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A. G. Ramm

Publications and source records attributed to A. G. Ramm.

At least 163 records · Page 9Linked to original sources
arXiv2004

Low-Pass Filters: Commentary on a Remark by Feynman

In Feynman's lectures there is a remark about the limiting value of the impedance of an n-section ladder consisting of purely reactive elements (capacitances and inductances). The remark is that this limiting impedance $z=\lim_{n\to\infty}z_n$ has a positive real part. He notes that this is surprising since the real part of each $z_n$ is zero, therefore it is impossible for the limit to have a positive real part. A recent article in this journal offered an explanation of this paradox based on the fact that realistic impedances have a non-negative real part, but the authors noted that their argument was incomplete. We use the same physical idea, but give a simple argument which shows that the sequence $z_n$ converges like a geometric series. We also calculate the finite speed at which energy is propagated out into the infinite ladder.

math-ph↗
arXiv2004

Modified Rayleigh Conjecture for scattering by periodic structures

This paper contains a self-contained brief presentation of the scattering theory for periodic structures. Its main result is a theorem (the Modified Rayleigh Conjecture, or MRC), which gives a rigorous foundation for a numerical method for solving the direct scattering problem for periodic structures. A numerical example illustrating the procedure is presented.

math.NA↗
arXiv2004

Dynamical systems method (DSM) for unbounded operators

Let $L$ be an unbounded linear operator in a real Hilbert space $H$, a generator of $C_0$ semigroup, and $g:H\to H$ be a $C^2_{loc}$ nonlinear map. The DSM (dynamical systems method) for solving equ$ $F(v):=Lv+gv=0$ consists of solving the Cauchy problem $\dot {u}=Φ(t,u)$, $u(0)=u_0$, where $Φ$ is a suitable operator, and proving that i) $\exists u(t) \quad \forall t>0$, ii) $\exists u(\infty)$, and iii) $F(u(\infty))=0$.

math.FA↗
arXiv2004

Dynamical systems method for solving nonlinear equations with non-smooth monotone operators

Consider an operator equation (*) $B(u)+\ep u=0$ in a real Hilbert space, where $\ep>0$ is a small constant. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this solution tends to a limit as time tends to infinity, 3) the limit solves the equation $B(u)=0$. Existence of the unique solution is proved by the DSM for equation (*) with monotone hemicontinuous operators $B$ defined on all of$ If $\ep=0$ and equation (**) $B(u)=0$ is solvable, the DSM yields$ solution to (**).

math.FA↗
arXiv2003

Comments on the letter of P.Sabatier "An erroneous statement"

In his letter Sabatier writes that my statement {\bf "The Newton-Sabatier (NS) method for inverting the fixed-energy phase shifts for a potential is fundamentally wrong"} is erroneous. In [2] my statement is proved. It is explained in [2] that the foundations of the NS method are wrong, and, in particular, that the basic integral equation of this method, in general is not uniquely solvable for some $r>0$, contrary to the implicit assumption of the NS method. If this equation is not uniquely solvable for some $r>0$, then the NS method breaks down: it produces a potential which is not locally integrable. Consequently, my conclusion was: {\it the NS theory as an inverse scattering theory is fundamentally wrong in the sense that its foundations are wrong}. This conclusion is fully justified in [2]. Additional comments on the Sabatier's letter are given.

math-ph↗
arXiv2003

On the discrepancy principle

A simple proof of the convergence of the variational regularization, with the regularization parameter, chosen by the discrepancy principle, is given for linear operators under suitable assumptions. It is shown that the discrepancy principle, in general, does not yield uniform with respect to the data convergence. An a priori choice of the regularization parameter is proposed and justified for some nonlinear operator equations.

math-ph↗
arXiv2003

Applic. Analysis, 81, N4, (2002), 929-937

Completeness of the set of products of the derivatives of the solutions to the equation $(av')'-łv=0, v(0,ł)=0$ is proved. This property is used to prove the uniqueness of the solution to an inverse problem of finding conductivity in the heat equation $\dot{u}=(a(x)u')', u(x,0)=0, u(0,t)=0,\ u(1,t)=f(t)$ known for all $t>0$, from the heat flux $a(1)u'(1,t)=g(t)$. Uniqueness of the solution to this problem is proved. The proof is based on Property C. It is proved the inverse that the inverse problem with the extra data (the flux) measured at the point, where the temperature is kept at zero, (point $x=0$ in our case) does not have a unique solution, in general.

math-ph↗
arXiv2003

Analytical solution of a new class of integral equations

Let $(1) Rh=f$, $0\leq x\leq L$, $Rh=\int^L_0 R(x,y)h(y) dy$, where the kernel $R(x,y)$ satisfies the equation $QR=Pδ(x-y)$. Here $Q$ and $P$ are formal differential operators of order $n$ and $m 0$, $m\geq 0$, $Qu:=q_n(x)u^{(n)} + \sum^{n-1}_{j=0} q_j(x) u^{(j)}$, $Ph:=h^{(m)} +\sum^{m-1}_{j=0} p_j(x) h^{(j)}$, $q_n(x)\geq c>0$, the coefficients $q_j(x)$ and $p_j(x)$ are smooth functions defined on $\R$, $δ(x)$ is the delta-function, $f\in H^α(0,L)$, given. Here $\dot H^{-α}(0,L)$ is the dual space to $H^α(0,L)$ with respect to the inner product of $L^2(0,L)$. Under suitable assumptions it is proved that $R:\dot H^{-α}(0,L) \to H^α(0,L)$ is an isomorphism. Equation (1) is the basic equation of random processes estimation theory. Some of the results are generalized to the case of multidimensional equation (1), in which case this is the basic equation of random fields estimation theory. $α:=\frac{n-m}{2}$, $H^α$ is the Sobolev space. An algorithm for finding analytically the unique solution $h\in\dot H^{-α} (0,L)$ to (1) of minimal order of singularity is

math.CA↗
arXiv2003

An essay on some problems of approximation theory

Several questions of approximation theory are discussed: 1) can one approximate stably in $L^\infty$ norm $f^\prime$ given approximation $f_δ, \parallel f_δ- f \parallel_{L^\infty} < δ$, of an unknown smooth function $f(x)$, such that $\parallel f^\prime (x) \parallel_{L^\infty} \leq m_1$? 2) can one approximate an arbitrary $f \in L^2(D), D \subset \R^n, n \geq 3$, is a bounded domain, by linear combinations of the products $u_1 u_2$, where $u_m \in N(L_m), m=1,2,$ $L_m$ is a formal linear partial differential operator and $N(L_m)$ is the null-space of $L_m$ in $D$, $3) can one approximate an arbitrary $L^2(D)$ function by an entire function of exponential type whose Fourier transform has support in an arbitrary small open set? Is there an analytic formula for such an approximation? N(L_m) := \{w: L_m w=0 \hbox{in\} D\}$?

math.CA↗
arXiv2003

Inequalities for solutions to some nonlinear equations

Let $F$ be a nonlinear Frechet differentiable map in a real Hilbert space. Condition sufficient for existence of a solution to the equation $F(u)=0$ is given, and a method (dynamical systems method, DSM) to calculate the solution as the limit of the solution to a Cauchy problem is justified under suitable assumptions.

math.DS↗
arXiv2003

Inequalities for the transformation operators and applications

Inequalities for the transformation operator kernel $A(x,y)$ in terms of $F$-function are given, and vice versa. These inequalities are applied to inverse scattering on half-line. Characterization of the scattering data corresponding to the usual scattering class $L_{1,1}$ of the potentials, to the class of compactly supported potentials, and to the class of square integrable potentials is given. Invertibility of each of the steps in the inversion procedure is proved.

math.CA↗
arXiv2003

Inverse scattering with fixed-energy data

The Newton-Sabatier method for solving inverse scattering problem with fixed-energy phase shifts for a sperically symmetric potential is discussed. It is shown that this method is fundamentally wrong: in general it cannot be carried through, the basic ansatz of R.Newton is wrong: the transformation kernel does not have the form postulated in this ansatz, in general, the method is inconsistent, and some of the physical conclusions, e.g., existence of the transparent potentials, are not proved. A mathematically justified method for solving the three-dimensional inverse scattering problem with fixed-energy data is described. This method is developed by A.G.Ramm for exact data and for noisy discrete data, and error estimates for this method are obtained. Difficulties of the numerical implementation of the inversion method based on the Dirichlet-to-Neumann map are pointed out and compared with the difficulty of the implementation of the Ramm's inversion method.

math.AP↗
arXiv2003

A characterization of unbounded Fredholm operators

This paper is a continuation of the paper (A.G.Ramm, Amer. Math. Monthly, 108, N 9, (2001), 855-860), where bounded Fredholm operators are studied. The theory of bounded linear Fredholm-type operators is presented in many texts. This paper is written for a broad audience and the author tries to give simple and short arguments.

math.FA↗
arXiv2003

On the discrepancy principle for the dynamical systems method

Assume that $$ Au=f,\quad (1) $$ is a solvable linear equation in a Hilbert space, $||A||<\infty$, and $R(A)$ is not closed, so problem (1) is ill-posed. Here $R(A)$ is the range of the linear operator $A$. A DSM (dynamical systems method) for solving (1), consists of solving the following Cauchy problem: $$ \dot u= -u +(B+\ep(t))^{-1}A^*f, \quad u(0)=u_0, \quad (2) $$ where $B:=A^*A$, $\dot u:=\frac {du}{dt}$, $u_0$ is arbitrary, and $\ep(t)>0$ is a continuously differentiable function, monotonically decaying to zero as $t\to \infty$. A.G.Ramm has proved that, for any $u_0$, problem (2) has a unique solution for all $t>0$, there exists $y:=w(\infty):=\lim_{t\to \infty}u(t)$, $Ay=f$, and $y$ is the unique minimal-norm solution to (1). If $f_\d$ is given, such that $||f-f_\d||\leq \d$, then $u_\d(t)$ is defined as the solution to (2) with $f$ replaced by $f_\d$. The stopping time is defined as a number $t_\d$such that $\lim_{\d \to 0}||u_\d (t_\d)-y||=0$, and $\lim_{\d \to 0}t_\d=\infty$. A discrepancy principle is proposed and proved in this paper. This principle yields $t_\d$ as the unique solution to the equation: $$ ||A(B+\ep(t))^{-1}A^*f_\d -f_\d||=\d, \quad (3) $$ where it is assumed that $||f_\d||>\d$ and $f_\d\perp N(A^*)$. For nonlinear monotone $A$ a discrepancy principle is formulated and justified.

math.DS↗
arXiv2003

Continuous modified Newton's-type method for nonlinear operator equations

A nonlinear operator equation $F(x)=0$, $F:H\to H,$ in a Hilbert space is considered. Continuous Newton's-type procedures based on a construction of a dynamical system with the trajectory starting at some initial point $x_0$ and becoming asymptotically close to a solution of $F(x)=0$ as $t\to +\infty$ are discussed. Well-posed and ill-posed problems are investigated.

math.NA↗
arXiv2003

Dynamical systems method for solving operator equations

Consider an operator equation $F(u)=0$ in a real Hilbert space. The problem of solving this equation is ill-posed if the operator $F'(u)$ 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.

math.DS↗
arXiv2003

Modified Rayleigh Conjecture and applications

Modified Rayleigh conjecture (MRC) in scattering theory is proposed and justified. MRC allows one to develop numerical algorithms for solving direct scattering problems related to acoustic wave scattering by soft and hard obstacles of arbitrary shapes. It gives an error estimate for solving the direct scattering problem. It suggests a numerical method for finding the shape of a starshaped obstacle from the scattering data.

math-ph↗