arXiv subjects
Publications and source records attributed to A. G. Ramm.
Let $G(k)=\int_0^1g(x)e^{kx}dx$, $g\in L^1(0,1)$. The main result of this paper is the following theorem. {\bf Theorem}. {\it If $\limsup_{k\to +\infty}|G(k)|<\infty$, then $g=0$. There exists $g\not\equiv 0$, $g\in L^1(0,1)$, such that $G(k_j)=0$, $k_j 0$ is.}
Let $F(u)=h$ be an operator equation in a Banach space $X$, $\|F'(u)-F'(v)\|\leq ω(\|u-v\|)$, where $ω\in C([0,\infty))$, $ω(0)=0$, $ω(r)>0$ if $r>0$, $ω(r)$ is strictly growing on $[0,\infty)$. Denote $A(u):=F'(u)$, where $F'(u)$ is the Fréchet derivative of $F$, and $A_a:=A+aI.$ Assume that (*) $\|A^{-1}_a(u)\|\leq \frac{c_1}{|a|^b}$, $|a|>0$, $b>0$, $a\in L$. Here $a$ may be a complex number, and $L$ is a smooth path on the complex $a$-plane, joining the origin and some point on the complex $a-$plane, $0<|a|<ε_0$, where $ε_0>0$ is a small fixed number, such that for any $a\in L$ estimate (*) holds. It is proved that the DSM (Dynamical Systems Method) \bee \dot{u}(t)=-A^{-1}_{a(t)}(u(t))[F(u(t))+a(t)u(t)-f],\quad u(0)=u_0,\ \dot{u}=\frac{d u}{dt}, \eee converges to $y$ as $t\to +\infty$, where $a(t)\in L,$ $F(y)=f$, $r(t):=|a(t)|$, and $r(t)=c_4(t+c_2)^{-c_3}$, where $c_j>0$ are some suitably chosen constants, $j=2,3,4.$ Existence of a solution $y$ to the equation $F(u)=f$ is assumed. It is also assumed that the equation $F(w_a)+aw_a-f=0$ is uniquely solvable for any $f\in X$, $a\in L$, and $\lim_{|a|\to 0,a\in L}\|w_a-y\|=0.$
A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In this paper a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. This paper also gives an a posteriori stopping rule for a modified iterative scheme developed in A.G.Ramm, JMAA,330 (2007),1338-1346, and proves convergence of the solution obtained by the iterative scheme.
The purpose of this paper is to give a convergence analysis of the iterative scheme: \bee u_n^\dl=qu_{n-1}^\dl+(1-q)T_{a_n}^{-1}K^*f_\dl,\quad u_0^\dl=0,\eee where $T:=K^*K,\quad T_a:=T+aI,\quad q\in(0,1),\quad a_n:=α_0q^n, α_0>0,$ with finite-dimensional approximations of $T$ and $K^*$ for solving stably Fredholm integral equations of the first kind with noisy data.
In this paper a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function $f(t)$ is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to $f(t)$, are proposed in this paper.
Convergence of the classical Newton's method and its DSM version for solving operator equations $F(u)=h$ is proved without any smoothness assumptions on $F'(u)$. It is proved that every solvable equation $F(u)=f$ can be solved by Newton's method if the initial approximation is sufficiently close to the solution and $||[F'(y)]^{-1}||\leq m$, where $m>0$ is a constant.
The many-body wave scattering problem is studied in the case where the bodies are small, $ka \ll 1$, where $a$ is the characteristic size of a body. The limiting case when $a \to 0$ and the total number of the small bodies is $M = O (a^{2-κ})$ is studied, $κ\in (0,1)$ is a parameter, the distance $d$ between neighboring bodies is $d=O(a^{(2-κ)/3}$.
Let $q(x)$ be real-valued compactly supported sufficiently smooth function, $q\in H^\ell_0(B_a)$, $B_a:=\{x: |x|\leq a, x\in R^3$ . It is proved that the scattering data $A(-β,β,k)$ $\forall β\in S^2$, $\forall k>0$ determine $q$ uniquely. here $A(β,α,k)$ is the scattering amplitude, corresponding to the potential $q$.
Sufficient conditions are given for a hard implicit function theorem to hold. The result is established by an application of the Dynamical Systems Method (DSM). It allows one to solve a class of nonlinear operator equations in the case when the Fréchet derivative of the nonlinear operator is a smoothing operator, so that its inverse is an unbounded operator.
A 3D singular integral equation is derived for electromagnetic wave scattering by bodies of arbitrary shape. Its numerical solution by a projection method is outlined.
Boundary integral equation is derived for the problem of scattering of electromagnetic waves by 3D homogeneous body of arbitrary shape.
An evolution equation, arising in the study of the Dynamical Systems Method (DSM) for solving equations with monotone operators, is studied in this paper. The evolution equation is a continuous analog of the regularized Newton method for solving ill-posed problems with monotone nonlinear operators $F$. Local and global existence of the unique solution to this evolution equation are proved, apparently for the firs time, under the only assumption that $F'(u)$ exists and is continuous with respect to $u$. The earlier published results required more smoothness of $F$. The Dynamical Systems method (DSM) for solving equations $F(u)=0$ with monotone Fréchet differentiable operator $F$ is justified under the above assumption apparently for the first time.
Let $\ell_j:=-\frac{d^2}{dx^2}+k^2q_j(x),$ $k=const>0, j=1,2,$ $0 0,$ where $p\in M$ is an arbitrary fixed function, and $u_j$ solves the problem $\ell_ju_j=0,\quad 0\leq x\leq 1,\quad u'_j(0,k)=0,\quad u_j(0,k)=1.$ If $(*)$ implies $h=0$, then the pair $\{\ell_1,\ell_2\}$ is said to have property $C$ on the set $M$. This property is proved for the pair $\{\ell_1,\ell_2\}$. Applications to some inverse problems for a heat equation are given. the set $M$. This property is proved for the pair
It is proposed to create materials with a desired refraction coefficient in a bounded domain $D\subset \R^3$ by embedding many small balls with constant refraction coefficients into a given material. The number of small balls per unit volume around every point $x\in D$, i.e., their density distribution, is calculated, as well as the constant refraction coefficients in these balls. Embedding into $D$ small balls with these refraction coefficients according to the calculated density distribution creates in $D$ a material with a desired refraction coefficient.
A method is given for creating material with a desired refraction coefficient. The method consists of embedding into a material with known refraction coefficient many small particles of size $a$. The number of particles per unit volume around any point is prescribed, the distance between neighboring particles is $O(a^{\frac{2-κ}{3}})$ as $a\to 0$, $0<κ<1$ is a fixed parameter. The total number of the embedded particle is $O(a^{κ-2})$. The physical properties of the particles are described by the boundary impedance $ζ_m$ of the $m-th$ particle, $ζ_m=O(a^{-κ})$ as $a\to 0$. The refraction coefficient is the coefficient $n^2(x)$ in the wave equation $[\nabla^2+k^2n^2(x)]u=0$.
Let $A(β,α,k)$ be the scattering amplitude corresponding to a real-valued potential which vanishes outside of a bounded domain $D\subset \R^3$. The unit vector $α$ is the direction of the incident plane wave, the unit vector $β$ is the direction of the scattered wave, $k>0$ is the wave number. The governing equation for the waves is $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$. For a suitable class of potentials it is proved that if $A_{q_1}(-β,β,k)=A_{q_2}(-β,β,k)$ $\forall β\in S^2,$ $\forall k\in (k_0,k_1),$ and $q_1,$ $q_2\in M$, then $q_1=q_2$. This is a uniqueness theorem for the solution to the inverse scattering problem with backscattering data. It is also proved for this class of potentials that if $A_{q_1}(β,α_0,k)=A_{q_2}(β,α_0,k)$ $\forall β\in S^2_1,$ $\forall k\in (k_0,k_1),$ and $q_1,$ $q_2\in M$,then $q_1=q_2$. Here $S^2_1$ is an arbitrarily small open subset of $S^2$, and $|k_0-k_1|>0$ is arbitrarily small.
The governing equation is $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$. It is shown that any desired potential $q(x)$, vanishing outside a bounded domain $D$, can be obtained if one embeds into D many small scatterers $q_m(x)$, vanishing outside balls $B_m:=\{x: |x-x_m|<a\}$, such that $q_m=A_m$ in $B_m$, $q_m=0$ outside $B_m$, $1\leq m \leq M$, $M=M(a)$. It is proved that if the number of small scatterers in any subdomain $Δ$ is defined as $N(Δ):=\sum_{x_m\in Δ}1$ and is given by the formula $N(Δ)=|V(a)|^{-1}\int_Δn(x)dx [1+o(1)]$ as $a\to 0$, where $V(a)=4πa^3/3$, then the limit of the function $u_{M}(x)$, $\lim_{a\to 0}U_M=u_e(x)$ does exist and solves the equation $[\nabla^2+k^2-q(x)]u=0$ in $\R^3$, where $q(x)=n(x)A(x)$,and $A(x_m)=A_m$. The total number $M$ of small inhomogeneities is equal to $N(D)$ and is of the order $O(a^{-3})$ as $a\to 0$. A similar result is derived in the one-dimensional case.
The governing equation is $u_t = (a(x)u_x)_x$, $0\le x\le 1$, $t>0$, $u(x,0)=0$, $u(0,t)=0$, $a(1)u'(1,t)=f(t)$. The extra data are $u(1,t)=g(t)$. It is assumed that $a(x)$ is a piecewise-constant function, and $f\not\equiv 0$. It is proved that the function $a(x)$ is uniquely defined by the above data. No restrictions on the number of discontinuity points of $a(x)$ and on their locations are made. The number of discontinuity points is finite, but this number can be arbitrarily large. If $a(x)\in C^2[0,1]$, then a uniqueness theorem has been established earlier for multidimensional problem, $x\in \mathbb{R}^n, n>1$ (see MR1211417 (94e:35004)) for the stationary problem with infinitely many boundary data. The novel point in this work is the treatment of the discontinuous piecewise-constant function $a(x)$ and the proof of Property C for a pair of the operators $\{\ell_1, \ell_2 \}$, where $\ell_j:= -\frac{d^2}{dx^2} + k^2 q_j^2(x)$, $j=1,2$, and $q_j^2(x)>0$ are piecewise-constant functions, and for the pair $\{L_1, L_2 \}$, where $L_ju:=-[a_j(x)u'(x)]'+λu$, $j=1,2$, and $a_j(x)>0$ are piecewise-constant functions. Property C stands for completeness of the set of products of solutions of homogeneous differential equations (see MR1759536 (2001f:34048))