Related papers: Analytical approximation schemes for solving exact…
We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…
A general procedure is presented how to improve the effective potential by using the renormalization group equation (RGE) in MS bar scheme. If one knows the L-loop effective potential and the RGE coefficient functions up to (L+1)-loop…
We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the…
Certain Petrov-Galerkin schemes are inherently stable formulations of variational problems on a given mesh. This stability is primarily obtained by computing an optimal test basis for a given approximation space. Furthermore, these…
This paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform…
Analytical solutions are presented for eigenvalues, eigenfunctions of {\color{red} D-dimensional Schrodinger equation having Eckart potential} within Nikiforov-Uvarov method. This uses a new, improved approximation for centrifugal term,…
We study the optimisation of exact renormalisation group (ERG) flows. We explain why the convergence of approximate solutions towards the physical theory is optimised by appropriate choices of the regularisation. We consider specific…
We study the derivative expansion for the effective action in the framework of the Exact Renormalization Group for a single component scalar theory. By truncating the expansion to the first two terms, the potential $U_k$ and the kinetic…
I discuss functional renormalization group (fRG) schemes, which allow for non-perturbative treatment of the self-energy effects and do not rely on the one-particle irreducible functional. In particular, I consider Polchinski or Wick-ordered…
In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a…
We introduce a numerical method for the approximation of functions which are analytic on compact intervals, except at the endpoints. This method is based on variable transforms using particular parametrized exponential and…
We develop approximate estimation methods for exponential random graph models (ERGMs), whose likelihood is proportional to an intractable normalizing constant. The usual approach approximates this constant with Monte Carlo simulations,…
Linear fixed point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random…
This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework…
The regularity of the solution of elliptic partial differential equa- tions in a polygonal domain with re-entrant corners is, in general, reduced compared to the one on a smooth convex domain. This results in a best approximation property…
We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for…
We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of convergence for spatially…
Following the previously developed approach to the calculation of quantum corrections to the effective potential in arbitrary scalar field theories in the leading logarithmic approximation, we extended it to the next-to-leading order. Based…
The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of…
A framework for performing dynamic mesh adaptation with the discontinuous Galerkin method (DGM) is presented. Adaptations include modifications of the local mesh step size (h-adaptation) and the local degree of the approximating polynomials…