University:
Boston University
Course:
MET CS 201 | Introduction to Programming
Academic year:

2022
Views:

359

Pages:

16
Author:

debatehl4dcq

Outline 1 PyCUDA 2 Automatic GPU Programming 3 GPU-DG: Challenges and Solutions Automating GPU Programming GPU programming can be time-consuming, unintuitive and error-prone. Obvious idea: Let the computer do it. One way: Smart compilers Automating GPU Programming GPU programming can be time-consuming, unintuitive and error-prone. Obvious idea: Let the computer do it. One way: Smart compilers GPU programming requires complex tradeoffs Tradeoffs require heuristics Heuristics are fragile Automating GPU Programming GPU programming can be time-consuming, unintuitive and error-prone. Obvious idea: Let the computer do it. One way: Smart compilers GPU programming requires complex tradeoffs Tradeoffs require heuristics Heuristics are fragile Another way: Dumb enumeration Enumerate loop slicings Enumerate prefetch options Choose by running resulting code on actual hardware Loo.py Example Empirical GPU loop optimization: a, b, c, i , j , k = [var(s) for s in ”abcijk ”] n = 500 k = make loop kernel([ LoopDimension(”i”, n), LoopDimension(”j”, n), LoopDimension(”k”, n), ], [ (c[ i +n∗j], a[ i +n∗k]∗b[k+n∗j]) ]) gen kwargs = { ”min threads”: 128, ”min blocks”: 32, } → Ideal case: Finds 160 GF/s kernel without human intervention. Loo.py Status Limited scope: Require input/output separation Kernels must be expressible using “loopy” model (i.e. indices decompose into “output” and “reduction”) Enough for DG, LA, FD, . . . Loo.py Status Limited scope: Require input/output separation Kernels must be expressible using “loopy” model (i.e. indices decompose into “output” and “reduction”) Enough for DG, LA, FD, . . . Kernel compilation limits trial rate Non-Goal: Peak performance Good results currently for dense linear algebra and (some) DG subkernels Outline 1 PyCUDA 2 Automatic GPU Programming 3 GPU-DG: Challenges and Solutions Introduction Challenges Benefits of Metaprogramming GPU-DG: Performance and Generality Viscous Shock Capture Outline 1 PyCUDA 2 Automatic GPU Programming 3 GPU-DG: Challenges and Solutions Introduction Challenges Benefits of Metaprogramming GPU-DG: Performance and Generality Viscous Shock Capture Discontinuous Galerkin Method Let Ω := S i Dk ⊂ Rd . Discontinuous Galerkin Method Let Ω := S i Dk ⊂ Rd . Goal Solve a conservation law on Ω: ut + ∇ · F (u) = 0 Discontinuous Galerkin Method Let Ω := S i Dk ⊂ Rd . Goal ut + ∇ · F (u) = 0 Solve a conservation law on Ω: Example Maxwell’s Equations: EM field: E (x, t), H(x, t) on Ω governed by j 1 ∂t E − ∇ × H = − , ε ε ρ ∇·E = , ε ∂t H + 1 ∇ × E = 0, µ ∇ · H = 0. Discontinuous Galerkin Method Multiply by test function, integrate by parts: ˆ 0= ut ϕ + [∇ · F (u)]ϕ dx Dk ˆ ˆ = ut ϕ − F (u) · ∇ϕ dx + (n̂ · F )∗ ϕ dSx , Dk ∂Dk Subsitute in basis functions, introduce elementwise stiffness, mass, and surface mass matrices matrices S, M, MA : X ∂t u k = − D ∂ν ,k [F (u k )] + Lk [n̂ · F − (n̂ · F )∗ ]|A⊂∂Dk . ν For straight-sided simplicial elements: Reduce D ∂ν and L to reference matrices. Decomposition of a DG operator into Subtasks DG’s execution decomposes into two (mostly) separate branches: Flux Gather Flux Lifting uk ∂t u k F (u k ) Local Differentiation Green: Element-local parts of the DG operator. DG on GPUs: Possible Advantages DG on GPUs: Why? GPUs have deep Memory Hierarchy The majority of DG is local. Compute Bandwidth Memory Bandwidth DG is arithmetically intense. GPUs favor dense data. Local parts of the DG operator are dense. DG on the GPU: What are we trying to achieve? Objectives: Main: Speed Reduce need for compute-bound clusters Secondary: Generality Be applicable to many problems Tertiary: Ease-of-Use Hide complexity of GPU hardware Setting (for now): Specialize to straight-sided simplices Optimize for (but don’t specialize to) tetrahedra (ie. 3D) Optimize for “medium” order (3. . . 5)