FMM CMSC 878R/AMSC 698R Lecture 20 Nail Gumerov & Ramani DuraiswamiAdaptive Multilevel FMM Some History: • J. Carrier, L. Greengard & V. Rokhlin, A fast adaptive multipole algorithm for particle simulations. SIAM J. Sci. Statist. Comput. 9, 669, 1988. • L. Van Dommelen & E.A. Rundensteiner, Fast, adaptive summation of point sources in the two-dimensional Poisson equation. J. Comp. Physics, 83, 126-147, 1989. • H. Cheng, L. Greengard & V. Rokhlin, A fast adaptive multipole algorithm in three dimensions. Here we present a new version and interpretation of AMLFMMWhat Does “Adaptive” Mean? • The number of operations depends on the point data. • Skipping boxes that do not contain sources and evaluation points in FMM is the first step of adaptation • A fully adaptive method should at run time decide the the computational process depending on the distribution of the Source and Evaluation Points. • The regular multilevel FMM uses pyramid data structure, while for the adaptive method we use tree (and/or forest) data structures.General Idea 1). Potential at any level of hierarchical space subdivision can be computed as Contribution of neighborhood Contribution of all other sources 2).General Idea (2) 5). 4). 3).General Idea (3)Hierarchical Spatial Domains Creating Data Structure. Step 0. Generate the same data structure as for regular MLFMM.Setting Data Structure. Step 1. Determine the Set of Target Boxes. sourceExample of Determination of Target Boxes 2222222233333422222222333334Evaluation Point Source Point Box level number In this example target boxes are shaded. Each of them contains not more than 3 source points in the neighborhood.Setting Data Structure. Step 2. Build D-tree. Consider Step 2 of the MLFMM Downward Pass. To compute D for a box we need D for the Parent of this boxSetting Data Structure. Step 2 (2). Build D-tree. AlgorithmSetting Data Structure. Step 2 (3). Build D-tree. Example D-tree Target BoxesSetting Data Structure. Step 2 (4). Build D-tree=D-tree. Consider again Step 2 of the MLFMM Downward Pass. To compute D for a box we need D for the Same box ~ ~Setting Data Structure. Step 3 (1). Build C-set. Consider Step 1 of the MLFMM Downward Pass. To compute D for a box we need C for the I4 neighborhood ~Setting Data Structure. Step 3 (2). Build C-set.Setting Data Structure. Step 4 (1). Build C-forest.Setting Data Structure. Step 4 (2). Build C-forest. AlgorithmSetting Data Structure. Step 4 (3). Build C-forest. C-setD-tree˜C-forestC-setD-tree˜D-tree˜C-forestUpward Pass (1) • For each Tree in the forest can be computed independently; • Skipping unnecessary translations, computing all leaves directly; • Skipping unnecessary translations when going Upward.Upward Pass (2)Upward Pass (3) 0123012 30123012 3Downward PassFinal SummationComparison of the Regular and Adaptive FMM. Sources (blue) distributed uniformly over the sphere. Targets (red) distributed uniformly inside a larger sphere.Comparison of the Regular and Adaptive FMM. Gaussian clusters of sources (blue) distributed uniformly in side a cube. Targets (red) distributed uniformly inside the same cube.Adaptivity• Parameter s is determined by the box with most particles. – Most boxes will be empty (ok) – or have much fewer than s particles • Reducing number of levels is good • Alternate strategy – Continue subdivision so that each parent box has > ℓ points and each leaf-box has ≤ ℓ points • Consider an algorithm due to Cheng et al (J. Comput. Phys. 1999).• At the finest level consider evaluation at box b. • It is surrounded by boxes – Coarser, Same or Finer level – Already considered, in a coarser S|R translation • Divide surrounding boxes not already considered in to L1, L2, L3, L4 domains – Boxes L1 share a boundary with box b, or they lie within the sphere of box b and need to be evaluated directly. – Boxes L2 are separated by at least one box of size of b S|R translate these – Boxes in L3 contain sources that lie outside the sphere of b and are closer than boxes L2 R-preFMM these – Box b lies outside the sphere of boxes in L4. but is too close to S|R. S-preFMM these© Gumerov & Duraiswami, 2002 - 2004 x* xi R S R* r* r* < R* S-expansion: |y - x*| > R* > |xi - x*| R-expansion: |y - x*| < r* < |xi - x*| For xi in L3 we can build R expansions in bAlgorithm parameters & optimization • Suggest that P and l should be balanced but do not provide an optimal choice • A good project – to implement this and find out optimal choices. – compare this scheme with our scheme and check efficiency – Perhaps develop a combined