arXiv subjects
Publications and source records attributed to A. G. Ramm.
Various versions of the Dynamical Systems Method (DSM) are proposed for solving linear ill-posed problems with bounded and unbounded operators. Convergence of the proposed methods is proved. Some new results concerning discrepancy principle for choosing regularization parameter are obtained.
Three open problems in analysis are formulated: 1. Injectivity of Radon transform 2. A uniqueness problem for an elliptic system of PDE 3. A problem in operator theory
Let $Ay=f$, $A$ is a linear operator in a Hilbert space $H$, $y\perp N(A):=\{u:Au=0\}$, $R(A):=\{h:h=Au,u\in D(A)\}$ is not closed, $\|f_δ-f\|\leqδ$. Given $f_δ$, one wants to construct $u_δ$ such that $\lim_{δ\to 0}\|u_δ-y\|=0$. A version of the DSM (dynamical systems method) for finding $u_δ$ consists of solving the problem \bee \dotu_δ(t)=-u_δ(t)+T^{-1}_{a(t)} A^\ast f_δ, \quad u(0)=u_0, \eqno{(\ast)}\eee where $T:=A^\ast A$, $T_a:=T+aI$, and $a=a(t)>0$, $a(t)\searrow 0$ as $t\to\infty$ is suitably chosen. It is proved that $u_δ:=u_δ(t_δ)$ has the property $\lim_{δ\to 0}\|u_δ-y\|=0$. Here the stopping time $t_δ$ is defined by the discrepancy principle: \bee \eqno{(\ast\ast)}\eee $c\in(1,2)$ is a constant. Equation $(\ast)$ defines $t_δ$ uniquely and $\lim_{δ\to 0}t_δ=\infty$. Another version of the discrepancy principle is also proved in this paper.
If $F:H\to H$ is a map in a Hilbert space $H$, $F\in C^2_{loc}$, and there exists $y$, such that $F(y)=0$, $F'(y)\not= 0$, then equation $F(u)=0$ can be solved by a DSM (dynamical systems method). This method yields also a convergent iterative method for finding $y$, and converges at the rate of a geometric series. It is not assumed that $y$ is the only solution to $F(u)=0$. Stable approximation to a solution of the equation $F(u)=f$ is constructed by a DSM when $f$ is unknown but $f_\d$ is known, where $||f_\d-f||\leq \d$.
An improvement of the author's result, proved in 1961, concerning necessary and sufficient conditions for the compactness of embedding operators is given. A counterexample to a published statement concerning compactness of embedding operators is constructed.
The Rayleigh conjecture about convergence up to the boundary of the series representing the scattered field in the exterior of an obstacle $D$ is widely used by engineers in applications. However this conjecture is false for some obstacles. AGR introduced the Modified Rayleigh Conjecture (MRC), which is an exact mathematical result. In this paper we review the theoretical basis for the MRC method for 2D and 3D obstacle scattering problems, for static problems, and for scattering by periodic structures. We also present successful numerical algorithms based on the MRC for various scattering problems. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. Various direct and inverse scattering problems require finding global minima of functions of several variables. The Stability Index Method (SIM) combines stochastic and deterministic method to accomplish such a minimization.
A standard way to solve linear algebraic systems $Au=f,\,\,(*)$ with ill-conditioned matrices $A$ is to use variational regularization. This leads to solving the equation $(A^*A+aI)u=A^*f_\d$, where $a$ is a regularization parameter, and $f_\d$ are noisy data, $||f-f_\d||\leq \d$. Numerically it requires to calculate products of matrices $A^*A$ and inversion of the matrix $A^*A+aI$ which is also ill-conditioned if $a>0$ is small. We propose a new method for solving (*) stably, given noisy data $f_\d$. This method, the DSM (Dynamical Systems Method) is developed in this paper for selfadjoint $A$. It consists in solving a Cauchy problem for systems of ordinary differential equations.
A convergent iterative process is constructed for solving any solvable linear equation in a Hilbert space.
Consider the equation $-\ve^2Δu_\ve+q(x)u_\ve=f(u_\ve)$ in $\R^3$, $|u(\infty)|<\infty$, $\ve=const>0$. Under what assumptions on $q(x)$ and $f(u)$ can one prove that the solution $u_\ve$ exists and $\lim_{\ve\to 0} u_\ve=u(x)$, where $u(x)$ solves the limiting problem $q(x)u=f(u)$? These are the questions discussed in the paper.
Let $A=A^*$ be a linear operator in a Hilbert space $H$. Assume that equation $Au=f \quad (1)$ is solvable, not necessarily uniquely, and $y$ is its minimal-norm solution. Assume that problem (1) is ill-posed. Let $f_\d$, $||f-f_d||\leq \d$, be noisy data, which are given, while $f$ is not known. Variational regularization of problem (1) leads to an equation $A^*Au+\a u=A^*f_\d$. Operation count for solving this equation is much higher, than for solving the equation $(A+ia)u=f_\d \quad (2)$. The first result is the theorem which says that if $a=a(\d)$, $\lim_{\d \to 0}a(\d)=0$ and $\lim_{\d \to 0}\frac \d {a(\d)}=0$, then the unique solution $u_\d$ to equation (2), with $a=a(\d),$ has the property $\lim_{\d \to 0}||u_\d-y||=0$. The second result is an iterative method for stable calculation of the values of unbounded operator on elements given with an error.
It is proved that the measurement of the acoustic pressure on the ear membrane allows one to determine the shape of the ear $ uniquely.
Let $u_t-u_{xx}=h(t)$ in $0\leq x \leq π, t\geq 0.$ Assume that $u(0,t)=v(t)$, $u(π,t)=0$, and $u(x,0)=g(t)$. The problem is: {\it what extra data determine the three unknown functions $\{h, v, g\}$ uniquely?}. This question is answered and an analytical method for recovery of the above three functions is proposed.
An improvement of the author's result, proved in 1961, concerning necessary and sufficient conditions for the compactness of an imbedding operator is given.
Formulas for stable differentiation of piecewise-smooth functions are given. The data are noisy values of these functions. The locations of discontinuity points and the sizes of the jumps across these points are not assumed known, but found stably from the noisy data.
Assume that $Au=f,\quad (1)$ is a solvable linear equation in a Hilbert space $H$, $A$ is a linear, closed, densely defined, unbounded operator in $H$, which is not boundedly invertible, so problem (1) is ill-posed. It is proved that the closure of the operator $(A^*A+\a I)^{-1}A^*$, with the domain $D(A^*)$, where $\a>0$ is a constant, is a linear bounded everywhere defined operator with norm $\leq 1$. This result is applied to the variational problem $F(u):= ||Au-f||^2+\a ||u||^2=min$, where $f$ is an arbitrary element of $H$, not necessarily belonging to the range of $A$. Variational regularization of problem (1) is constructed, and a discrepancy principle is proved.
A method is given for calculation of a distribution of small particles, embedded in a medium, so that the resulting medium would have a desired radiation pattern for the plane wave scattering by this medium.
withdrawn Several symmetry problems are discussed. These include the Pompeiu problem and similar conjectures for the heat and wave equations.