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.

Cerebral aneurysms and arteriovenous malformations are life-threatening hemodynamic pathologies of the brain. While surgical intervention is often essential to prevent fatal outcomes, it carries significant risks both during the procedure and in the postoperative period, making the management of these conditions highly challenging. Parameters of cerebral blood flow, routinely monitored during medical interventions or with modern noninvasive high-resolution imaging methods, could potentially be utilized in machine-learning-assisted protocols for risk assessment and therapeutic prognosis. To this end, we developed a linear oscillatory model of blood velocity and pressure for clinical data acquired from neurosurgical operations. Using the method of Sparse Identification of Nonlinear Dynamics (SINDy), the parameters of our model can be reconstructed online within milliseconds from a short time series of the hemodynamic variables. The identified parameter values enable automated classification of the blood-flow pathologies by means of logistic regression, achieving a balanced accuracy of 74%. Our results demonstrate the potential of this model for both diagnostic and prognostic applications, providing a robust and interpretable framework for assessing cerebral blood vessel conditions.

In this work, the inverse problem of reconstructing the squared slowness function in the two-dimensional eikonal equation has been developed and numerically investigated. The inverse problem is reduced to the minimization of a travel-time misfit functional by gradient method and employs the Fast Marching Method for an efficient solution of the direct eikonal problem. A gradient of the functional is calculated by direct and adjoint problems, enabling the computational cost of the gradient comparable to that of a single direct solve. Numerical experiments on synthetic models which parameters are close to the human body with inclusions show that the method can accurately recover the spatial structure and contrast of the slowness distribution under different grid resolutions and source configurations. A comparative analysis of simple gradient and heavy ball method has shown that the use of adaptive relaxation of steps accelerates the convergence of gradient methods, including accelerated gradient methods.

The Cauchy problem for multi-dimensional nonlinear elliptic equations is reformulated as a least squares problem and then stabilized by Tikhonov regularization method. The last is solved in the framework of artificial neural networks, while the regularization parameter is chosen by the discrepancy principle or the L-curve method. Numerical simulations show that the approach is efficient for this severely ill-posed, multi-dimensional and nonlinear problem.

[This corrects the article DOI: 10.1371/journal.pone.0319837.].

In this paper, we develop the explicit finite difference method (FDM) to solve an ill-posed Cauchy problem for the 3D acoustic wave equation in a time domain with the data on a part of the boundary given (continuation problem) in a cube. FDM is one of the numerical methods used to compute the solutions of hyperbolic partial differential equations (PDEs) by discretizing the given domain into a finite number of regions and a consequent reduction in given PDEs into a system of linear algebraic equations (SLAE). We present a theory, and through Matlab Version: 9.14.0.2286388 (R2023a), we find an efficient solution of a dense system of equations by implementing the numerical solution of this approach using several iterative techniques. We extend the formulation of the Jacobi, Gauss–Seidel, and successive over-relaxation (SOR) iterative methods in solving the linear system for computational efficiency and for the properties of the convergence of the proposed method. Numerical experiments are conducted, and we compare the analytical solution and numerical solution for different time phenomena.

Street protests have been a common feature of human society for many centuries. They often act as a driver of social changes but they may also disrupt everyday life and lead to considerable economic losses. Understanding of factors that may affect the duration of street protests and the number of participants is a problem of pivotal importance. Mathematical modelling is an efficient research approach to study this problem. Here we present a novel modelling framework that takes into account heterogeneity of protesters behaviour and the effect of policing. Using the 2018-2019 Yellow Vest Movement in France as a case study, we show that our model is in a very good agreement with data. We also show that a moderate increase in the efficiency of police actions on particular days may have a significant effect on protest's intensity and duration. Our findings open a possibility for a more efficient protests management.

Let X be a Banach space with norm || center dot || Let A : D(A) subset of X -> X be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that epsilon > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u : [ 0, T] -> X satisfying u(t) + Au = 0, 0 < t < T, ||u(T)-phi|| <= epsilon, for phi in X, is regularized by the generalized sobolev equation where 0 < alpha < 1 and A alpha = A(I + alpha A(b))(-1) with b >= 1. Error estimate of the method with respect to the noise level are proved.

The article aimed to show the fundamental possibility of constructing a computational digital twin of the acoustic tomograph within the framework of a unified physics–mathematical model based on the Navier–Stokes equations. The authors suggested that the size of the modeling area is quite small, sound waves are waves of “small” disturbance, and given that a person consists of more than 60% water, human organs can be modeled using a liquid model, taking into account their density. During numerical experiments, we obtained the pressure registered in the receivers that are located on the side walls of the tomograph. The differences in pressure values are shown depending on the configuration of inclusions in the mannequin imitating internal organs. The results show that the developed technology can be used to probe the human body in medical acoustic tomographs and determine the acoustic parameters of the human body to detect neoplasms.

Cerebral aneurysms and arteriovenous malformations are life-threatening hemodynamic pathologies of the brain. While surgical intervention is often essential to prevent fatal outcomes, it carries significant risks both during the procedure and in the postoperative period, making the management of these conditions highly challenging. Parameters of cerebral blood flow, routinely monitored during medical interventions or with modern noninvasive high-resolution imaging methods, could potentially be utilized in machine learning-assisted protocols for risk assessment and therapeutic prognosis. To this end, we developed a linear oscillatory model of blood velocity and pressure for clinical data acquired from neurosurgical operations. Using the method of Sparse Identification of Nonlinear Dynamics (SINDy), the parameters of our model can be reconstructed online within milliseconds from a short time series of the hemodynamic variables. The identified parameter values enable automated classification of the blood-flow pathologies by means of logistic regression, achieving an accuracy of 73 \%}. Our results demonstrate the potential of this model for both diagnostic and prognostic applications, providing a robust and interpretable framework for assessing cerebral blood vessel conditions.

The paper discusses the features of constructing numerical schemes for solving coefficient inverse problems for nonlinear partial differential equations of the reaction–diffusion–advection type with data of various types. As input data for the inverse problem, we consider (1) data at the final moment of time, (2) data at the spatial boundary of a domain, (3) data at the position of the reaction front. To solve the inverse problem in all formulations, the gradient method of minimizing the target functional is used. In this case, when constructing numerical minimization schemes, both an approach based on discretization of the analytical expression for the gradient of the functional and an approach based on differentiating the discrete approximation of the functional to be minimized are considered. Features of the practical implementation of these approaches are demonstrated by the example of solving the inverse problem of reconstructing the linear gain coefficient in a nonlinear Burgers-type equation.

This paper is devoted to solving the inverse problem (determining the parameters of a system of ordinary differential equations based on additional information determined at discrete points in time) and analyzing its solution for a mathematical model describing the dynamics of changes in the population and capital of two regions of the world. The inverse problem is reduced to the problem of minimizing the target functional and is solved by the method of differential evolution. A numerical method for solving direct and inverse problems is implemented. The developed method was tested on model and real data for countries such as Russia, China, India and the USA.

We investigate deep learning approach in 2D dynamic ultrasound acoustic tomography. The mathematical model of acoustic tomography is described by a first -order hyperbolic system PDE and is based on conservation laws. This model guarantees us that the training sets of dynamic data are close to the physical solution. We train a neural network consisting of an encoder and a decoder with this data (they contain only one inclusion) and associate the data with a velocity coefficient. Numerical results show that we recover not only single inclusions, but also homogeneities consisting of two inclusions.

We consider the coefficient inverse problem for the 2D system of acoustics. Our goal is to recover the coefficient of acoustic attenuation by using the additional information of the wave-field in the number of receivers. We obtain the gradient of the cost functional and implement the numerical algorithm for solving the inverse problem, based on a optimization approach. We provide the numerical results of recovering the absorption coefficient and study its influence on the efficiency of reconstructing other parameters of the system. By taking into account the absorption of the sounding wave we aim to bring the mathematical model closer to the applications, related to the ultrasound tomography of the human tissue.

We consider the coefficient inverse problem for the 2D acoustic equation. The problem is recovering the speed of sound in the medium (which depends only on the depth) and the density (function of both variables). We describe the method, based on the Gelfand–Levitan–Krein approach, which allows us to obtain both functions by solving two sets of integral equations. The main advantage of the proposed approach is that the method does not use the multiple solution of direct problems, and thus has quite low CPU time requirements. We also consider the variation of the method for the 1D case, where the variation of the wave equation is considered. We illustrate the results with numerical experiments in the 1D and 2D case and study the efficiency and stability of the approach.

In this article we consider a mathematical model of interpretation of a geoelectric section based on georadar data. One of the reasons preventing the spread of GPR technologies is the complexity of data interpretation, which requires the involvement of highly qualified specialists. In this regard, the study of the mathematical model and comparison with real georadar data expands the possibilities of interpretation of the georadar. To test the algorithm for solving the inverse problem of determining the electrical conductivity of layered media, data from exploration wells of the Karaganda coal basin field were used. We assume that the medium is horizontally layered and the electromagnetic pa- rameters of the medium depend only the depth. The problem of determining the conductivity is reduced to the problem of minimizing the cost functional by the gra- dient method. The gradient of the functional is calculated through the solution of the adjoint problem, the results of numerical calculations are given.

Abstract The horizontally diagonalize and fit (HDF) method is proposed to solve the ill-posed Cauchy problem for the three-dimensional Poisson equation with data given on the part of the boundary (a continuation problem). The HDF method consists in discretization over horizontal variables and transformation of the system of differential equations to a diagonal form. This allows to uncouple the original three-dimensional continuation problem into a moderate number of one-dimensional problems in the vertical dimension. The problem size reduction can be carried taking into account the noise level, so that the number k of one-dimensional problems appears to be a regularization parameter. Our experiments show that HDF is applicable to large-scale problems and for n ≤ 2500 {n\leq 2500} is significantly more efficient than Landweber iteration.

In this paper, we consider the Gelfand–Levitan–Marchenko–Krein approach. It is used for solving a variety of inverse problems, like inverse scattering or inverse problems for wave-type equations in both spectral and dynamic formulations. The approach is based on a reduction of the problem to the set of integral equations. While it is used in a wide range of applications, one of the most famous parts of the approach is given via the inverse scattering method, which utilizes solving the inverse problem for integrating the nonlinear Schrodinger equation. In this work, we present a short historical review that reflects the development of the approach, provide the variations of the method for 1D and 2D problems and consider some aspects of numerical solutions of the corresponding integral equations.

This paper considers a model for the accumulation of mutations in a population of mice with a weakened function of polymerases responsible for correcting DNA copying errors during cell division. The model uses the results of the experiment published by Japanese scientists, which contain data on the accumulation of phenotypic differences in three isolated groups of laboratory mice. We have developed a model for the accumulation of negative mutations. Since the accumulation of phenotypic differences in each of the three groups of mice occurred in its own way, we assumed that these differences were associated with genotypic differences in the zeroth generation and set the inverse problem of determining the initial distribution of these differences. Additional information for solving the inverse problem was a set of experimental data on the number of mutant lines and the number of individuals in each group of mice. The results obtained confirmed our assumption.

ФЕДЕРАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ ЦЕНТР ФУНДАМЕНТАЛЬНОЙ И ТРАНСЛЯЦИОННОЙ МЕДИЦИНЫ ФЕДЕРАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ ЦЕНТР «КРАСНОЯРСКИЙ НАУЧНЫЙ ЦЕНТР СИБИРСКОГО ОТДЕЛЕНИЯ РОССИЙСКОЙ АКАДЕМИИ НАУК» ИНСТИТУТ ВЫЧИСЛИТЕЛЬНОГО МОДЕЛИРОВАНИЯ СО