Related papers: Methods of Solving Ill-Posed Problems
We explore the problem of stabilization of unstable periodic orbits in discrete nonlinear dynamical systems. This work proposes the generalization of predictive control method for resolving the stabilization problem. Our method embodies the…
We consider the method of quasi-solutions (also referred to as Ivanov regularization) for the regularization of linear ill-posed problems in non-reflexive Banach spaces. Using the equivalence to a metric projection onto the image of the…
Using the damped pendulum system we introduce the averaging method to study the periodic solutions of a dynamical system with small perturbation. We provide sufficient conditions for the existence of periodic solutions with small amplitude…
Further development of the method of computational experiments for solving ill-posed problems is given. The effective (unoverstated) estimate for solution error of the first-kind equation is obtained using the truncating singular numbers…
Classically, to solve differential equation problems, it is necessary to specify sufficient initial and/or boundary conditions so as to allow the existence of a unique solution. Well-posedness of differential equation problems thus involves…
In this paper, we consider the solution of ill-conditioned systems of linear algebraic equations that can be determined imprecisely. To improve the stability of the solution process, we "immerse" the original imprecise linear system in an…
It's well-known that inverse problems are ill-posed and to solve them meaningfully one has to employ regularization methods. Traditionally, the most popular regularization approaches are Variational-type approaches, i.e.,…
In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e., the solution…
We consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for…
Solving inverse partial differential equation (PDE) problems is a fundamental topic in scientific research due to its broad significance across a wide range of real-world applications. Inverse PDE problems arise across medical imaging,…
Variational-hemivariational inequalities are an important mathematical framework for nonsmooth problems. The framework can be used to study application problems from physical sciences and engineering that involve non-smooth and even…
Several N-body problems in ordinary (3-dimensional) space are introduced which are characterized by Newtonian equations of motion (``acceleration equal force;'' in most cases, the forces are velocity-dependent) and are amenable to exact…
For the first time, a globally convergent numerical method is presented for ill-posed Cauchy problems for quasilinear PDEs. The key idea is to use Carleman Weight Functions to construct globally strictly convex Tikhonov-like cost…
In limited data computerized tomography, the 2D or 3D problem can be reduced to a family of 1D problems using the differentiated backprojection (DBP) method. Each 1D problem consists of recovering a compactly supported function $f \in…
In this paper, we investigate the Cauchy problem for both linear and semi-linear elliptic equations. In general, the equations have the form \[ \frac{\partial^{2}}{\partial…
We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have…
We consider a mixed dimensional elliptic partial differential equation posed in a bulk domain with a large number of embedded interfaces. In particular, we study well-posedness of the problem and regularity of the solution. We also propose…
In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where the nonlinear term depends on $u$ and $\partial_t u$. We prove a ill-posedness result for the "defocusing" case, and…
Regularization is a core component of modern inverse problems, as it helps establish the well-posedness of the solution of interest. Popular regularization approaches include variational regularization and iterative regularization. The…
The note focuses on the differential geometric approach to the study of nonlinear systems that are affine in control. We first develop normal forms for nonlinear system affine in control. Based on these normal forms, we then address the…