Scientific Computing on Graphical Processors: FMM, Flagon, Signal

Processing, Plasma

Ramani Duraiswami and Nail GumerovComputer Science & UMIACS

University of Maryland, College Park

Joint work with Yuancheng Luo, Adam O’Donovan, Bill Dorland and students of CMSC 828 E (Scientific Computing on Graphical Processors)www.umiacs.umd.edu/~ramani/cmsc828e_gpusci

FMM on the GPU N.A. Gumerov and R. Duraiswami, Fast multipole methods on

graphics processors. Journal of Computational Physics, 227, 8290-8313, 2008.

N-body problems - important in stellar dynamics, molecular modeling, etc.

Several papers implement quadratic algorithms on the GPU (but restricted to O(104) particles)

To go to O(106) and beyond we need the FMM Reduces quadratic complexity to linear order Complex algorithm which relies on a balance between

local interactions (brute force) and tree-based far field

Direct summation on GPU

Cost =A1 N+B1 N/s+C1 Ns.Compare GPU final summation complexity:

and total FMM complexity:Cost = AN+BN/s+CNs.

sopt = (B1 /C1 )1/2,Optimal cluster size for direct summation step of the FMM

This leads to

Cost =(A+A1 )N+(B+B1 )N/s+C1 Ns,and sopt = ((B+B1 )/C1 )1/2 .

FMM requires a balance between direct summation and the rest of the algorithm

Performance on a 8800 GTX

661.761 s116.1 sp=12721.227 s88.09 sp=8290.979 s28.37 sp=4

RatioGPUserial CPU

(potential only) N=1,048,576

481.395 s66.56 sp=12560.908 s51.17 sp=8330.683 s22.25 sp=4

RatioGPUserial CPU

(potential+forces (gradient)) N=1,048,576

Performance

Computations of the potential and forces:

Peak performance of GPU for direct summation 290 Gigaflops, while for the FMM on GPU effective rates in range 25-50 Teraflops are observed (following the citation below).

M.S. Warren, J.K. Salmon, D.J. Becker, M.P. Goda, T. Sterling & G.S. Winckelmans. “Pentium Pro inside: I. a treecode at 430 Gigaflops on ASCI Red,” Bell price winning paper at SC’97, 1997.

CPU

GPU

direct

dir

FMM

FMM

Performance

p=4 p=8 p=12

What is more accurate for solution of large problems on GPU: direct summation or FMM?

1.E-09

1.E-08

1.E-07

1.E-06

1.E-05

1.E-04

1.E-03

1.E-02

1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07Number of Sources

L2-r

elat

ive

erro

r

p=4

p=8

p=12

Direct

CPU

GPU

FMM

FMM

FMM

Filled = GPU, Empty = CPU

Error in computations of potential

Error computedover a grid of 729 sampling points, relative to “exact” solution, which is direct summation with double precision.

Possible reason why the GPU error in direct summation grows: systematic roundoff error in computation of function 1/sqrt(x).(still a question).

Flagon: Use GPUs via extensible libraries GPUs are great as we all have heard But require you to program in extended version of C Need NVIDIA toolchain What if you have an application that is

In Fortran 9x/2003, Matlab, C/C++ Too large to fit on the GPU and needs to use the CPU cores, MPI, etc. as part

of a larger application, but take advantage of GPU Offload computations which have good speedups on the GPU to it using

library calls in your programming environment Enter the FLAGON

An extensible open source library and a middleware framework that allows use of GPU

Implemented currently for Fortran-9X, and preliminarily for C++ and MATLAB

Programming on the GPU GPU organized as 2-30 groups of

multiprocessors (8 relatively slow processors) with small amount of own memory and fast access to common shared memory, and slow access to global memory

Factor of 100s difference in speed as one goes up the memory hierarchy

To achieve gains problems must fit the SPMD paradigm and manage memory

Research issues: Identifying important tasks and mapping them to the

architecture Making it convenient for programmers to call GPU code

from host code

Local memory~50kB

GPU global memory

~1GB

Host memory~2-32 GB

Approach to use GPU: Flagon Middleware

Defines Module/Class that provides pointers on CPU to Device Variables on the GPU

Execute small, well written, CU functions to perform primitive operations on device avoid data transfer overhead by Initially using pinned memory copies and pointers Subsequently transfer data to CPU only when necessary

Provide wrappers to BLAS, FFT, and other software (random number, sort, screen dump, etc.)

Allow incorporation of existing mechanisms for doing distributed programming (OpenMP, MPI, etc.) to handle clusters

Allow relatively easy conversion of existing code

Sample scientific computing applications Radial basis function fitting Plasma turbulence computations Fast Multipole Force calculation in particle systems Machine Learning Numerical Relativity Space Turbulence Signal Processing Integral Equations

FLAGON Device Variables User instantiates device

variables in Fortran Encapsulates parameters

and attributes of the data structure transferred between host and device

Tracks (via pointers) allocated memory on the device

Stores data attributes (type and dimensions) on the host and device

FLAGON Structure

devVar

Device Pointer

Device Dimensions

Device Leading

Dimensions

Device Status

Device Data Type

Pointer to device memory

address

Data type stored on device

Allocation status on device

X, Y, Z dimensions of vector

or matrix on host

XL, XY

Lleading dimensions

of vector or matrix on

device

FLAGON Work-Cycle Compiling and link library

to user Fortran code Load library into memory Allocate device variables

and copy host data to device Work-cycle allows

subsequent computations to be performed solely on the device

Data transfer from device to host when done

Discard/free data on the device

FLAGON Work Cycle

Load FLAGON Library

Allocate Device

Variable(s )

Memory Transfer

Host to Device

Work

Memory Transfer

Device to Host

Allocates and pads

memory on GPU Device

Transfer host data from

Fortran to CUDA global

memory

Call CUBLAS, CUFFT,

CUDPP, CUDA functions

and perform all

calculations on the GPU

Transfer data back from

device to host

Specify GPU device,

load CUBLAS library

FLAGON Functions Initialization functions

open_devObjects, close_devObjects Memory functions

Allocation/deallocation allocate_dv(chartype, nx, ny, nz) deallocate_dv(devVar)

Memory transfer transfer_[i, r, c]4(hostVar, devVar, c2g) transfer_[i, r, c] (hostVar, devVar, c2g)

Memory copy copy(devVar1,devVar2) function cloneDeepWData(devVarA) function cloneDeepWOData(devVarA)

Misc. swap(devVar1, devVar2) part(deviceVariable,i1,i2,j1,j2,k1,k2) get_[i, s, c] set_[I, s, c]

Point-wise Functions Arithmetic

devf_[hadamardf, divide, addition, subtraction] (devVar3, devVar1, devVar2, option)

Scaling devf_[i,s,c]scal(deviceVariable, a, b),

devf_cscalconj(deviceVariable, a, b) Misc.

devf_zeros(deviceVariable), devf_conjugate(deviceVariable), devf_partofcmplx(whichpart,deviceVariable)

CUBLAS Functions: BLAS 1, BLAS 2, BLAS 3 (with

shorter call strings) CUFFT Functions:

FFT Plans devf_fftplan(devVariable, fft_type,

batch) devf_destroyfftplan(plan)

FFT Functions devf_fft(input, plan, output) devf_bfft(input, plan, output) devf_ifft(input, plan, output) devf_fftR2C(input, plan, output) devf_fftC2R(input, plan, output)

CUDPP Functions: devf_ancCUDPPSortScan(devVarIn, devVarOut,

operation, dataType, algorithm, option) devf_ancCUDPPSortSimple(devVarIn, devVarOut) Ancillary Functions: devf_ancMatrixTranspose(devVarIn, devVarOut) devf_ancBitonicSort(devVar1)

Extensible

Example of code conversion

Plasma turbulence computations spectral code, solved via a standard Runge-Kutta time advance, coupled with a

pseudo-spectral evaluation of NL terms. Derivatives are evaluated in k−space, while multiplications in Eq. (2) are carried

out in real space. standard 2/3 rule for dealiasing is applied, and small “hyperviscous” damping

terms are added to provide stability at the grid scale. results agree with analytic expectations and same on both CPU & GPU.

32x speedup!with Bill Dorland

Audio Camera spherical array of microphones Use beamforming algorithms we developed can find sounds coming from

particular directions Run several beamformers, one “look

direction” and assign output to an “Audio pixel”

Compose audio image. Transform the spherical array into

a camera for audio images Requires significant processing to

form pixels from all directions in a frame before the next frame is ready

Azimuth

Elevation

Azimuth

Adam O’Donovan

O’Donovan et al. : Several papers in IEEE CVPR, IEEE ICASSP, WASPAA (2007-2008) Movies at www.umiacs.umd.edu/~odonovan/Audio_Camera

Plasma Computations via PIC

Image courtesy: George Stanchev and Bill Dorland

Data structures for coalesced access Particles modeling a density or real particles Right hand side of evolution equation controlled by

a PDE for field solved on a regular grid Either spectrally or via finite differences Before/After time step require interpolation of field

quantities at grid nodes to/from particles Organized particles in a box using octrees created

via bit interleaving resulting in a Morton curve layout

Update procedures at the end of each time step

George Stantchev, William Dorland, Nail Gumerov “Fast parallel particle-to-grid interpolation for plasma PIC simulations on the GPU,” J. Parallel Distrib. Comput., 2008

Numerical relativity Beginning collaboration with Prof. Tiglio's group Student (John Dickerson) project in CMSC 828 E Spectral –element computations of Kerr tails in

numerical relativity accelerated using FLAGON

Kernel Methods on Balaji V. GPUs Srinivasan Kernel methods are very popular in computational

statistics and computational ML kernel density estimation (KDE), Renyi entropy based distances

between distributions (KRD) Gaussian process regresion Acceleration of 10x to 100x on

a GT240

Optimized bandwith based KDE

Map Reduce framework Aparnafor large scale video Kothaanalysis Video data is extremely large and ubiquitous Particular motivation 30,000 hours of biological

video (courtship rituals of Australian bowebirds) Algorithm framework – reduce frames to a few

features and compare frame-based features Ripe for Map-reduce type operations Simple bird-locator and activity detector

3 X speed-up More complex video processing: larger speedups

LVIS Data Analysis – Shravya Konda NASA’s Laser Vegetation Imaging Sensor LIDAR based Analyze the

returned pulsefor peaks and mode charac-teristics

Achieved 25X Speed up on a 8800GTX

Work ongoing Thanks to Michelle Hofton, (Geography, UMD) and J. Brian Blair (NASA Goddard) for data and discussion

Adding QR, LU, random Lipinginitialization to FLAGON Liu Flagon allows Fortran-9X users to define GPU based variables

as pointers, copy data to them, and use GPU Allows custom functions for extensibility Lightweight no-overhead GPU use Added dense matrix decompositions to FLAGON

LU for linear systems QR for least squares

Random number initialization (uniform and normal) Port of work of Volkov/Demmel (Berkley) and Giles (Oxford) Achieved speed ups reported by these authors but in Flagon

framework

Displaying Flagon objects Adam during simulation O’Donovan A much discussed application of GPUs –

monitoring computations as they proceed Perhaps use it for computational steering A mechanism to throw up

line-graphs (vector data), matrix data (colour maps) and slice data on screen

Issues: OpenGL thread model and interaction with CUDA computations

Other CMSC 828E projects Implementing a caching scheme for GPU computing

Kapil Anand Accelerating the Approximate Nearest Neighbor

library Daniel Hakim Adding MultiGPU MPI capabilities to Flagon

Kate DeSpain Support Vector Machines for Speaker ID on CUDA

Samuel Lamphier