This paper presents a review of both classical and modern results pertaining to partial inverse spectral problems for differential operators. Such problems consist in the recovery of differential expression coefficients in some part of the domain (a...
This note reviews certain classical applications of microlocal analysis in inverse problems. The text is based on lecture notes for a postgraduate level minicourse on applications of microlocal analysis in inverse problems, given in Helsinki and Shan...
A spectral inverse problem concerns the reconstruction of parameters of a parent graph from prescribed spectral data of subgraphs. Also referred to as the P–NP Isomorphism Problem, Reconstruction or Exact Graph Matching, the aim is to seek sets of pa...
In this paper, we study inverse nodal problems for a boundary value problem. A uniqueness result for the potential function and a reconstruction method are obtained. By using the nodal points as input data, we compute the approximation solution of th...
Ultrafast pump–probe spectroscopic studies allow for deep insights into the mechanisms and timescales of photophysical and photochemical processes. Extracting valuable information from these studies, such as reactive intermediates’ lifeti...
Most inverse problems in the industry (and particularly in geophysical exploration) are highly underdetermined because the number of model parameters too high to achieve accurate data predictions and because the sampling of the data space is scarce a...
In this paper we examine an Information-Theoretic method for solving noisy linear inverse estimation problems which encompasses under a single framework a whole class of estimation methods. Under this framework, the prior information about the unknow...
This article proposes the application of a new mathematical model of spots for solving inverse problems using a learning method, which is similar to using deep learning. In general, the spots represent vague figures in abstract “information spa...
The unique solvability in the sense of classical solutions for nonlinear inverse problems to differential equations, solved for the oldest Dzhrbashyan–Nersesyan fractional derivative, is studied. The linear part of the equation contains a bound...
Inverse problems arise anywhere we have an indirect measurement. In general, they are ill-posed to obtain satisfactory solutions, which needs prior knowledge. Classically, different regularization methods and Bayesian inference-based methods have bee...
This paper deals with inverse problems related to degenerate fractional integro-differential equations in Banach spaces. We study existence, uniqueness and regularity of solutions to the problem, claiming to extend well known studies for the case of...
This paper aims at discussing the resolution achievable in the reconstruction of both circumference sources from their radiated far-field and circumference scatterers from their scattered far-field observed for the 2D scalar case. The investigation i...
We are concerned with a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct prob...
In this paper, solvability of some inverse problems for a nonlocal analog of a pseudoparabolic equation is studied. The nonlocal analog of a pseudoparabolic equation is formed using transformations that have the involution property. Two types of inve...
In the ever-evolving landscape of tomographic imaging algorithms, this literature review explores a diverse array of themes shaping the field’s progress. It encompasses foundational principles, special innovative approaches, tomographic impleme...
One of the key tools in an organization’s performance management is the goal tree, which is used for solving both direct and inverse problems. This research deals with goal setting based on a model of the future by presenting the goal and subgo...
Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as...
The present study is concerned with the numerical solution of external boundary conditions in inverse problems for one-dimensional multilayer diffusion, using the difference method. First, we formulate multispecies parabolic problems with three types...
Inverse inclusion problems arise in several applications such as electrical impedance tomography, nondestructive testing, and geophysical imaging. The purpose of this review is to present some recent advances concerning stability estimates for the de...
In this paper, we consider the single-machine scheduling problem with given release dates and the objective to minimize the maximum penalty which is NP-hard in the strong sense. For this problem, we introduce a dual and an inverse problem and show th...
Due to the fundamental solutions are employed as basis functions, the localized method of fundamental solution can obtain more accurate numerical results than other localized methods in the homogeneous problems. Since the inverse Cauchy problem is il...
For many scientific inverse problems, we are required to evaluate an expensive forward model. Moreover, the model is often given in such a form that it is unrealistic to access its gradients. In such a scenario, standard Markov Chain Monte Carlo algo...
The Landweber iteration method is one of the most popular methods for the solution of linear discrete ill-posed problems. The diversity of physical problems and the diversity of operators that result from them leads us to think about updating the mai...
In this paper, we study inverse interface problems with unknown boundary conditions, using point observations for parabolic equations with cylindrical symmetry. In the one-dimensional, two-layer interface problem, the left interval
In this paper, a meshfree weighted radial basis collocation method associated with the Newton’s iteration method is introduced to solve the nonlinear inverse Helmholtz problems for identifying the parameter. All the measurement data can be incl...
In this paper, we propose an approach to inverse spectral problems for the n-th order
In this paper, we consider a class of matrix functions that contains regularization matrices of Mirzoev and Shkalikov for differential operators with distribution coefficients of order
In this paper, we study two inverse eigenvalue problems (IEPs) of constructing two special acyclic matrices. The first problem involves the reconstruction of matrices whose graph is a path, from given information on one eigenvector of the required ma...
Markov chain Monte Carlo (MCMC) stands out as an effective method for tackling Bayesian inverse problems. However, when dealing with computationally expensive forward models and high-dimensional parameter spaces, the challenge of repeated sampling be...
This paper is concerned with estimating the achievable resolution in the reconstruction of strip sources from the knowledge of its radiated field and strip objects from the knowledge of its scattered field. In particular, the study focuses on the eva...
Inverse scattering problems (ISPs) stand at the center of many important imaging applications, such as geophysical explorations, industrial non-destructive testing, bio-medical imaging, etc. Recently, a new type of contraction integral equation for i...
Inverse Physics-Informed Neural Networks (inverse PINNs) offer a robust framework for solving inverse problems governed by partial differential equations (PDEs), particularly in scenarios with limited or noisy data. However, conventional inverse PINN...
In this paper, the inverse problems for the boundary value and initial value in a heat equation are posed and solved. It is well known that those problems are ill posed. The problems are reformulated as integral equations of the first kind by using t...
This article addresses a two-dimensional inverse problem of magnetotelluric sounding under the assumption of E-polarized electromagnetic fields. The main focus is on the construction of a forward numerical model based on the Helmholtz equation with a...
Classical methods for inverse problems are mainly based on regularization theory, in particular those, that are based on optimization of a criterion with two parts: a data-model matching and a regularization term. Different choices for these two term...
We study inverse problems of identification of lower-order coefficients in a second-order parabolic equation. The coefficients are sought in the form of a finite series segment with unknown coefficients, depending on time. The linear case is also con...
This paper presents a regularization framework that aims to improve the fidelity of Tikhonov inverse solutions. At the heart of the framework is the data-informed regularization idea that only data-uninformed parameters need to be regularized, while...
Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Netwo...
We address the inverse extremal eigenvalue problem (IEEP) for multi-arrowhead pentadiagonal matrices, a structured class that combines pentadiagonal bandwidth with an alternating arrowhead structure. We identify four distinct structural classes based...
Some inverse problems of Stokes flow, including noisy boundary conditions, unknown angular velocity, and dynamic viscous constant identification are studied in this paper. The interpolation equations for those inverse problems are constructed using t...
The paper considers the solvability of some inverse problems for fractional differential equations with a nonlocal biharmonic operator, which is introduced with the help of involutive transformations in two space variables. The considered problems ar...
The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In thi...
In this paper, an approach to solving direct and inverse scattering problems on the half-line for a one-dimensional Schrödinger equation with a complex-valued potential that is exponentially decreasing at infinity is developed. It is based on a...
In this paper Lie group method in combination with Magnus expansion is utilized to develop a universal method applicable to solving a Sturm–Liouville problem (SLP) of any order with arbitrary boundary conditions. It is shown that the method has...
We firstly prove the Horváth-type theorem for Sturm–Liouville operators on a star-shaped graph and then solve a new partial inverse nodal problem for this operator. We give some algorithms to recover this operator from a dense nodal subs...
This work presents a technique called Time Traveling Regularization (TTR) applied to an optimization technique in order to solve ill-posed problems. This new methodology does not interfere in the minimization technique process. The Golden Section met...
We propose a class of coupled dynamical systems for solving linear inverse problems, treating both the unknown variable and an auxiliary variable representing measurement dynamics as state variables. This framework does not rely on probabilistic mode...
We address the inverse medium scattering problem with phaseless data motivated by nondestructive testing for optical fibers. As the phase information of the data is unknown, this problem may be regarded as a standard phase retrieval problem that cons...
Computational techniques allow breaking the limits of traditional imaging methods, such as time restrictions, resolution, and optics flaws. While simple computational methods can be enough for highly controlled microscope setups or just for previews,...
In this article, a newly developed multiple-source meshless method (MSMM) capable of solving inverse heat conduction problems in two dimensions is presented. Evolved from the collocation Trefftz method (CTM), the MSMM approximates the solution by usi...
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