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Lecture_14_Poisson_1_jd2005

# Lecture_14_Poisson_1_jd2005 - CS 267 Applications of...

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03/09/2005 CS267 Lecture 14 CS 267: Applications of Parallel Computers Solving Linear Systems arising from PDEs - I James Demmel www.cs.berkeley.edu/~demmel/cs267_Spr05

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03/09/2005 CS267 Lecture 14 Outlin e ° Review Poisson equation ° Overview of Methods for Poisson Equation ° Jacobi’s method ° Red-Black SOR method ° Conjugate Gradients ° FFT ° Multigrid (next lecture) Reduce to sparse-matrix-vector multiply Need them to understand Multigrid
03/09/2005 CS267 Lecture 14 Recap of “Sources of Parallelism” Lecture ° Discrete event systems: Examples: “Game of Life,” logic level circuit simulation. ° Particle systems: Examples: billiard balls, semiconductor device simulation, galaxies. ° Lumped variables depending on continuous parameters: ODEs, e.g., circuit simulation (Spice), structural mechanics, chemical kinetics. ° Continuous variables depending on continuous parameters: PDEs, e.g., heat, elasticity, electrostatics. ° A given phenomenon can be modeled at multiple levels. ° Many simulations combine more than one of these techniques.

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03/09/2005 CS267 Lecture 14 Recap, cont: Solving PDEs ° Hyperbolic problems (waves): Sound wave(position, time) Use explicit time-stepping Solution at each point depends on neighbors at previous time ° Elliptic (steady state) problems: Electrostatic Potential (position) Everything depends on everything else This means locality is harder to find than in hyperbolic problems ° Parabolic (time-dependent) problems: Temperature(position,time) Involves an elliptic solve at each time-step ° Focus on elliptic problems Canonical example is the Poisson equation 2 u/ x 2 + 2 u/ y 2 + 2 u/ z 2 = f(x,y,z)
03/09/2005 CS267 Lecture 14 Poisson’s equation arises in many models ° Electrostatic or Gravitational Potential: Potential(position) ° Heat flow: Temperature(position, time) ° Diffusion: Concentration(position, time) ° Fluid flow: Velocity,Pressure,Density(position,time) ° Elasticity: Stress,Strain(position,time) ° Variations of Poisson have variable coefficients 3D: 2 u/ x 2 + 2 u/ y 2 + 2 u/ z 2 = f(x,y,z) 2D: 2 u/ x 2 + 2 u/ y 2 = f(x,y) 1D: d 2 u/dx 2 = f(x) f represents the sources; also need boundary conditions

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03/09/2005 CS267 Lecture 14 Relation of Poisson’s equation to Gravity, Electrostatics ° Force on particle at (x,y,z) due to particle at 0 3D: -(x,y,z)/r 3 , where r = (x 2 +y 2 +z 2 ) 2D: -(x,y)/r 2 , where r = (x 2 + y 2 ) ° Force is also gradient of potential V 3D: V = -1/r, V = ( V/ x, V/ y, V/ z) 2D: V = log r, V = ( V/ x, V/ y) ° V satisfies Poisson’s equation (try it!)
03/09/2005 CS267 Lecture 14 Poisson’s equation in 1D: 2 u/ x 2 = f(x) 2 -1 -1 2 -1 -1 2 -1 -1 2 -1 -1 2 T = 2 -1 -1 Graph and “ stencil Discretize d 2 u/dx 2 = f(x) on regular mesh u i = u(i*h) to get [ u i+1 – 2*u i + u i-1 ] / h 2 = f(x) Write as solving Tu = -h 2 * f for u where

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03/09/2005 CS267 Lecture 14 2D Poisson’s equation ° Similar to the 1D case, but the matrix T is now ° 3D is analogous 4 -1 -1 -1 4 -1 -1 -1 4 -1 -1 4 -1 -1 -1 -1 4 -1 -1 -1 -1 4 -1 -1 4 -1 -1 -1 4 -1 -1 -1 4 T = 4 -1 -1 -1 -1 Graph and “ stencil
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Lecture_14_Poisson_1_jd2005 - CS 267 Applications of...

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