Low-rank Properties, Tree Structure, and Recursive Algorithms with Applications. Jingfang Huang Department of Mathematics UNC at Chapel Hill

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1 Low-rank Properties, Tree Structure, and Recursive Algorithms with Applications Jingfang Huang Department of Mathematics UNC at Chapel Hill

2 Fundamentals of Fast Multipole (type) Method Fundamentals: Low rank? Low dimensional? Simple rules? Process the low rank/ low dimensional or simple data on the tree structure? Recursive Implementations and parallelization

3 Example: (real parallel code) RECFMM Joint work with Bo Zhang, Xiaobai Sun, and Nikos Pitsianis Open source code available. Cilk Plus runtime used.

structure based algorithms. E.g., Cilk plus runtime scheduler.

4 Parallelizing Recursive Algorithms: Cilk Plus Runtime Scheduler Existing results on parallelizing hierarchy tree structure based algorithms. E.g., Cilk plus runtime scheduler. (Joint work with B. Zhang, X. Sun, N. Pitsianis.)

5 Computational Software section in CiCP 1. Computational Software: Simple FMM Libraries for Electrostatics, Slow Viscous Flow, and Frequency-Domain Wave Propagation, by Zydrunas Gimbutas and Leslie Greengard 2. Computational Software: PVFMM: A Parallel Kernel Independent FMM for Particle and Volume Potentials, by Dhairya Malhotra and George Biros Associate Editors: Prof. Benzhuo Lu (Chinese Academy of Sciences) and J. Huang.

6 L 2 Low rank: Mathematical Analysis vs Numerical Approach

7 Matrix Analysis: PCA or SVD

8 A simple Matlab code clear all; clc; m=400;n=400;x=rand(n); y=rand(m)+2; for i=1:m, for j=1:n, A(i,j)=1/(y(i)-x(j)); end end [U,S,V]=svd(A); format long e; pca=diag(s); pca(1:15)

392747101949028e-02 1.790209461350161e-03 3.

151573821662682e-08 1.982373081735641e-10 3.

9 And the results (numerical rank=9) e e e e e e e e e e e e-13

10 Mathematical analysis: Low rank? Taylor expansion:

11 Multipole coefficients: compressed data Note that P=15 for single precision and p=30 for double precision.

12 Low Rank and Low Dimensional Representations Separation of Variables: Low rank and low dimensional data

13 Question: Generalization?

14 Low Rank? Tree? Recursive Algorithm Apply the fundamentals to model and algorithm design. Today s topic 1: Recursive Tree Algorithms for Orthogonal Matrix Generation and Matrix-Vector Multiplications in Rigid Body Dynamics Simulations

15 Background: Rigid body dynamics Consider a molecular system modeled by rigid bodies, each rigid body consists of n beads Given the force on and location of each bead The resultant force (size 3) and torque (size 3)of the rigid body are given by

16 Matrix Language

17 Orthogonal Form

18 Problem Statement

19 Background: Brownian Dynamics with Hydrodynamic Interactions The bead model 1. Molecular system is modeled by m rigid bodies, each with given external forces and torques, j=1,,m. 2. Each rigid body contains n j beads. 3. The bead-to-bead hydrodynamics interactions satisfy where D is the Rotne-Prager-Yamakawa tensor (translation invariant Green s function, particle method for the Stokes equation) Reference: JA McCammon and G Huber (UCSD)

20 Bead model in Brownian dynamics For the given force field on each rigid body, the unknown rigid body velocity vectors satisfy: Or equivalently, Note: Need to solve this at each time marching step

21 Numerical Difficulty Numerical difficulty: calculation of Reminder: Question (1) (Alternative approaches: Schur complement)

22 Reformulation

Fast direct solver based preconditioner for each rigid body (Codes from Jianlin

23 Next Generation Brownian Dynamics Solver We already have 1. Fast multipole method for evaluating 2. Fast direct solver based preconditioner for each rigid body (Codes from Jianlin Xia, et. al. ) Building blocks needed: stable and efficient algorithms for evaluating A nice project for undergrad students (Fuhui Fang, UNC)

24 So the problems:

25 Divide-and-conquer: Recursive Algorithms on the Tree Structure Consider the divide-and-conquer strategy Observations: and

26 Divide-and-conquer: Recursive Algorithms on the Tree Structure

27 Algorithm Complexity and Storage How to go from Tree code to FMM?

28 An upward pass: Algorithm for computing Qv Info-set = multipole coefficients Multiple->Multipole: Parent P and Children X and Y, then,

29 Translation Operator: Inertia Matrix Parent s inertia matrix and Children s inertia matrices then,

30 Residue Vectors We want to compute without explicit forming

31 How to compute residue vector? Then,

32 Computing R from inertia matrix R is readily available from the inertia matrix

33 Recursive Algorithm

34 Downward Pass: Algorithm for Q T v Local Expansions?

35 A downward pass Recursive Algorithm

36 Pseudo-code

37 Numerical Results

38 Storage

39 Implicit Q v CPU time

40 Details: backward stable algorithm? How to make sure Q T Q v v is small? In progress

41 Summary of Topic 1: Use the fundamentals Low rank (comes from rigid body assumption) Tree structure Recursive implementation. Efficient algorithms can be developed.

42 Topic 2: Half-space impedance Green s function for the Helmholtz equation

43 Layered media Green s function On the efficient representation of the half-space impedance Green's function for the Helmholtz equation, by Michael O'Neil, Leslie Greengard, Andras Pataki

44 How to evaluate the convolution? Integral equation formulation where is the Green s function for the half-space with homogeneous impedance boundary conditions, and the resulting Fredholm 2 nd kind integral equation become

45 Method of images Using the free space Green s function

46 Modified fast multipole method Multipole-to-multipole: same as free space FMM, as if the images do NOT exist. Multipole-to-local: modified translation which include all the image contributions. Local-to-local: No change! Error analysis: Low rank? Note that source images are always well-separated from the target box. M2L Translation operator: analytical formula is possible. Either precomputed, or by designing special functions.

47 Generalization to multiple layers? Can we develop the corresponding approximation theory for general multi-layer Green s function in the form of Sommerfeld integral representation? Answer: Yes! Approximation using special basis, current research.

48 Summary (Maybe it is a good idea to) consider recursively (hierarchically) on the tree structure when designing models and algorithms Mathematical analysis may help us to find the special matrix structures.

49 Thanks!

A Kernel-independent Adaptive Fast Multipole Method

A Kernel-independent Adaptive Fast Multipole Method Lexing Ying Caltech Joint work with George Biros and Denis Zorin Problem Statement Given G an elliptic PDE kernel, e.g. {x i } points in {φ i } charges