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Alexander G. Ramm

Publications and source records attributed to Alexander G. Ramm.

At least 37 records · Page 2Linked to original sources
arXiv2012

Scattering of electromagnetic waves by many thin cylinders

Electromagnetic wave scattering by many parallel infinite cylinders is studied asymptotically as $a\to 0$. Here $a$ is the radius of the cylinders. It is assumed that the points $\hat{x}_m$ are distributed so that $\mathcal{N}(Δ)=\frac{1}{a}\int_ΔN(x)dx[1+o(1)], $ where $\mathcal{N}(Δ)$ is the number of points $\hat{x}_m=(x_{m1},x_{m2})$ in an arbitrary open subset of the plane $xoy$, the axes of the cylinders are passing through points $\hat{x}_m$, these axes are parallel to the z-axis. The function $N(x)\geq 0$ is a given continuous function. An equation for the self-consistent (efficient) field is derived as $a\to 0$. The cylinders are assumed perfectly conducting. Formula is derived for the effective refraction coefficient in the medium in which many cylinders are distributed.

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arXiv2011

A collocation method for solving some integral equations in distributions

A collocation method is presented for numerical solution of a typical integral equation Rh :=\int_D R(x, y)h(y)dy = f(x), x ε D of the class R, whose kernels are of positive rational functions of arbitrary selfadjoint elliptic operators defined in the whole space R^r, and D \subset R^r is a bounded domain. Several numerical examples are given to demonstrate the efficiency and stability of the proposed method.

math.NA↗
arXiv2011

Scattering of scalar waves by many small particles

Formulas are derived for solutions of many-body wave scattering problems by small particles in the case of acoustically soft, hard, and impedance particles embedded in an inhomogeneous medium. The limiting case is considered, when the size $a$ of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived.

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arXiv2010

Creating materials with a desired refraction coefficient: numerical experiments

A recipe for creating materials with a desired refraction coefficient is implemented numerically. The following assumptions are used: \bee ζ_m=h(x_m)/a^κ,\quad d=O(a^{(2-κ)/3}),\quad M=O(1/a^{2-κ}),\quad κ\in(0,1), \eee where $ζ_m$ and $x_m$ are the boundary impedance and center of the $m$-th ball, respectively, $h(x)\in C(D)$, Im$h(x)\leq 0$, $M$ is the number of small balls embedded in the cube $D$, $a$ is the radius of the small balls and $d$ is the distance between the neighboring balls. An error estimate is given for the approximate solution of the many-body scattering problem in the case of small scatterers. This result is used for the estimate of the minimal number of small particles to be embedded in a given domain $D$ in order to get a material whose refraction coefficient approximates the desired one with the relative error not exceeding a desired small quantity.

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arXiv2004

Analysis of a method for identification of obstacles

Some difficulties are pointed out in the methods for identification of obstacles based on the numerical verification of the inclusion of a function in the range of an operator. Numerical examples are given to illustrate theoretical conclusions. Alternative methods of identification of obstacles are mentioned: the Support Function Method (SFM) and the Modified Rayleigh Conjecture (MRC) method.

math-ph↗
arXiv2004

Numerical Solution of Obstacle Scattering Problems

Some novel numerical approaches to solving direct and inverse obstacle scattering problems (IOSP) are presented. Scattering by finite obstacles and by periodic structures is considered. The emphasis for solving direct scattering problem is on the Modified Rayleigh Conjecture (MRC) method, recently introduced and tested by the authors. This method is used numerically in scattering by finite obstacles and by periodic structures. Numerical results it produces are very encouraging. The support function method (SFM) for solving the IOSP is described and tested in some examples. Analysis of the various versions of linear sampling methods for solving IOSP is given and the limitations of these methods are described.

math.AP↗
arXiv2004

Optimization methods in direct and inverse scattering

In many Direct and Inverse Scattering problems one has to use a parameter-fitting procedure, because analytical inversion procedures are often not available. In this paper a variety of such methods is presented with a discussion of theoretical and computational issues. The problem of finding small subsurface inclusions from surface scattering data is solved by the Hybrid Stochastic-Deterministic minimization algorithm. A similar approach is used to determine layers in a particle from the scattering data. The Inverse potential scattering problem for spherically symmetric potentials and fixed-energy phase shifts as the scattering data is described. It is solved by the Stability Index Method. This general approach estimates the size of the minimizing sets, and gives a practically useful stopping criterion for global minimization algorithms. The 3D inverse scattering problem with fixed-energy data and its solution by the Ramm's method are discussed. An Obstacle Direct Scattering problem is treated by a Modified Rayleigh Conjecture (MRC) method. A special minimization procedure allows one to inexpensively compute scattered fields for 2D and 3D obstacles having smooth as well as nonsmooth surfaces. A new Support Function Method (SFM) is used for Inverse Obstacle Scattering problems. Another method for Inverse scattering problems, the Linear Sampling Method (LSM), is analyzed.

math.NA↗
arXiv2004

Integral Operators Basic in Random Fields Estimation Theory

The paper deals with the basic integral equation of random field estimation theory by the criterion of minimum of variance of the error estimate. This integral equation is of the first kind. The corresponding integra$ operator over a bounded domain $Ω$ in ${\Bbb R}^{n}$ is weakly singular. This operator is an isomorphism between appropriate Sobolev spaces. This is proved by a reduction of the integral equ$ an elliptic boundary value problem in the domain exterior to $Ω.$ Extra difficulties arise due to the fact that the exterior boundary value problem should be solved in the Sobolev spaces of negative order.

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arXiv2003

One-dimensional inverse scattering and spectral problems

Inverse scattering and spectral one-dimensional problems are discussed systematically in a self-contained way. Many novel results, due to the author are presented. The classical results are often presented in a new way. Several highlights of the new results include: Analysis of the invertibility of the steps in the Gel'fand-Levitan and Marchenko inversion procedures, Theory of the inverse problem with I-function as the data and its applications; Proof of the property C for ordinary differential operators, numerous applications of property C; Inverse problems with "incomplete" data; Spherically symmetric inverse scattering problem with fixed-energy data: analysis of the Newton-Sabatier (NS) scheme for inversion of fixed-energy phase shifts is given. This analysis shows that the NS scheme is fundamentally wrong, and is not a valid inversion method. Complete presentation of the Krein inverse scattering theory is given. Consistency of this theory is proved. Quarkonium systems; Some new inverse problems for the heat and wave equations. A study of inverse scattering problem for an inhomogeneous Schrödinger equation;

math-ph↗
arXiv2003

On deconvolution methods

Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation ${\mathbf k} u:=\int_0^t k(t-s)u(s)ds=g(t),\quad 0\leq t\leq T$. The data, $g(t)$, are noisy. Of special practical interest is the case when the data are noisy and known at a discrete set of times. A general approach to the deconvolution problem is proposed: represent ${\mathbf k}=A(I+S)$, where a method for a stable inversion of $A$ is known, $S$ is a compact operator, and $I+S$ is injective. This method is illustrated by examples: smooth kernels $k(t)$, and weakly singular kernels, corresponding to Abel-type of integral equations, are considered. A recursive estimation scheme for solving deconvolution problem with noisy discrete data is justified mathematically, its convergence is proved, and error estimates are obtained for the proposed deconvolution method.

math.NA↗
arXiv2001

Linear ill-posed problems and dynamical systems

A linear equation Au=f (1) with a bounded, injective, but not boundedly invertible linear operator in a Hilbert space H is studied. A new approach to solving linear ill-posed problems is proposed. The approach consists of solving a Cauchy problem for a linear equation in H, which is a dynamical system, proving the existence and uniqueness of its global solution u(t), and establishing that u(t) tends to a limit y, as t tends to infinity, and this limit y solves equation (1). The case when f in (1) is given with some error is also studied.

math-ph↗
arXiv2001

Continuous regularized Gauss-Newton-type algorithm for nonlinear ill-posed equations with simultaneous updates of inverse derivative

A new continuous regularized Gauss-Newton-type method with simultaneous updates of the operator $(F^{\pr*}(x(t))F'(x(t))+\ep(t) I)^{-1}$ for solving nonlinear ill-posed equations in a Hilbert space is proposed. A convergence theorem is proved. An attractive and novel feature of the proposed method is the absence of the assumptions about the location of the spectrum of the operator $F'(x)$. The absence of such assumptions is made possible by a source-type condition.

math-ph↗
arXiv2000

Property C for ODE and applications

An overview of the authors results is given. Property C for ODE is defioned, It is proved that the pair of Sturm-Liouville operators has property C. This property is applied to many inverse problems. Some well-known results, such as uniqueness of the recovery of the potential from scattering data, from the spectral function and from two spectra are proved in a new short way based on property C. Many new results are obtained. In particular, it is proved that a compactly supported potential can be uniquely recovered from the phase shift of s-wave alone, known at all energies, without knowledge of bound states and norming constants. The fixed-energy phase shifts known at some values of the angular momentum determine a compactly supported potential uniquely. Mixed data inverse problems are studied. Analysis of the Newton-Sabatier procedure is given. It is shown that this procedure is a parameter-fitting procedure rather than an inversion method. Other results on inverse problems are obtained.

math-ph↗
arXiv2000

Application of the hybrid stochastic-deterministic minimization method to a surface data inverse scattering problem

A method for the identification of small inhomogeneities from a surface data is presented in the framework of an inverse scattering problem for the Helmholtz equation. Using the assumptions of smallness of the scatterers one reduces this inverse problem to an identification of the positions of the small scatterers. These positions are found by a global minimization search. Such a search is implemented by a novel Hybrid Stochastic-Deterministic Minimization method. The method combines random tries and a deterministic minimization. The effectiveness of this approach is illustrated by numerical experiments. In the modeling part our method is valid when the Born approximation fails. In the numerical part, an algorithm for the estimate of the number of the small scatterers is proposed.

math-ph↗
arXiv2000

Piecewise-constant potentials with practically the same fixed-energy phase shifts

It has recently been shown that spherically symmetric potentials of finite range are uniquely determined by the part of their phase shifts at a fixed energy level $k^2>0$. However, numerical experiments show that two quite different potentials can produce almost identical phase shifts. It has been guessed by physicists that such examples are possible only for "less physical" oscillating and changing sign potentials. In this note it is shown that the above guess is incorrect: we give examples of four positive spherically symmetric compactly supported quite different potentials having practically identical phase shifts. The note also describes a hybrid stochastic-deterministic method for global minimization used for the construction of these potentials.

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arXiv2000

Stable identification of piecewise-constant potentials from fixed-energy phase shifts

An identification of a spherically symmetric potential by its phase shifts is an important physical problem. Recent theoretical results assure that such a potential is uniquely defined by a sufficiently large subset of its phase shifts at any one fixed energy level. However, two different potentials can produce almost identical phase shifts. That is, the inverse problem of the identification of a potential from its phase shifts at one energy level $k^2$ is ill-posed, and the reconstruction is unstable. In this paper we introduce a quantitative measure $D(k)$ of this instability. The diameters of minimizing sets $D(k)$ are used to study the change in the stability with the change of $k$, and the influence of noise on the identification. They are also used in the stopping criterion for the nonlinear minimization method IRRS (Iterative Random Reduced Search). IRRS combines probabilistic global and deterministic local search methods and it is used for the numerical recovery of the potential by the set of its phase shifts. The results of the identification for noiseless as well as noise corrupted data are presented.

math-ph↗
arXiv2000

Stability of solutions to inverse scattering problems with fixed-energy data

A review of the author's results is given. Inversion formulas and stability estimates for the solutions to 3D inverse scattering problems with fixed-energy data are obtained. Inversions of exact and noisy data are stidied. The inverse potential scattering problem is discussed in detail, inversion formulas are derived and error estimates are obtained. Inverse obstacle scattering problem with data at a fixed frequency is studied. Uniqueness theorems and stability estimates are obtained. Inverse geophysical scattering problem is discussed. An algorithm for computing the Dirichlet-to-Neumann map from the scattering amplitude and vicxe versa is obtained. An analytical example of non-uniqueness of the solution to a 3D inverse geophysical problem is constructed. An inverse problem for a parabolic equation is discussed.

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