arXiv subjects
Publications and source records attributed to A. G. Ramm.
A new understanding of the notion of regularizer is proposed. It is argued that this new notion is more realistic than the old one and better fits the practical computational needs. An example of the regularizer in the new sense is given. A method for constructing regularizers in the new sense is proposed and justified.
Variational regularization and the quasisolutions method are justified for unbounded closed, possibly nonlinear, operators. The argument is quite simple and yields general results.
An inverse problem of identifying locations and certain properties of small dielectric inhomogeneities in a homogeneous background medium from boundary measurements on a part of the boundary is studied. Using as weights particular background solutions constructed by solving a minimization problem, an asymptotic (variational) method based on appropriate averaging of the partial boundary measurements is developed.
A novel numerical method for solving inverse scattering problem with fixed-energy data is proposed. The method contains a new important concept: the stability index of the inversion problem. This is a number, computed from the data, which shows how stable the inversion is. If this index is small, then the inversion provides a set of potentials which differ so little, that practically one can represent this set by one potential. If this index is larger than some threshold, then practically one concludes that with the given data the inversion is unstable and the potential cannot be identified uniquely from the data. Inversion of the fixed-energy phase shifts for several model potentials is considered. The results show practical efficiency of the proposed method. The method is of general nature and is applicable to a very wide variety of the inverse problems.
It is proved that the Newton-Sabatier (NS) procedure does not solve the inverse scattering problem with fixed-energy data and is not a valid inversion method, in the following sense: 1) the basic integral equation, introduced by R. Newton without derivation, in general, may be not solvable for some $r>0$, and in this case NS procedure breaks down: it produces a potential which is not locally integrable. 2) the ansatz $(\ast)$ $K(r,s) = \sum^\infty_{l=0} c_l ϕ_l (r) u_l (s)$, used by R. Newton, is incorrect: the transformation operator $I-K$, corresponding to a generic does not have $K$ of the form $(\ast),$ and 3) the set of potentials $q \in L_{1,1},$ that can possibly be obtained by NS procedure, is not dense in the set of all $L_{1,1}$ potentials in the norm of $L_{1,1}$. Therefore one cannot justify NS procedure even for approximate solution of the inverse scattering problem with fixed-energy phase shifts as data. Thus, the NS procedure, if considered as a method for solving the inverse scattering problem, is based on an incorrect ansatz, the basic integral equation of NS procedure is, in general, not solvable for some $r>0$, and in this case this procedure breaks down, and NS procedure is not an inversion theory: it cannot recover generic potentials $q \in L_{1,1}$ from their fixed-energy phase shifts. Suppose now that one considers another problem: given fixed-energy phase shifts, corresponding to some potential, find a potential which generates the same phase shifts. Then NS procedure does not solve this problem either: the basic integral equation, in general, may be not solvable for some $r>0$, and then NS procedure breaks down.
A counterexample is given to the uniqueness result given in the paper by J.Cox and K.Thompson, "Note on the uniqueness of the solution of an equation of interest in inverse scattering problem", J. Math.Phys., 11, N3, (1970), 815-817.
The inverse problem of finding the coefficient $\g$ in the equation $\dot{u}=A(t)u+\g(t)u+f(t)$ from the extra data of the form $ϕ(t)=u(t),w$ is studied. The problem is reduced to a Volterra equation of the second kind. Applications are given to parabolic equations with second order differential operators.
A version of the projection method for solving the scattering problem for acoustic and electromagnetic waves is proposed and shown to be more efficient numerically than the earlier ones because the corresponding matrix is not ill-conditioned.
If $f(x,k)$ is the Jost solution and $f(x) = f(0,k)$, then the $I$-function is $I(k) := \frac{f^\prime(0,k)}{f(k)}$. It is proved that $I(k)$ is in one-to-one correspondence with the scattering triple ${\mathcal S} :=\{S(k), k_j, s_j, \quad 1 \leq j \leq J\}$ and with the spectral function $ρ(λ)$ of the Sturm-Liouville operator $l= -\frac{d^2}{dx^2} + q(x)$ on $(0, \infty)$ with the Dirichlet condition at $x=0$ and $q(x) \in L_{1,1} := \{q: q= \bar q, \int^\infty_0 (1+x) |q(x) dx < \infty\}$. Analytical methods are given for finding $\mathcal S$ from $I(k)$ and $I(k)$ from $\mathcal S$, and $ρ(λ)$ from $I(k)$ and $I(k)$ from $ρ(λ)$. Since the methods for finding $q(x)$ from $\mathcal S$ or from $ρ(λ)$ are known, this yields the methods for finding $q(x)$ from $I(k)$.
Let $D\subset \R^n$, $n\geq 3,$ be a bounded domain with a $C^{\infty}$ boundary $S$, $L=-\nabla^2+q(x)$ be a selfadjoint operator defined in $H=L^2(D)$ by the Neumann boundary condition, $θ(x,y,λ)$ be its spectral function, $θ(x,y,λ):=\ds\sum_{λ_j<λ} ϕ_j(x)ϕ$ where $Lϕ_j=λ_jϕ_j$, $ϕ_{j N}|_S=0,$ $\|ϕ_j\|_{L^2(D)}=1$, $j=1,2,...$. The potential $q(x)$ is a real-valued function, $q\in C^\infty(D)$. It is proved that $q(x)$ is uniquely determined by the data $θ(s,s,λ) \forall s\in S$, $\forall λ\in \R_+$ if all the eigenvalues of $L$ are simple.
Let $u_t = u_{xx} - q(x) u, 0 \leq x \leq 1$, $t>0$, $u(0, t) = 0, u(1, t) = a(t), u(x,0) = 0$, where $a(t)$ is a given function vanishing for $t>T$, $a(t) \not\equiv 0$, $\int^T_0 a(t) dt < \infty$. Suppose one measures the flux $u_x (0,t) := b_0 (t)$ for all $t>0$. Does this information determine $q(x)$ uniquely? Do the measurements of the flux $u_x (1,t) := b(t)$ give more information about $q(x)$ than $b_0 (t)$ does? The above questions are answered in this paper.
Let $lu = -u^{\prime \prime} + q(x)u$, where $q(x)$ is a real-valued $L^2_{loc}(0, \infty)$ function. H. Weyl has proved in 1910 that for any $z$, $Imz \neq 0$, the equation $(l - z)w=0$, $x>0$, has a solution $w \in L^2(0, \infty)$. We prove this classical result using a new argument.
In this paper we review the results of the author on the theory of scalar and vector wave scattering by small bodies of an arbitrary shape with the emphasis on practical applicability of the formulas obtained and on the mathematical rigor of the theory. For the scalar wave scattering by a single body, our main results can be described as follows: (1) Analytical formulas for the scattering amplitude for a small body of an arbitrary shape are obtained; dependence of the scattering amplitude on the boundary conditions is described. (2) An analytical formula for the scattering matrix for electromagnetic wave scattering by a small body of an arbitrary shape is given. Applications of these results are outlined (calculation of the properties of a rarefied medium; inverse radio measurement problem; formulas for the polarization tensors and capacitance). (3) The multi-particle scattering problem is analyzed and interaction of the scattered waves is taken into account. For the self-consistent field in a medium consisting of many particles ~ 10^{23}, integral-differential equations are found. The equations depend on the boundary conditions on the particle surfaces. These equations offer a possibility of solving the inverse problem of finding the medium properties from the scattering data. For about 5 to 10 bodies the fundamental integral equations of the theory can be solved numerically to study the interaction between the bodies.
Let $A$ be a linear bounded operator in a Hilbert space $H$, $N(A)$ and $R(A)$ its null-space and range, and $A^*$ its adjoint. The operator $A$ is called Fredholm iff $dim N(A)= dim N(A^*):=n<\infty$ and $R(A)$ and $R(A^*)$ are closed subspaces of $H$. A simple and short proof is given of the following known result: $A$ is Fredholm iff $A=B+F$, where $B$ is an isomorphism and $F$ is a finite-rank operator. The proof consists in reduction to a finite-dimensional linear algebraic system which is equivalent to the equation $Au=f$ in the case of Fredholm operators.
A nonlinear equation in a Banach space is written as a linear equation with a linear operator depending on the unknown solution. This method, which we call a global linearization method, differs essentially from the local linearization methods of the Newton-type. Inverting the above linear operator by the methods known for linear operators one gets an equation which sometimes is much better for numerical solution than the original one. Some theorems about convergence of the proposed iterative process for solving the transformed equation are given. Examples of applications are considered.
The inverse problem of finding the refraction coefficient of a shallow ocean from the observation of the extra Cauchy data for the acoustic field at the surface of the ocean is studied. Uniqueness theorem is proved and a reconstruction algorithm is obtained. Comments are given on the related results in the literature.
It is proved that if the scattering amplitudes at a fixed wavenumber for two obstacles from a certain large class of obstacles differ a little, than the obstacles differ a little. Error estimate is given. It is proved that there is an inversion formula for finding the obstacle from a fixed-energy scattering data.
The inverse problem of finding the velocity profile from the surface measurements of acoustic field is considered. The dimensionalities of the data and of the unknown velocity profile are equal, both are equal 3. Analytical explicit examples are given showing vast non-uniqueness of the solution to this inverse problem.