Linear Algebra on GPUs

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Linear Algebra on GPUs

• Jens Krüger Technische Universität München

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Linear algebra?

• Why are we interested in Linear Algebra?
• It is THE machinery to solve PDEs
• PDEs are at the core of many graphics
  applications
• Physics based simulation, Animation, Mesh
  fairing

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LA on GPUs?

• and why put LA on GPU?
• A perfect couple
• GPUs are fast stream processors,
• and many LA operations are streamable
• which goes hand in hand
• The solution is already on the GPU and ready for
  display

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Getting started
Computer graphics applications
GPU as workhorse for numerical computations

Programmable GPUs
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Getting started
Computer graphics applications
GPU as workhorse for numerical computations

Programmable GPUs
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Internal affairs

• Per-pixel vs. per-vertex operations
• 6 gigapixels/second vs. 0.7 gigavertices/second
• Efficient texture memory cache
• Texture read-write access

• 2D Textures are even better

• 2D RGBA textures really rock

• Textures best we can do

Vector representation
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Representation (cont.)

• Dense Matrix representation
• Treat a dense matrix as a set of column vectors
• Again, store these vectors as 2D textures

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Representation (cont.)

• Banded Sparse Matrix representation
• Treat a banded matrix as a set of diagonal
  vectors
• Combine opposing vectors to save space

Matrix
N
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Operations 1

• Vector-Vector Operations
• Reduced to 2D texture operations
• Coded in pixel shaders
• Example Vector1 Vector2 ? Vector3

Render Texture
Static quad
Vertex Shader
Pixel Shader
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Operations 2 (reduce)

• Reduce operation for scalar products

Reduce m x n region in fragment shader
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The single float on GPUs

• Some operations generate single float values e.g.
  reduce
• Read-back to main-mem is slow
• ? Keep single floats on the GPU as 1x1 textures

...
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Operations (cont.)

• Matrix-Vector Operations
• Split it up into Vector Vector operations

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Operations

• In depth example Vector / Banded-Matrix
  Multiplication

A
x
b

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Example (cont.)

• Vector / Banded-Matrix Multiplication

A
b
A
x
b

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Example (cont.)

• Compute the result in 2 Passes

A
Pass 2
Pass 1
b
x
b
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Building a Framework

• Presented so far
• Representations on the GPU for
• Single float values
• Vectors
• Matrices
• Dense
• Banded
• Random sparse (see SIGGRAPH 03)
• Operations on these representations
• Add, multiply, reduce,
• Upload, download, clear, clone,

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Framework Classes (UML)
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Framework Example CG

• Encapsulate into Classes for more complex
  algorithms
• Example use Conjugate Gradient Method, complete
  source

void clCGSolversolveInit() Matrix-gtmatrixVect
orOp(CL_SUB,X,B,R) // R Ax-b R-gtmultiply(-1)
// R -R R-gtclone(P) // P R
R-gtreduceAdd(R, Rho) // rho
sum(RR) void clCGSolversolveIteration()
Matrix-gtmatrixVectorOp(CL_NULL,P,NULL,Q) // Q
Ap P-gtreduceAdd(Q,Temp) // temp
sum(PQ) Rho-gtdiv(Temp,Alpha) // alpha
rho/temp X-gtaddVector(P,X,1,Alpha) // X X
alphaP R-gtsubtractVector(Q,R,1,Alpha) // R R
- alphaQ R-gtreduceAdd(R,NewRho) // newrho
sum(RR) NewRho-gtdivZ(Rho,Beta) // beta
newrho/rho R-gtaddVector(P,P,1,Beta) // P
RbetaP clFloat temp tempNewRho NewRhoRho
Rhotemp // swap rho and newrho
pointers void clCGSolversolve(int maxI)
solveInit() for (int i 0ilt maxIi)
solveIteration() int clCGSolversolve(float
rhoTresh, int maxI) solveInit()
Rho-gtclone(NewRho) for (int i 0ilt maxI
NewRho.getData() gt rhoTreshi)
solveIteration() return i
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Example 12D Waves (explicit)

• Finite difference discretization
• You could write a custom shader for this filter
• Think about this as a matrix-vector operation

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2D Waves (cont.)

• One Time Matrix Initialization

for (isYiltsXsYi) datai ß
// setup diagonal-sY matrix-gtgetRow(sX(sY-1))
-gtsetData(data) for (i0iltsXsYi) datai
(isX) ? ß 0 // setup diagonal-1 matrix-gt
getRow(sXsY-1)-gtsetData(data) for
(i0iltsXsYi) datai 2-4ß
// setup diagonal matrix-gtgetRow(sXsY)-gtsetData(d
ata) for (i0iltsXsYi) datai ((i1)sX)
? ß 0 // setup diagonal1 matrix-gtgetRow(sXs
Y1)-gtsetData(data) for (i0iltsX(sY-1)i)
datai ß // setup
diagonalsY matrix-gtgetRow(sX(sY1))-gtsetData(dat
a)
Per Frame Iteration
clMatrix-gtmatrixVectorOp(CL_SUB,clCurrent,clLast,c
lNext) // next matrixcurrent-last clLast-gtcop
yVector(clCurrent) //
save for next iteration clCurrent-gtcopyVector(clNe
xt) // save for next
iteration cluNext-gtunpack(clNext)
// unpack for rendering renderHF(cl
uNext-gtm_pVectorTexture) // render as
heightfield
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Example 22D Waves (implicit)

• Key Idea
• Use different time discretization (e.g. Crank
  Nicholson)
• Results in system of linear equations
• Iterative solution using CG

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2D Waves (cont.)

• One Time Matrix Initialization

for (isYiltsXsYi) datai -alpha
// setup diagonal-sY matrix-gtgetRow(sX(sY-1
))-gtsetData(data) for (i0iltsXsYi) datai
(isX) ? - alpha 0 // setup
diagonal-1 matrix-gtgetRow(sXsY-1)-gtsetData(data)
for (i0iltsXsYi) datai 4alpha1
// setup diagonal matrix-gtgetRow(sXsY)-gtse
tData(data) for (i0iltsXsYi) datai
((i1)sX) ? -alpha0 // setup
diagonal1 matrix-gtgetRow(sXsY1)-gtsetData(data)
for (i0iltsX(sY-1)i) datai -alpha
// setup diagonalsY matrix-gtgetRow(sX(sY
1))-gtsetData(data)
Per Frame Iteration
cluRHS-gtcomputeRHS(cluLast, cluCurrent) //
generate c(i,j,t) clRHS-gtrepack(cluRHS)
// encode into RGBA iSteps
pCGSolver-gtsolve(iMaxSteps) // solve using
CG cluLast-gtcopyVector(cluCurrent) // save
for next iteration clNext-gtunpack(cluCurrent)
// unpack for rendering renderHF(cluCur
rent-gtm_pVectorTexture) // render as heightfield
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Demos
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The End

• Thank you!
• Questions?

For more infos, browse to http//wwwcg.in.tum.de/
GPU http//www.gpgpu.org