On deconvolution problems: numerical aspects

An optimal algorithm is described for solving the deconvolution problem of the form ${\bf k}u:=\int_0^tk(t-s)u(s)ds=f(t)$ given the noisy data $f_\delta$, $||f-f_\delta||\leq \delta.$ The idea of the method consists of the representation ${\bf k}=A(I+S)$, where $S$ is a compact operator, $I+S$ is injective, $I$ is the identity operator, $A$ is not boundedly invertible, and an optimal regularizer is constructed for $A$. The optimal regularizer is constructed using the results of the paper MR 40#5130.Comment: 7 figure

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.Comment: 21p

Reconstructing dynamical networks via feature ranking

Empirical data on real complex systems are becoming increasingly available. Parallel to this is the need for new methods of reconstructing (inferring) the topology of networks from time-resolved observations of their node-dynamics. The methods based on physical insights often rely on strong assumptions about the properties and dynamics of the scrutinized network. Here, we use the insights from machine learning to design a new method of network reconstruction that essentially makes no such assumptions. Specifically, we interpret the available trajectories (data) as features, and use two independent feature ranking approaches -- Random forest and RReliefF -- to rank the importance of each node for predicting the value of each other node, which yields the reconstructed adjacency matrix. We show that our method is fairly robust to coupling strength, system size, trajectory length and noise. We also find that the reconstruction quality strongly depends on the dynamical regime

An FFT method for the numerical differentiation

We consider the numerical differentiation of a function tabulated at equidistant points. The proposed method is based on the Fast Fourier Transform (FFT) and the singular value expansion of a proper Volterra integral operator that reformulates the derivative operator. We provide the convergence analysis of the proposed method and the results of a numerical experiment conducted for comparing the proposed method performance with that of the Neville Algorithm implemented in the NAG library