A Scalable Tridiagonal Solver For GPUs

Team:WenMin Xiao&ChaoQun Li

Institute of information science and technology of Hunan University

Outline

Introduction Algorithms

Classic algorithms Design algorithm for gpu architecture

Performance Summary

What is a tridiagonal system?

What is it used for?

Scientific and engineering application Numerical ocean models Semi-coarsening for multi-grid solvers Spectral Poisson Solvers Cubic Spline Approximation

Video and computer-animated films Depth of field blurs Fluid simulation

Two Applications on GPU

Depth of field blur, Michael Kass et al. Shallow water simulation

OpenGL and Shader language CUDA

Cyclic reduction Cyclic reduction

2006 2007

A Classic Serial Algorithm

Gussian elimination in tridiagonal case(Thomas algorithm)

Phase 1:Forword Reduction

Phase 2:Backward Substitution

Elimination steps?

Complexity?

2n-1

O(n)=2(n-1)+1

Parallel Algorithms Coarse-grained algorithms(multi-core CPU)

Two-way Gussian elimination Sub-structing method

Fine-grained algorithms (many-core GPU) Cyclic Reduction (CR) Parallel Cyclic Reduction (PCR) Recursive Doubling (RD) Hybrid Thomas-PCR algorithm

A set of equations

mapped to one thread

A single equation mapped to one thread

Cyclic Reduction2-4 threads working

Forward Reduction

Backward

Substitution

8-unkown system

4-unkown system

2-unkown system

Solve 2 unkowns

Solve the rest 2 unkowns

Solve the rest 4 unkonws

2*log2(8)-1 = 2*3 -1 = 5 steps

Parallel Cyclic Reduction(PCR)

Forward

Redution

No Backward

Substitution

One 8-unkown system

Two 4-unkown systems

Four 2-unkown systems

Solve all unkowns

4 threads working

log2(8)=3 steps

Advantages of Previous Algorithms

Thoms It is easy to parallel solve the

independent system CR

Every step we reduce the system by half PCR

Fewer steps required

Hybird Algorithm

Improved PCR(Tiled PCR) P-Thomas

One 8-unkown system

One PCR step

Parallel Thomas

GPU Implementation•Linear systems mapped to

multiprocessors (blocks)

•Equations mapped to

processors (threads)

Tiled PCR A variant of incomplete PCR in a sense that it stops breaking

down the system before the algorithm reaches the smallest possible matrics

Redundancy of naive tiling of PCR

Redundancy

Anthor type of Redundancy

K-step PCR: the number of redundant memory access per tile boundary

∑−

=

=1

0

2)(k

i

ikf

Dependency & ParallelismHow to Reduce Redundancy?

A set of elements being an object of PCR operation

Cached results

Solution 1

Redundancy is also exist!

Dependency & Parallelism contFine-grained tiling

A set of elements being an object of PCR operation

Cached results

Solution 2

Without redundancy

Sequential Computation

Cache DesignBuffered Sliding Window

Illustration of the buffered sliding window

1.Immedicate results are

cached2.Each tile are processed parallel

3.Each of tile has multiple

sub tiles

4.Sub tiles are processed

sequentially using cache

Components ofBuffered Sliding Window

Bottom bufferThe input element are just loaded from global memory and ready to be processed

Middle buffermostly interacts with the elements in the bottom by providing denpendency to them at the same time referring them

Top buffercaches elements from the middle buffer for the last step of PCR

Example

Advantages of TPCR

Fewer steps Better memory latency hiding(less

idle time of GPU execution) Minimizing redundant global memory

access Arbitrary tiling size

Thread-level Parallel Thomas Algorithm

Solves multiple independent system that each thread solves a different system

Global memory should be coalesced

64B aligned segment

128B aligned segment

Performance EvaluationTest-Platform

Nvidia GTX480 GPU with 1.5GB memory bandwith

3.33GHZ Intel quad-core i7 975 CPU Fedora 12 Linux CUDA 3.2

Performance ResultsParameter M and N: number of systems and system size

8.3x and 49x speedups

5x and 30x speedups

Performance Analysis

Factors that determine performance Size of intermeidate results cache Global/shared memory access Overhead for synchronization Bank conflick

Summary

We studied 3 kinds of algorithm for addressing tridiagonal solver

We learned Sophisticatedly designed tiling and the buffer sliding window in TPCR algorithm

Reference

http://en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm

Fast Tridiagonal Solvers on GPU CUDA Programming Guide 高性能运算之 CUDA

Question?