PPT-2.7.6 Conjugate Gradient Method for a Sparse System

Shi amp Bo What is sparse system A system of linear equations is called sparse if only a relatively small number of its matrix elements are nonzero It is wasteful to use general methods . This method in corporates a restarting scheme to automatical ly estimate the strong convexity parameter and achieves a nearly optimal iteration complexi ty Then we consider the regularized least squares LS problem in the highdimensional setting Alt This short note is on the derivation and convergence of a popular algorithm for minimization of quadratic functionals or solving linear systems known as the method of Conjugate Gradients CG To the best of the knowledge of the author of this short n . Siddharth. . Choudhary. What is Bundle Adjustment ?. Refines a visual reconstruction to produce jointly optimal 3D structure and viewing parameters. ‘bundle’ . refers to the bundle of light rays leaving each 3D feature and converging on each camera center. . The min and max of a function. Michael . Sedivy. Daniel . Eiland. Introduction. Given a function F(x), how do we determine the location of a local extreme (min or max value)?. Two standard methods exist :. to. Numerical Analysis . I. MATH/CMPSC 455. Conjugate Gradient Methods. A-Orthogonal Basis. . . form a basis of , where. is the . i-th. row of the identity matrix. They are orthogonal in the following sense:. Conjugate Gradient . 1) CG is a numerical method to solve a linear system of equations . 2) CG is used when A is Symmetric and Positive definite matrix (SPD). 3) CG of . Hestenes. and . :. Application to Compressed Sensing and . Other Inverse . Problems. M´ario. A. T. . Figueiredo. Robert . D. . Nowak. Stephen . J. Wright. Background. Previous Algorithms. Interior-point method. . CG Method. Non-linear CG. Solving Linear System of Equations. Preconditioned CG and Regularization. Outline. Kiss point. x. g. x*. dx. ’. dx _ . g’. Quasi-Newton Condition: . g. ’ . – . g. Conjugate . Gradient Method for a Sparse System. Shi & Bo. What is sparse system. A system of linear equations is called sparse if . only a relatively small . number of . its matrix . elements . . G.Anuradha. Review of previous lecture-. Steepest Descent. Choose the next step so that the function decreases:. For small changes in . x. we can approximate . F. (. x. ):. where. If we want the function to decrease:. :. Application to Compressed Sensing and . Other Inverse . Problems. M´ario. A. T. . Figueiredo. Robert . D. . Nowak. Stephen . J. Wright. Background. Previous Algorithms. Interior-point method. . Contents. Problem Statement. Motivation. Types . of . Algorithms. Sparse . Matrices. Methods to solve Sparse Matrices. Problem Statement. Problem Statement. The . solution . of . the linear system is the values of the unknown vector . Michael . Sedivy. Daniel . Eiland. Introduction. Given a function F(x), how do we determine the location of a local extreme (min or max value)?. Two standard methods exist :. F(x) with global minimum D and local minima B and F. First order methods For convex optimization J. Saketha Nath (IIT Bombay; Microsoft) Topics Part – I Optimal methods for unconstrained convex programs Smooth objective Non-smooth objective Part – II