Distributed parameter problems

Mostly PDEs:

Different types involve different communication:

• Global dependence $\implies$ lots of communication
• Finite wave speed $\implies$ local dependence

Example: 1D heat equation

Consider flow (e.g. of heat) in a uniform rod

• Heat ($Q$) $\propto$ temperature ($u$) $\times$ mass ($\rho h$)
• Heat flow $\propto$ temperature gradient (Fourier’s law)

Example: 1D heat equation

$$\begin{aligned} \frac{\partial Q}{\partial t} \propto h \frac{\partial u}{\partial t} &\approx C \left[ \left( \frac{u(x-h)-u(x)}{h} \right) + \left( \frac{u(x)-u(x+h)}{h} \right) \right] \\ \frac{\partial u}{\partial t} &\approx C \left[ \frac{u(x-h)-2u(x)+u(x+h)}{h^2} \right] \rightarrow C \frac{\partial^2 u}{\partial x^2} \end{aligned}$$

Spatial discretization

Heat equation with $u(0) = u(1) = 0$ $$\frac{\partial u}{\partial t} = C \frac{\partial^2 u}{\partial x^2}$$

Spatial semi-discretization: $$\frac{\partial^2 u}{\partial x^2} \approx \frac{u(x-h)-2u(x)+u(x+h)}{h^2}$$

Spatial discretization

Yields a system of ODEs $$ \begin{align} \frac{du}{dt} & = C h^{-2} (-T) u \\ &= -C h^{-2} \begin{bmatrix} 2 & -1 & & & & \\ -1 & 2 & -1 & & & \\ & \ddots & \ddots & \ddots & \\ & & -1 & 2 & -1 \\ & & & -1 & 2 \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_{n-2} \\ u_{n-1} \end{bmatrix} \end{align}$$

Explicit time stepping

Approximate PDE by ODE system (“method of lines”): $$\frac{du}{dt} = C h^{-2} T u$$ Now need a time-stepping scheme for the ODE:

• Simplest scheme is Euler: $$u(t+\delta) \approx u(t) + u'(t) \delta = \left( I - C \frac{\delta}{h^2} T \right) u(t)$$
• Taking a time step $\equiv$ sparse matvec with $\left( I - C \frac{\delta}{h^2} T \right)$
• This may not end well...

Explicit time stepping data dependence

Nearest neighbor interactions per step $\implies$
finite rate of numerical information propagation

Explicit time stepping in parallel

for t = 1 to N
  communicate boundary data ("ghost cell")
  take time steps locally
end

Overlap communication + computation

for t = 1 to N
  start boundary data sendrecv
  compute new interior values
  finish sendrecv
  compute new boundary values
end

Batching time steps

for t = 1 to N by B
  start boundary data sendrecv (B values)
  compute new interior values
  finish sendrecv (B values)
  compute new boundary values
end

Explicit pain

Unstable for $\delta > O(h^2)$!

Explicit and implicit

• Propagates information at finite rate
• Steps look like sparse matvec (in linear case)
• Stable step determined by fastest time scale
• Works fine for hyperbolic PDEs

Explicit and implicit

• No need to resolve fastest time scales
• Steps can be long... but expensive
  • Linear/nonlinear solves at each step
  • Often these solves involve sparse matvecs
• Critical for parabolic PDEs

Poisson problems

Consider 2D Poisson $$-\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f $$

• Prototypical elliptic problem (steady state)
• Similar to a backward Euler step on heat equation

Poisson problem discretization

$$ u_{i,j} = h^{-2} \left( 4u_{i,j}-u_{i-1,j}-u_{i+1,j}-u_{i,j-1}-u_{i,j+1} \right) $$

Poisson problem discretization

$$L = \left[ \begin{array}{ccc|ccc|ccc} 4 & -1 & & -1 & & & & & \\ -1 & 4 & -1 & & -1 & & & & \\ & -1 & 4 & & & -1 & & & \\ \hline -1 & & & 4 & -1 & & -1 & & \\ & -1 & & -1 & 4 & -1 & & -1 & \\ & & -1 & & -1 & 4 & & & -1 \\ \hline & & & -1 & & & 4 & -1 & \\ & & & & -1 & & -1 & 4 & -1 \\ & & & & & -1 & & -1 & 4 \end{array} \right]$$

Poisson solvers in 2D

Demmel, Applied Numerical Linear Algebra.

Remember: best MFlop/s $\neq$ fastest solution!