Overview. Overview. Mathematical Primitives. Robert Strzodka. Fragment Processor Functionality as seen from a High Level Language

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1 Overview Matematical GPU Fuctioality Matematical Primitives Robert Strzoda Partial Differetial Equatios o GPUs Discretizatio Grids Discretizatio Scemes Quatizatio Efficiet Liear Algebra o GPUs Assembly of te Matrix Matrix ector Product Local ad Global Data Access 2 GPU as a Data-Parallel Computer Data specificatio Kerel specificatio Geeral executio quad Write Data To Texture Load Fragmet Program Cofigure OpeGL for : Rederig Bid Textures Bid Fragmet Program Textures Fragmet program Draw sigle large Draw Large Quad slide from Te GPGPU Programmig Model Write results to texture Fragmet Processor Fuctioality as see from a Hig Level Laguage Float data types: 6-bit & 32-bit (IDIA, 24-bit (ATI ectors, structs ad arrays: float4, float vec[6], float3x4, float arr[0][20], struct {} Aritmetic ad logic operators: +, -, *, /; &&,,! Trigoometric, expoetial fuctios: si, asi, exp, log, pow, User defied fuctios max3(float a, float b, float c { retur max(a,max(b,c; } Coditioal statemets by predicatio, urollable loops: if, for, wile, dyamic bracig i PS3 Arbitrary texture positios ca be accessed presetatio by Aaro Lefo 3 4 Overview PDE Examples Matematical GPU Fuctioality cloud dyamics Partial Differetial Equatios o GPUs Discretizatio Grids Discretizatio Scemes Quatizatio fluid dyamics Efficiet Liear Algebra o GPUs Assembly of te Matrix Matrix ector Product Local ad Global Data Access boilig 5 images courtesy of Mar Harris 6

2 PDE Examples Diffusio Example Iitial image u0 : Ω [0,], uow u : (Ω, ℜ + ℜ t u (G ( u u = 0 visualizatio i ℜ + Ω u (0 = u0 i Ω ν u = 0 o ℜ + Ω liear G (v := g(x isotropic o-liear image processig G (v := g ( v scalar aisotropic ( G (v := B T (v g ( 0 v 0 g 2 ( v B(v x images courtesy of 7 Marc Drose, Tobias Preusser Diffusio Example Deoisig by a Liear ad a oo-liear Diffusio 8 Diffusio Example Multiscale Flow isualizatio liear diffusio Iitial image Step Step 2 Step 3 Step 4 Step 5 o-liear diffusio Iitial image Step 7 Step 6 Step 8 images courtesy of 9 Discretizatio Grids o GPUs Tobias Preusser 0 Discretizatio Grids o GPUs Ustructured grid Equidistat grid Good performace for static, poor for dyamic grid topology Easy to implemet Oe texture olds all values A idirectio texture is eeded Deformed tesor grid Adaptive grid Parallel dyamic updates Ca adle coeretly cagig dyamic grid topology Oe texture for values, secod for deformatio A as, tree or page table is eeded 2 2

3 Discretizatio Scemes o GPUs Quatizatio - Ucotrolled ad Cotrolled Roudoff Effects o Diffusio i 8 bit Fiite Differeces Iterpolative approac: simple ad fast Usually iteractio wit direct eigbors Fiite olumes olumetric approac: mass coservatio Good at discotiuities, less for smoot data Iteractio over elemet boudaries Fiite Elemets Approximative approac: error miimizatio Good adlig of deformed, ustructured grids Iteractio of basis fuctios (all eigbors 3 4 Quatizatio Roudoff i Floats Roudoff examples for te float s23e8 format additive roudoff a= = fl multiplicative roudoff b=.0002 * = fl cacellatio c=a,b (c- * 0 8 = fl 0 Cacellatio promotes te small error to te absolute error 4 ad a relative error of order oe. Order of operatios ca be crucial: = fl = fl Cacellatio caot be avoided automatically, so watc out! 5 Quatizatio - Preservig Accuracy Watc out for cacellatio a b, r = c*a-c*b r = c(a-b Maximize operatios o te same scale a [0,], b,c [0-3, 0-4 ], r = a+b+c r = a+(b+c Mae implicit relatios betwee costats explicit a i =0.0, i=0..99 r = Σ i<00 a i a 99 =-(Σ i<99 a i, r = Σ i<00 a i = Use symmetric itervals for multiplicatio a ~ [-, ], r = 0.34*(a+ r = 0.34a+0.34 Miimize te umber of multiplicatios r = 0.25a + 0.b + 0.5c r = 0.(a+b+0.5(a+c 6 Overview Matematical GPU Fuctioality Partial Differetial Equatios o GPUs Discretizatio Grids Discretizatio Scemes Quatizatio Efficiet Liear Algebra o GPUs Assembly of te Matrix Matrix ector Product Local ad Global Data Access 7 Diffusio Example Discretizatio cotiuous model time disc. (semi-implicit space disc. (Fiite Differeces liear equatio system ] U + = U Typical situatio i semiimplicit scemes Matrix A depeds oliearly o explicit data Liear equatio system must be solved, u u 0 u = t + u u u + U + u τ = U τ = + ]: = τ + Solvers o GPUs ave similar requiremets as o parallel computers Parallel processig of matrix etries o direct write-read depedecies Examples: Jacobi solver, cojugate gradiet, bloc-sor 8 3

4 Diffusio Example Liear Algebra liear equatio system ] U + = U, ]: = τ + two mai computatioal tass. Derivatives ad oliear fuctios >>> Assembly of te matrix 2. Iterative liear equatio solver >>> Matrix vector product structure of te ormal matrix vector product ( AU A U = A U + A U + A U + =,, 0 0,, 2 2 For eac pair of data items (A,U 2 floats must be read A GPU pipelie ca perform 8 MADs (multiply ad add ops Tus to eep 6 pipelies busy 256 floats must be read per cloc But a GPU ca read oly 6 floats >>> 6.25% of pea performace 9 Efficiet Matrix Assembly o GPUs Pure matrix vector product is badwidt boud o GPUs umber of processig elemets agaist badwidt icreases Try to occupy te may processig elemets wit computatios Tree possibilities for a matrix vector product A if A depeds o some data ad must be computed itself O-te-fly: compute etries of A for eac A applicatio Lowest memory requiremet Good for simple etries or seldom use of A Partial assembly: precompute oly some itermediate results Allows to balace computatio ad badwidt requiremets Good coice of precomputed results requires also little memory Full assembly: precompute all etries of A, use tese i A Good if oter computatios ide badwidt problem i A Oterwise try to use partial assembly 20 Diffusio Example Partial Assembly ( : = X Y A ] : = τ [ G ( Perform ] Partial Assembly: Precompute ( (,0 ( X = G U Y : ( : = (0, ( = ( ( x x + (,0 ( X X + Y Y + (,0 + (0, o-te-fly ( x ( x +(,0 (,0 ( ( X Y +(,0 ( X ( X +(,0 Matrix ector Product - Local Gater Step Step + ( C ( F A C, : C Bad Matrix Represetatio Treat a bad matrix as a set of diagoal vectors Combie opposig vectors to save space i Matrix 2 ectors Matrix ector Product - Global Gater ( F γ 2 2D-Textures 2 γ A, -i 2 ( γ slide courtesy of Jes Krüger

5 Irregular Matrix Represetatio i Broo Store all matrix elemets i oe vector elem Store idices Global idex to first elemet i a row start Colum idex of te elemet ipos umber of elemets i a row le Matrix ector Product Global Static Scatter Step Step + Te idex of te updated ode caot be caged dyamically i te loop body. Slow remedy: scatterig of poits. image courtesy of Ia Buc Gater/Scatter i Matrix ector Products Iteractio types betwee ode values (vector compoets ~ γ A = T ( γ γ A ~ γ A = ( A,. A : = ( A γ, A,. : = ( A, X Y : = X Y ( ( Tγ ( X : = X + γ Bot types are itercageable i matrix vector products Easy coversio if gater ad scatter positios are static Dyamic (positio gater is o for GPU, dyamic scatter rater slow A scatterig matrix vector product is good for global regular etries. Summary PDE Decisios Discretizatio grid (equidistat, tesor, adaptive Discretizatio sceme (fiite differeces, elemets Quatizatio (umber format, sceme desig, Liear Algebra Decisios Type of Matrix Assembly (o-te-fly, partial, full Type of Matrix ector Product (MP local >>> gater MP wit bad matrix global irregular >>> gater MP wit irregular matrix global regular >>> scatter MP wit bad matrix images courtesy of Mar Harris