Symmetry problems 2
Some symmetry problems are formulated and solved. New simple proofs are given for the earlier studied symmetry problems.
math.CA↗
arXiv subjects
A. G. Ramm
Publications and source records attributed to A. G. Ramm.
A discrepancy principle for equations with monotone continuous operators
A discrepancy principle for solving nonlinear equations with monotone operators given noisy data is formulated. The existence and uniqueness of the corresponding regularization parameter $a(δ)$ is proved. Convergence of the solution obtained by the discrepancy principle is justified. The results are obtained under natural assumptions on the nonlinear operator.
math.NA↗
A nonlinear inequality and applications
A nonlinear inequality is formulated in the paper. An estimate of the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations. It can be applied to the study of global existence of solutions to nonlinear PDE.
math.CA↗
Dynamical Systems Gradient method for solving nonlinear equations with monotone operators
A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications.
math.NA↗
Dynamical systems method for solving nonlinear equations with monotone operators
A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators in a Hilbert space is studied in this paper. An a posteriori stopping rule, based on a discrepancy-type principle is proposed and justified mathematically. The results of two numerical experiments are presented. They show that the proposed version of DSM is numerically efficient. The numerical experiments consist of solving nonlinear integral equations.
math.NA↗
Dimension reduction in representation of the data
Suppose the data consist of a set $S$ of points $x_j$, $1\leq j \leq J$, distributed in a bounded domain $D\subset R^N$, where $N$ is a large number. An algorithm is given for finding the sets $L_k$ of dimension $k\ll N$, $k=1,2,...K$, in a neighborhood of which maximal amount of points $x_j\in S$ lie. The algorithm is different from PCA (principal component analysis)
stat.ML↗
Dynamical systems gradient method for solving ill-conditioned linear algebraic systems
A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An algorithm for computing the solution using a spectral decomposition of the left-hand side matrix is proposed. Numerical results show that when a spectral decompositon of the left-hand side matrix is available or not computationally expensive to obtain the new method can be considered as an alternative to the Variational Regularization.
math.NA↗
The Dynamical Systems Method for solving nonlinear equations with monotone operators
A review of the authors's results is given. Several methods are discussed for solving nonlinear equations $F(u)=f$, where $F$ is a monotone operator in a Hilbert space, and noisy data are given in place of the exact data. A discrepancy principle for solving the equation is formulated and justified. Various versions of the Dynamical Systems Method (DSM) for solving the equation are formulated. These methods consist of a regularized Newton-type method, a gradient-type method, and a simple iteration method. A priori and a posteriori choices of stopping rules for these methods are proposed and justified. Convergence of the solutions, obtained by these methods, to the minimal norm solution to the equation $F(u)=f$ is proved. Iterative schemes with a posteriori choices of stopping rule corresponding to the proposed DSM are formulated. Convergence of these iterative schemes to a solution to equation $F(u)=f$ is justified. New nonlinear differential inequalities are derived and applied to a study of large-time behavior of solutions to evolution equations. Discrete versions of these inequalities are established.
math.NA↗
Preparing materials with a desired refraction coefficient
A recipe is given for creating material with a desired refraction coefficient by embedding many small particles in a given material. To implement practically this recipe, some technological problems are to be solved. These problems are formulated.
cond-mat.mtrl-sci↗
Does negative refraction make a perfect lens?
A discussion of a question, studied earlier by V.Veselago in 1967 and by J. Pendry in 2000, is given. The question is: can a slab of the material with negative refraction make a perfect lens? Pendry's conclusion was: yes, it can. Our conclusion is: no, in practice it cannot, because of the fluctuations of the refraction coefficient of the slab. Resolution ability of linear isoplanatic optical instruments is discussed.
cond-mat.mtrl-sci↗
A nonlinear inequality
A quadratic inequality is formulated in the paper. An estimate on the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations.
math.DS↗
Creating Wave-Focusing Materials
Basic ideas for creating wave-focusing materials by injecting small particles in a given material are described. The number of small particles to be injected around any point is calculated. Inverse scattering problem with fixed wavenumber and fixed incident direction of the plane acoustic wave is formulated and solved.
math-ph↗
An iterative scheme for solving nonlinear equations with monotone operators
An iterative scheme for solving ill-posed nonlinear operator equations with monotone operators is introduced and studied in this paper. A Dynamical Systems Method (DSM) algorithm for stable solution of ill-posed operator equations with monotone operators is proposed and its convergence is proved. A new discrepancy principle is proposed and justified. A priori and a posteriori stopping rules for the DSM algorithm are formulated and justified.
math.NA↗
A DSM proof of surjectivity of monotone nonlinear mappings
We prove that if $F$ is twice Frechet differentiable and coercivity conditions hold, then $F$ is surjective, i.e., the equation $F(u)=h$ is solvable for every $h\in H$. This is a basic result in the theory of monotone operators. Our aim is to give a simple and short proof of this result based on the Dynamical Systems Method (DSM), developed in the monograph A.G. Ramm, Dynamical systems method, Elsevier, Amsterdam, 2007.
math.NA↗
Electromagnetic wave scattering by small bodies
A reduction of the Maxwell's system to a Fredholm second-kind integral equation with weakly singular kernel is given for electromagnetic (EM) wave scattering by one and many small bodies. This equation is solved asymptotically as the characteristic size of the bodies tends to zero. The technique developed is used for solving the many-body EM wave scattering problem by rigorously reducing it to solving linear algebraic systems, completely bypassing the usage of integral equations. An equation is derived for the effective field in the medium, in which many small particles are embedded. A method for creating a desired refraction coefficient is outlined.
math-ph↗
Electromagnetic wave scattering by many conducting small particles
A rigorous theory of electromagnetic (EM) wave scattering by small perfectly conducting particles is developed. The limiting case when the number of particles tends to infinity is discussed.
math-ph↗
Dynamical systems method for solving linear finite-rank operator equations
A version of the Dynamical Systems Method (DSM) for solving ill-conditioned linear algebraic systems is studied in this paper. An {\it a priori} and {\it a posteriori} stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems.
math.NA↗
Solving ill-conditioned linear algebraic systems by the dynamical systems method (DSM)
An iterative scheme for the Dynamical Systems Method (DSM) is given such that one does not have to solve the Cauchy problem occuring in the application of the DSM for solving ill-conditioned linear algebraic systems. The novelty of the algorithm is that the algorithm does not have to find the regularization parameter $a$ by solving a nonlinear equation. Numerical experiments show that DSM competes favorably with the Variational Regularization.
math.NA↗