FMM for Electromagnetic & Potential Simulations: Error Bounds & Evaluation, Study Guides, Projects, Research of Computer Science

Download FMM for Electromagnetic & Potential Simulations: Error Bounds & Evaluation and more Study Guides, Projects, Research Computer Science in PDF only on Docsity! 1 CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 CMSC 878R/AMSC 698R Lecture 17 CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Projects Shreyas Ananthan Fast Multipole Methods for 2-D Coulombic Potentials Animated demonstration of the FMM algorithm Non Uniform Fast Fourier Transforms Same (joint project) Fast Multipole Methods for Particle Simulations in Vortex Dominated Flows Fast particle methods for simulation and control of wakes from lifting surfaces' The Fast Multipole Method for Diffuse Light Scattering Simulation of ferromagnetic particle chain formation in a fluid medium under the action of a magnetic field FMM for solving an eigenvalue problem for Laplace’s equation Weigong Zhang Yang Wang Zhiyun Li Sandeep Gupta Karthikeyan Duraisamy Alap Karapurkar Shaju John Zhenyu Zhang CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Preliminary report due in two weeks • Dimensionality of the problem • A paragraph about applications where the problem arises • Mother function Ф(x, y) • R expansion • S expansion • S|R, S|S, R|R translations and operators • Truncation error bound • What are your overall goals. • References • Plan for the project CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Review of MLFMM CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Prepare Data Structures • Convert data set into integers given some maximum number of bits allowed/dimensionality of space • Interleave • Sort • Go through the list and check at what bit position two strings differ – For a given s determine the number of levels of subdivision needed • Digression (what is the algorithm to determine the number of levels)? CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Hierarchical Spatial Domains Ε1 Ε3 Ε4 Ε2 2 CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 UPWARD PASS • Partition sources into a source hierarchy. • Stop hierarchy so that boxes at the finest level contain s sources • Let the number of levels be L • Consider the finest level • For non-empty boxes we create S expansion about center of the box Φ(xi,y)=∑P uiB(x*,xi) S(x*,y) • We need to keep these coefficients. C(n,l) for each level as we will need it in the downward pass • Then use S/S translations to go up level by level up to level 2. • Cannot go to level 1 (Why?) CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 UPWARD PASS • At the end of the upward pass we have a set of S expansions (i.e. we have coefficients for them) • we have a set of coefficients C(n,l) for n=1,…,2ld l=L,…,2 • Each of these expansions is about a center, and is valid in some domain • We would like to use the coarsest expansions in the downward pass (have to deal with fewest numbers of coefficients) • But may not be able to --- because of domain of validity CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 • S expansion is valid in the domain E_3 outside domain E_1 (provided d<9) Ε 1 Ε3 xi xc(n,L) y CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 DOWNWARD PASS • Starting from level 2, build an R expansion in boxes where R expansion is valid • Must to do S|R translation • The S expansion is not valid in boxes immediately surrounding the current box • So we must exclude boxes in the E4 neighborhood Ε 4 CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Downward Pass. Step 1. Level 2: Level 3: CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Downward Pass. Step 1. THIS MIGHT BE THE MOST EXPENSIVE STEP OF THE ALGORITHM 5 CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 A scheme for error evaluation (4) Consider computation of the final coefficients with p-truncated matrices These truncation operators can be skipped! (Pr2=Pr) So: CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 A scheme for error evaluation (5) The error comes only from truncation operator CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Truncated Translation Theorem CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Proof CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 A scheme for error evaluation (5) For uniformly and absolutely convergent series: CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 A scheme for error evaluation (6) For uniformly and absolutely convergent series it is possible to find such εmax(p) that for given minimum(maximum) translation distance the max abs difference between two subsequent functions is smaller than εmax(p). In this case the total error of FMM does not exceed: 6 CMSC878R/AMSC698R Fall 2003 © Gumerov & Duraiswami, 2002 - 2003 Different schemes for error estimate are possible.