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Variational method
About: Variational method is a research topic. Over the lifetime, 7154 publications have been published within this topic receiving 119573 citations.
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TL;DR: It is shown how the nonquadratic criterion to be minimized can be split into a sequence of half-quadratically problems easier to solve numerically.
Abstract: This paper is concerned with a classical denoising and deblurring problem in image recovery. Our approach is based on a variational method. By using the Legendre--Fenchel transform, we show how the nonquadratic criterion to be minimized can be split into a sequence of half-quadratic problems easier to solve numerically. First we prove an existence and uniqueness result, and then we describe the algorithm for computing the solution and we give a proof of convergence. Finally, we present some experimental results for synthetic and real images.
351 citations
TL;DR: The use of numerical solutions to atom-like single site potentials as a basis for molecular orbital calculations is investigated in this article, where the atomic Hamiltonian is modified by adding a potential well to induce additional discrete levels with the desired spatial characteristics.
Abstract: The use of numerical solutions to atomlike single site potentials as a basis for molecular orbital calculations is investigated. The atomic Hamiltonian is modified by addition of a potential well to induce additional discrete levels with the desired spatial characteristics. The discrete variational method is employed, in the Hartree‐Fock‐Slater model, to compare levels obtained for FeCl4 using multiple‐scattering, conventional Slater‐orbital, and numerical basis sets. The numerical technique is shown to be an accurate and efficient method for treating general (non‐muffin‐tin) molecular potentials. The errors in energy levels due to the muffin‐tin approximation are calculated.
348 citations
324 citations
TL;DR: In this article, a general theory of rates of convergence for the Rayleigh-Ritz variational method is given for the ground states of atoms and molecules, and the theory shows what functions should be added to the basis set to improve the rate of convergence.
Abstract: A general theory of rates of convergence for the Rayleigh–Ritz variational method is given for the ground states of atoms and molecules. The theory shows what functions should be added to the basis set to improve the rate of convergence, and gives explicit formulas for estimating corrections to variational energies and wave functions. An application of this general theory to a CI calculation on the ground state of the helium atom yields an explicit large L asymptotic formula for the ‘‘L’’‐limit energies E L . The increments are found to obey the formula E L −E L−1 =−3C 1(L+ 1/2 )− 4−4C 2(L+ 1/2 )− 5+O(L − 6), where the constants C 1 and C 2 are given by explicit integrals over the exact wave function evaluated at r 1 2=0. Numerical evaluation of these integrals yields 3C 1≅0.0741 and 4C 2≅0.0309, in excellent agreement with the empirical results 3C 1≅0.0740 and 4C 2≅0.031 found by Carroll, Silverstone, and Metzger.
323 citations
01 Jan 2000
TL;DR: This tutorial provides a few examples that highlight how to cast problems as optimization problems and how to find an approximate solution when the exact solution is not feasible.
Abstract: Tutorial topics • A bit of history • Examples of variational methods • A brief intro to graphical models • Variational mean field theory – Accuracy of variational mean field – Structured mean field theory • Variational methods in Bayesian estimation • Convex duality and variational factorization methods – Example: variational inference and the QMR-DT Variational methods • Classical setting: " finding the extremum of an integral involving a function and its derivatives " Example: finding the trajectory of a particle under external field • The key idea here is that the problem of interest is formulated as an optimization problem Variational methods cont'd • Variational methods have a long history in physics, statistics, control theory as well as economics. – calculus of variations (physics) – linear/non-linear moments problems (statistics) – dynamic programming (control theory) • Variational formulations appear naturally also in machine learning contexts: – regularization theory – maximum entropy estimation • Recently variational methods been used and further developed in the context of approximate inference and estimation Examples of variational methods • In classical examples the formulation itself is given but for us this is one of the key problems • We provide here a few examples that highlight 1. how to cast problems as optimization problems 2. how to find an approximate solution when the exact solution is not feasible • The examples we use involve a) finite element methods for solving differential equations b) large deviation methods (Chernoff bound)
317 citations