Administrivia: May 20, 2013 Course project progress reports due Wednesday. Reading in Multigrid Tutorial: Chapters 3-4: Multigrid cycles and implementation.

Slides:

Advertisements

Similar presentations

Mutigrid Methods for Solving Differential Equations Ferien Akademie 05 – Veselin Dikov.

Advertisements

Example 1 Matrix Solution of Linear Systems Chapter 7.2 Use matrix row operations to solve the system of equations  2009 PBLPathways.

A Discrete Adjoint-Based Approach for Optimization Problems on 3D Unstructured Meshes Dimitri J. Mavriplis Department of Mechanical Engineering University.

CS 240A: Solving Ax = b in parallel Dense A: Gaussian elimination with partial pivoting (LU) Same flavor as matrix * matrix, but more complicated Sparse.

1 Iterative Solvers for Linear Systems of Equations Presented by: Kaveh Rahnema Supervisor: Dr. Stefan Zimmer

1 Numerical Solvers for BVPs By Dong Xu State Key Lab of CAD&CG, ZJU.

CS 290H 7 November Introduction to multigrid methods

SOLVING THE DISCRETE POISSON EQUATION USING MULTIGRID ROY SROR ELIRAN COHEN.

An Efficient Multigrid Solver for (Evolving) Poisson Systems on Meshes Misha Kazhdan Johns Hopkins University.

CSCI-455/552 Introduction to High Performance Computing Lecture 26.

MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute of Science UCLA

Geometric (Classical) MultiGrid. Hierarchy of graphs Apply grids in all scales: 2x2, 4x4, …, n 1/2 xn 1/2 Coarsening Interpolate and relax Solve the large.

Numerical Algorithms ITCS 4/5145 Parallel Computing UNC-Charlotte, B. Wilkinson, 2009.

Image Reconstruction Group 6 Zoran Golic. Overview Problem Multigrid-Algorithm Results Aspects worth mentioning.

Ma221 - Multigrid DemmelFall 2004 Ma 221 – Fall 2004 Multigrid Overview James Demmel

Algebraic MultiGrid. Algebraic MultiGrid – AMG (Brandt 1982)  General structure  Choose a subset of variables: the C-points such that every variable.

Numerical Algorithms • Matrix multiplication

Influence of (pointwise) Gauss-Seidel relaxation on the error Poisson equation, uniform grid Error of initial guess Error after 5 relaxation Error after.

CS267 Poisson 2.1 Demmel Fall 2002 CS 267 Applications of Parallel Computers Solving Linear Systems arising from PDEs - II James Demmel

CS267 L12 Sources of Parallelism(3).1 Demmel Sp 1999 CS 267 Applications of Parallel Computers Lecture 12: Sources of Parallelism and Locality (Part 3)

Sparse Matrix Methods Day 1: Overview Day 2: Direct methods

03/09/06CS267 Lecture 16 CS 267: Applications of Parallel Computers Solving Linear Systems arising from PDEs - II James Demmel

The Landscape of Ax=b Solvers Direct A = LU Iterative y’ = Ay Non- symmetric Symmetric positive definite More RobustLess Storage (if sparse) More Robust.

Geometric (Classical) MultiGrid. Linear scalar elliptic PDE (Brandt ~1971)  1 dimension Poisson equation  Discretize the continuum x0x0 x1x1 x2x2 xixi.

CS267 L25 Solving PDEs II.1 Demmel Sp 1999 CS 267 Applications of Parallel Computers Lecture 25: Solving Linear Systems arising from PDEs - II James Demmel.

CS240A: Conjugate Gradients and the Model Problem.

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 10 Further Topics in Algebra.

Support-Graph Preconditioning John R. Gilbert MIT and U. C. Santa Barbara coauthors: Marshall Bern, Bruce Hendrickson, Nhat Nguyen, Sivan Toledo authors.

1 Numerical Integration of Partial Differential Equations (PDEs)

1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 11 Further Topics in Algebra.

Complexity of direct methods n 1/2 n 1/3 2D3D Space (fill): O(n log n)O(n 4/3 ) Time (flops): O(n 3/2 )O(n 2 ) Time and space to solve any problem on any.

Algebra I Chapter 10 Review

Solving Exponential and Logarithmic Equations Section 6.6 beginning on page 334.

1.3 The Intersection Point of Lines System of Equation A system of two equations in two variables looks like: – Notice, these are both lines. Linear Systems.

Introduction to Scientific Computing II From Gaussian Elimination to Multigrid – A Recapitulation Dr. Miriam Mehl.

CS 219: Sparse Matrix Algorithms

Big Ideas in Mathematics Chapter Three

Chapter 21 Exact Differential Equation Chapter 2 Exact Differential Equation.

Boundary Value Problems l Up to this point we have solved differential equations that have all of their initial conditions specified. l There is another.

Multigrid Computation for Variational Image Segmentation Problems: Multigrid approach  Rosa Maria Spitaleri Istituto per le Applicazioni del Calcolo-CNR.

MECH4450 Introduction to Finite Element Methods Chapter 9 Advanced Topics II - Nonlinear Problems Error and Convergence.

Introduction to Scientific Computing II Overview Michael Bader.

Introduction to Scientific Computing II Multigrid Dr. Miriam Mehl Institut für Informatik Scientific Computing In Computer Science.

Introduction to Scientific Computing II Multigrid Dr. Miriam Mehl.

Lecture 21 MA471 Fall 03. Recall Jacobi Smoothing We recall that the relaxed Jacobi scheme: Smooths out the highest frequency modes fastest.

Discretization for PDEs Chunfang Chen,Danny Thorne Adam Zornes, Deng Li CS 521 Feb., 9,2006.

MULTISCALE COMPUTATIONAL METHODS Achi Brandt The Weizmann Institute of Science UCLA

CS 290H Administrivia: May 14, 2008 Course project progress reports due next Wed 21 May. Reading in Saad (second edition): Sections

Partial Derivatives Example: Find If solution: Partial Derivatives Example: Find If solution: gradient grad(u) = gradient.

CS 290H 31 October and 2 November Support graph preconditioners Final projects: Read and present two related papers on a topic not covered in class Or,

MECH593 Introduction to Finite Element Methods

Chapter 8 Systems of Linear Equations in Two Variables Section 8.3.

Algebra Review. Systems of Equations Review: Substitution Linear Combination 2 Methods to Solve:

Conjugate gradient iteration One matrix-vector multiplication per iteration Two vector dot products per iteration Four n-vectors of working storage x 0.

The Landscape of Sparse Ax=b Solvers Direct A = LU Iterative y’ = Ay Non- symmetric Symmetric positive definite More RobustLess Storage More Robust More.

Xing Cai University of Oslo

Model Problem: Solving Poisson’s equation for temperature

CS 290N / 219: Sparse Matrix Algorithms

بسم الله الرحمن الرحيم.

CS 290H Administrivia: April 16, 2008

Parallel Algorithm Design using Spectral Graph Theory

Solving Linear Systems Algebraically

Numerical Algorithms • Parallelizing matrix multiplication

Linear Algebra Lecture 3.

Chapter Three: Expressions and Equations

Finite Difference Method for Poisson Equation

Administrivia: November 9, 2009

Ph.D. Thesis Numerical Solution of PDEs and Their Object-oriented Parallel Implementations Xing Cai October 26, 1998.

Solving a System of Linear Equations

Presentation transcript:

Administrivia: May 20, 2013 Course project progress reports due Wednesday. Reading in Multigrid Tutorial: Chapters 3-4: Multigrid cycles and implementation Chapter 5: Multigrid convergence (next time) Chapter 8: Algebraic multigrid (optional)

Complexity of linear solvers 2D3D Sparse Cholesky:O(n 1.5 )O(n 2 ) CG, exact arithmetic: O(n 2 ) CG, no precond: O(n 1.5 )O(n 1.33 ) CG, modified IC: O(n 1.25 )O(n 1.17 ) CG, support trees: O(n 1.20 )O(n 1.31 ) CG, Spielman/Teng: O(n 1+ε ) Multigrid:O(n) n 1/2 n 1/3 Time to solve model problem (Poisson’s equation) on regular mesh

Complexity of direct methods n 1/2 n 1/3 2D3D Space (fill): O(n log n)O(n 4/3 ) Time (flops): O(n 3/2 )O(n 2 ) Time and space to solve any problem on any well- shaped finite element mesh

Multigrid Find an approximate solution on a coarser mesh, and then improve it on a finer mesh Reduce smooth fine-mesh error by using coarse mesh again Use idea recursively on hierarchy of meshes Solves the model problem in linear time! Often useful when hierarchy of meshes can be built This is just the intuition – lots of theory and technology

5 Irregular mesh: Tapered Tube (Multigrid)