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N Body Problems using Message Parallel Approach

Given by Geoffrey C. Fox at CPS615 Basic Simulation Track for Computational Science on Fall Semester 97. Foils prepared 20 October 1997

This uses the simple O(N2) Particle Dynamics Problem as a motivator to discuss solution of ordinary differential equations
We discuss Euler, Runge Kutta and predictor corrector methods
Various Message parallel O(N2) algorithms are described with performance comments
There is a related data parallel module sharing the same initial foils

This mixed presentation uses parts of the following base foilsets which can also be looked at on their own!
CPS615 Foils -- set E: ODE's and Particle Dynamics
Overview of Fortran 90 and HPF Fall 96

Table of Contents for N Body Problems using Message Parallel Approach

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CPS 615 Lectures 1998 Fall Semester -- Message Parallel N Body
1 CPS 615 -- Computational Science in
Simulation Track
Message Parallel Module on ODE's and Particle Dynamics
October 20, 1997
2 Abstract of Message Parallel ODE and Particle Dynamics
Particle Dynamics as applications
3 Particle (N-Body) Applications and Ordinary Differential Equations (ODE's)
4 Particle Applications - Ordinary Differential Equations (ODE's)
5 Particle Applications - the N-body problem
6 Newton's First Law -- The Gravitational Force on a Particle
7 Equations of Motion -- Newton's Second Law
ODE Solution Techniques
8 Numerical techniques for solving ODE's
9 Second and Higher Order Equations
10 Basic Discretization of Single First Order Equation
11 Errors in numerical approximations
12 Runge-Kutta Methods: Euler's method
13 Estimate of Error in Euler's method
14 Relationship of Error to Computation
15 Example using Euler's method from the CSEP book
16 Approximate solutions at t=1,using Euler's method with different values of h
17 Runge-Kutta Methods: Modified Euler's method
18 Approximate solutions of the ODE for et at t=1, using modified Euler's method with different values of h
19 The Classical Runge-Kutta -- In Words
20 The Classical Runge-Kutta -- Formally
21 The Classical Runge-Kutta Pictorially
22 Predictor / Corrector Methods
23 Definition of Multi-step methods
24 Features of Multi-Step Methods
25 Comparison of Explicit and Implicit Methods
Setup of N Body Problem
26 Solving the N-body equations of motion
27 Representing the Message Parallel N-Body problem
28 Form of the Computation -- Message Parallel
29 Summary of Parallel N-Body Programming Methods and Algorithms
30 Status of Parallelism in Various N Body Cases
31 Other N-Body Like Problems - I
32 Other N-Body Like Problems - II
Message Parallel Approach
33 Essential Structure of Message Parallel O(N2) Algorithm - I
34 Essential Structure of Message Parallel O(N2) Algorithm - II
35 Structure of Runge Kutta Phases
36 The 9 Fortran Phases in Runge Kutta Update
37 Features of Message Parallel Computation
38 Message Parallel Force Computation MPGrav
3 Message Parallel Methods
39 Very Bad Naive Message Parallel Algorithm
40 Features of Very Bad Naive Message Parallel Algorithm
41 Much Better Message Parallel Algorithm
42 Blocking of Messages
43 Compilers, Caches and Data Locality
44 Communication and Computation Complexity
One Dimensional System Dimension
45 N-body Problem is a one dimensional Algorithm
46 Factor of Two in the Parallel O(N2) Algorithm
47 Best Message Parallel N Body Algorithm - I
48 Best Message Parallel N Body Algorithm - II
Decomposition Approaches Block Cyclic Scattered
49 Choice of Place to Compute Fi,j
50 The Two Basic Distributions in HPF
51 The Example of Matrix Inversion
52 Example of Graphics Rendering
Pipeline Approach
53 Final Remarks on Best Algorithm
54 Better Data Parallel Pipeline Algorithm for Computation of Accelerations,
taking 1/2 the time for iterations over force computation
55 Pipeline Algorithm in detail
56 Basic Message Parallel pipeline operation
57 Examples of Message Parallel Pipeline Algorithm
References
58 Notes and References
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