthermore, it is shown that the proposed method remains applicable even when only partial boundary Cauchy data are available.
A posteriori error estimates for approximate solutions of linear ill-posed inverse problems are under investigation. A brief overview of existing approaches to this problem is given. Special variational a posteriori error estimates are studied for various classes of function spaces. Obtaining all such estimates reduces to solving a certain canonical problem. For the canonical problem, a solution algorithm is given in various forms. This algorithm is several times faster than previously proposed ones. The estimates and algorithm are suitable for various Hilbert and Banach spaces. The obtaining of such estimates is illustrated using the example of a two-dimensional model inverse problem of heat conduction.
In this study, the ( 4 + 1 ) {(4\kern-1.0pt+\kern-1.0pt1)} -dimensional fractional Boiti-Leon-Manna-Pempinelli (BLMP) equation is considered to be an essential model for the incompressible fluid. The ( m + 1 G ' ) {(m+\frac{1}{G<^>{\prime}})} -expansion method which is an effective analytical method is employed to investigate the exact wave solutions of the equation mentioned above. The proposed scheme has provided various new types of exact solutions and it has not been applied before to the ( 4 + 1 ) {(4\kern-1.0pt+\kern-1.0pt1)} -dimensional BLMP equation with hyperbolic local derivative. These novel solutions can be observed as kink and singular solitons to describe wave propagation in fluid mechanics or continuum mechanics by selecting the suitable parameters and may find potential applications in modeling fluid motion phenomena such as tsunami propagation, ocean surface wave dynamics, and wave-current interactions in marine engineering. The exact wave structures have been revealed and compared employing the hyperbolic local derivative which is newly defined in the literature. The emerged solutions have also been corroborated via the computational software Mathematica and these results have been observed in 2D and 3D graphics.
We study a class of problems in which unknown constant coefficients appear in the free terms of non-autonomous systems of ordinary differential equations and must be identified. The basic and additional (overdetermining) conditions are, in general, nonlocal. They involve cumulative information about the state functions, including their values at selected points and their integrals over given subintervals. Depending on how the number of unknown coefficients compares with the number of additional conditions, different problem formulations and corresponding solution methods are discussed. We also present results of numerical experiments and analyze how errors in the given conditions influence the accuracy of the computed solutions for test problems.
In this paper, we consider a final value problem for the following time-fractional diffusion system: { D(t)(alpha)u + Au = F(u, v), D(t)(alpha)v + Bv = G(u, v), where & Ascr; and & Bernoullis; are symmetric uniformly elliptic operators defined on a bounded domain Omega in & Ropf;(d) with sufficiently smooth boundary and D alpha t refers to the Caputo fractional derivative of order 1 < alpha < 2. Under suitable assumptions, we establish the existence and uniqueness of the solution to the problem and prove that the problem is ill-posed in the sense of Hadamard. A filter regularization method is proposed to approximate the solution and estimate the error between the regularized solution and the exact solution.
In this paper, we proved two types of uniqueness results for solutions to parabolic equations whose traces at time T>0 , for large spatial variables satisfy exponential decay assumption. For solutions to parabolic equations in unbounded domains, we can state the main results as follows. The first uniqueness asserts that if |u(T,x)|<= Ce-|x|2/4T as |x|->infinity with some T>0 and a potential satisfies some restrictions, then u(t,x)=0 for 00 , x 'is an element of R-n -1 and t is an element of I : arbitrarily fixed interval in (0,infinity) , where theta>0 and delta>0 are some constants, then u=0 in I & times;R-n . The proof of the first result is based on the Radon transform, while the second relies on a Carleman estimate for a parabolic equation which is applicable to parabolic equations with memory terms and inverse problems.
The viscoelastic equations with the Lam & eacute; coefficients lambda and mu, density rho and viscoelastic coefficients lambda 1 {\lambda_{1}} and mu 1 {\mu_{1}} are considered. All these coefficients are functions of space variable x is an element of & Ropf; 3 {\mathbf{x}\in\mathbb{R}{3}} and they are known constants outside ball B R {B_{R}} of radius R centered at the origin. It is supposed that lambda, mu and rho are given smooth functions in & Ropf; 3 {\mathbb{R}{3}} , while lambda 1 {\lambda_{1}} and mu 1 {\mu_{1}} are unknown in B R {B_{R}} . The problem of recovering the latter functions from a given information about solutions of some the Cauchy problems for viscoelastic equations is studied. At first, running plain waves going in the homogeneous media from infinity in direction nu is an element of S 2 {\nu\in\mathbb{S}{2}} are considered. These waves are pure compressive or shear ones of the delta-image form. They are passed thorough ball B R {B_{R}} and their amplitudes are measured on the boundary of B R {B_{R}} for all nu is an element of S 2 {\nu\in\mathbb{S}{2}} . It is shown that this information allows to reduce the origin problem to two problems of the inversion of the ray transforms for a family of geodesic lines related to two Riemannian metrics. The latter problems are well studied, particularly, stability and uniqueness theorems are known for theirs. The reducing of the original problem of recovering desired coefficients to linear inversion problems opens a good way for its numerical solution.
Image reconstruction in medical emission tomography is an inverse ill-posed problem with Poisson data. Iterative regularization using a statistical stopping rule is studied in this paper. We suggest a new approach to estimate the optimal breakpoint for iterative image reconstruction algorithms in emission tomography. The new method is based on the use of the central limit theorem with a highly accurate normal approximation to the distribution of the suggested statistic under the null hypothesis. In the simulations, our method demonstrated accurate estimates for the optimal breakpoint for iterative algorithms.
We consider the Cauchy problems for operator analogues of the Poisson equations with data given in one case all around the boundary and the other on the part of the boundary. The problem is recast as an inverse boundary value problem for a system of coupled Beltrami-like equation associated with A-analytic and A-anti-analytic maps, which we solve by an extension of Bukhgeim's A-analytic theory.
Consider the Porous Medium Equation partial derivative(t)u = Delta u(gamma) subject to homogeneous Dirichlet boundary condition, where gamma > 1 is known as the polytropic exponent. Given the values of gamma and the solution u(T-e) at a sufficiently large but unknown time T-e, in this paper we provide an explicit formula for recovering the elapsed time T-e, without any knowledge of the initial data. Our method is based on an asymptotic inequality satisfied by u(T-e) and Te that leads to an asymptotic stability estimate for Te, allowing us to recover it as the global minimizer of some function. We finally validate our results numerically in two and three dimensions.
