Related papers: Self-similar renormalization for nonlinear problem…
This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a…
The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton-Maclaurin expansion. Several basic…
In statistical and nonlinear systems, two qualitatively distinct parameter regions are typically identified: the regular region, characterized by smooth behavior of key quantities, and the critical region, where these quantities exhibit…
This tutorial serves as an introduction to recently developed non-asymptotic methods in the theory of -- mainly linear -- system identification. We emphasize tools we deem particularly useful for a range of problems in this domain, such as…
Self-normalized processes are basic to many probabilistic and statistical studies. They arise naturally in the the study of stochastic integrals, martingale inequalities and limit theorems, likelihood-based methods in hypothesis testing and…
A method is suggested for treating the well-known deficiency in the use of Pade approximants that are well suited for approximating rational functions, but confront problems in approximating irrational functions. We develop the approach of…
Power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasi-periodic solutions the issue of convergence of the series is plagued of the…
Exact renormalization group techniques are applied to mass deformed N=4 supersymmetric Yang-Mills theory, viewed as a regularised N=2 model. The solution of the flow equation, in the local potential approximation, reproduces the one-loop…
We describe a strategy for solving nonlinear eigenproblems numerically. Our approach is based on the approximation of a vector-valued function, defined as solution of a non-homogeneous version of the eigenproblem. This approximation step is…
Approximately 10 years ago, the method of renormalization-group symmetries entered the field of boundary value problems of classical mathematical physics, stemming from the concepts of functional self-similarity and of the Bogoliubov…
The result from developing and applying the notions of functional self-similarity and the Bogoliubov renormalization group to boundary-value problems in mathematical physics during the last decade are reviewed. The main achievement is the…
We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal…
We consider the problem of finding approximate analytical solutions for nonlinear equations typical of physics applications. The emphasis is on the modification of the method of Pad\'e approximants that are known to provide the best…
Two different models exhibiting self-organized criticality are analyzed by means of the dynamic renormalization group. Although the two models differ by their behavior under a parity transformation of the order parameter, it is shown that…
Recently, the stochastic asymptotical regularization (SAR) has been developed in (\emph{Inverse Problems}, 39: 015007, 2023) for the uncertainty quantification of the stable approximate solution of linear ill-posed inverse problems. In this…
Some renormalization group approaches have been proposed during the last few years which are close in spirit to the Nightingale phenomenological procedure. In essence, by exploiting the finite size scaling hypothesis, the approximate…
A recently developed variational resummation technique, incorporating renormalization group properties consistently, has been shown to solve the scale dependence problem that plagues the evaluation of thermodynamical quantities, e.g.,…
A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series…
Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…
Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by…