5,149 research outputs found
On a new notion of the solution to an ill-posed problem
A new understanding of the notion of the stable solution to ill-posed
problems is proposed. The new notion is more realistic than the old one and
better fits the practical computational needs. A method for constructing stable
solutions in the new sense is proposed and justified. The basic point is: in
the traditional definition of the stable solution to an ill-posed problem
, where is a linear or nonlinear operator in a Hilbert space , it
is assumed that the noisy data are given,
, and a stable solution u_\d:=R_\d f_\d is defined
by the relation \lim_{\d\to 0}||R_\d f_\d-y||=0, where solves the
equation , i.e., . In this definition and are unknown. Any
f\in B(f_\d,\d) can be the exact data, where B(f_\d,\d):=\{f:
||f-f_\delta||\leq \delta\}.The new notion of the stable solution excludes the
unknown and from the definition of the solution
Scattering of electromagnetic waves by many thin cylinders
Electromagnetic wave scattering by many parallel infinite cylinders is
studied asymptotically as . Here is the radius of the cylinders. It
is assumed that the points are distributed so that
where
is the number of points in an
arbitrary open subset of the plane , the axes of the cylinders are passing
through points , these axes are parallel to the z-axis. The function
is a given continuous function. An equation for the
self-consistent (efficient) field is derived as . The cylinders are
assumed perfectly conducting. Formula is derived for the effective refraction
coefficient in the medium in which many cylinders are distributed
Completeness of the set of scattering amplitudes
Let be an arbitrary fixed function with small norm on the
unit sphere , and be an arbitrary fixed bounded domain.
Let and be fixed.
It is proved that there exists a potential such that the
corresponding scattering amplitude
approximates with
arbitrary high accuracy: \|f(\alpha')-A_q(\alpha')_{L^2(S^2)}\|\leq\ve where
\ve>0 is an arbitrarily small fixed number. This means that the set
is complete in . The results
can be used for constructing nanotechnologically "smart materials"
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 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
Wave scattering by small impedance particles in a medium
The theory of acoustic wave scattering by many small bodies is developed for
bodies with impedance boundary condition. It is shown that if one embeds many
small particles in a bounded domain, filled with a known material, then one can
create a new material with the properties very different from the properties of
the original material. Moreover, these very different properties occur although
the total volume of the embedded small particles is negligible compared with
the volume of the original material
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
A new discrepancy principle
The aim of this note is to prove a new discrepancy principle. The advantage
of the new discrepancy principle compared with the known one consists of
solving a minimization problem approximately, rather than exactly, and in the
proof of a stability result
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
Dynamical systems method for solving operator equations
Consider an operator equation in a real Hilbert space.
The problem of solving this equation is ill-posed if the operator is
not boundedly invertible, and well-posed otherwise.
A general method, dynamical systems method (DSM) for solving linear and
nonlinear ill-posed problems in a Hilbert space is presented.
This method consists of the construction of a nonlinear dynamical system,
that is, a Cauchy problem, which has the following properties:
1) it has a global solution,
2) this solution tends to a limit as time tends to infinity,
3) the limit solves the original linear or non-linear problem. New
convergence and discretization theorems are obtained. Examples of the
applications of this approach are given. The method works for a wide range of
well-posed problems as well.Comment: 21p
Scattering of electromagnetic waves by many thin cylinders: theory and computational modeling
Electromagnetic (EM) wave scattering by many parallel infinite cylinders is
studied asymptotically as a tends to 0, where a is the radius of the cylinders.
It is assumed that the centres of the cylinders are distributed so that their
numbers is determined by some positive function N(x). The function N(x) >= 0 is
a given continuous function. An equation for the self-consistent (limiting)
field is derived as a tends to 0. The cylinders are assumed perfectly
conducting. Formula for the effective refraction coefficient of the new medium,
obtained by embedding many thin cylinders into a given region, is derived. The
numerical results presented demonstrate the validity of the proposed approach
and its efficiency for solving the many-body scattering problems, as well as
the possibility to create media with negative refraction coefficients.Comment: 21 pages, 13 figure
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