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Multigrid. Some material from lectures of J. Demmel, UC Berkeley Dept of CS. Multigrid.

Multigrid

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Multigrid

Some material from lectures of J. Demmel, UC Berkeley Dept of CS

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- Multigrid is a “unite-and conquer” algorithm for solving the elliptic PDEs. Multigrid begins by obtaining (or improving) a solution on an NxN grid by using and (N/2)x(N/2) grid (and proceeding recursively). This strategy, in addition to reducing computational work, shifts the frequency structure of errors present in the approximate solution.
- For a more detailed mathematical introduction to the multigrid algorithm, see "A Multigrid Tutorial" by W. Briggs.

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- Consider an (2^m-1)-by-(2^m-1) grid of unknowns (a convenient number to work with). With boundary nodes, e add the nodes at the boundary, get a (2^m+1)-by-(2^m+1) grid on which the algorithm will operate. We let n=2^m+1.
- Let P(i) denote the problem of solving a discrete Poisson equation on a (2^i+1)-by-(2^i+1) grid, with (2^i-1)^2 unknowns.
- The problem is specified by the grid size i, the coefficient matrix T(i), and the right hand side b(i).
- Generate a sequence of related problems P(m), P(m-1), P(m-2), ... , P(1) on coarser and coarser grids, where the solution to P(i-1) is a good approximation to the solution of P(i).

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- Let b(i) be the right-hand-side of the linear system P(i), and x(i) an approximate solution to P(i). Thus, x(i) and b(i) are (2^i-1)-by-(2^i-1) arrays of values at each grid point.
- Next define operators that transfer information between grids.
- The restriction operator R(i) takes a pair (b(i),x(i)), consisting of a problem P(i) and its approximate solution x(i), and maps it to (b(i-1),x(i-1)), the companion problem on the next coarser grid. On grid (i-1), start with a guess at the solution x(i-1): ( b(i-1), x(i-1) ) = R(i)( b(i), x(i) ) We will see that restriction is implemented simply by computing a weighted average of each grid point value with its nearest neighbors.

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- The interpolation operator In(i-1) takes an approximate solution x(i-1) for P(i-1) and converts it to an approximation x(i) for the problem P(i) on the next finer grid: ( b(i), x(i) ) = In(i-1)( b(i-1), x(i-1) ) Its implementation also requires just a weighted average with nearest neighbors.
- The solution operator S(i) take a problem P(i) and approximate solution x(i), and computes an improved x(i). x_improved(i) = S(i)( b(i), x(i) ) We often a variation of Jacobi's method as the solution operator (recall symmetry of Jacobi, in contrast to, say, GS)

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- Starting with a problem on a fine grid ( b(i), x(i) ), we want to damp the high frequency error by coarsening the grid, and improve the low frequency error by a smoothing operator.
- First a cut at the high frequency error by smoothing/approximate solution:
x(i) = S(i)( b(i), x(i) ).

- Coarsen the grid: R(i)( b(i), x(i) ).
- Solve the coarser problem recursively
MGV( R(i)( b(i), x(i) ) ).

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- Coarse grid solution to fine grid In(i-1)( MGV( R(i)( b(i), x(i) ) ) ).
- Subtract the correction computed on the coarse grid from the fine grid solution x(i) = x(i) - d(i)
- Improve the solution more by smoothing: x(i) = S(i)( b(i), x(i) ).

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- The work at each point in the algorithm is proportional to the number of unknowns, since the value at each grid point is just averaged with its nearest neighbors. So each grid point on level i will cost (2^i-1)^2 = O(4^i) operations. If the finest grid is at level m, the total serial work will be given by the geometric sum
which is O(m) = O( log #unknowns ).

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- The full Multigrid algorithm uses the V-cycle as a building block.
- FMG( b(m), x(m) ) begins by solving the simplest problem P(1) exactly.
- Given a solution x(i-1) of the coarse problem P(i-1), prolongate to a starting guess x(i) for the next finer problem P(i): In(i-1)( b(i-1), x(i-1) ).
- Solve the finer problem using the Multigrid V-cycle with this starting guess: MGV( In(i-1)( b(i-1), x(i-1) ) )

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- To get a sense of the work, note there is one V cycle for each call to MGV in the inner loop of FMG. On a serial machine, this means that, at level i, it costs O(4^i) computations. Thus the total serial cost is again O( 4^m ) = O( # unknowns )
- The serial complexity is optimal, in that a constant amount of work per unknown is performed. Analysis of a parallel version is considered later.

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- Projection and prolongation are easier in 1D, and it is easier to get a sense of the algorithmic details.
- Consider P(i), the problem on a 1D grid with 2^i+1 points, with the middle 2^i-1 being the unknowns.

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- Let T(i) be the coefficient matrix of problem P(i), that is a scaled (2^i-1)-by-(2^i-1) tridiagonal matrix with 2s on the diagonal and -1s on the off-diagonal.
- The scale factor 4^(-i) is the 1/h^2 term that shows up because T(i) is approximating a second derivative on a grid with grid spacing h=2^(-i).

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[ 2 -1 ]

[ -1 2 -1 ]

T(i) = 4^(-i) * [ -1 2 -1 ]

[ ... ... ... ]

[ -1 2 -1 ]

[ -1 2 ]

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Recall from Fourier analysis, any function (in this case, the error in our approximate solution) can be considered as a linear combination of basis vectors - here, sine functions of different frequencies.

These sine functions are in fact the eigenvectors of the T(i) matrix. The following figure shows some of these sine-curves when i=5, so T(5) is 31-by-31. The top plot is a plot of the eigenvalues (frequencies) of 4^5*T(5), from lowest to highest. The subsequent plots are the eigenvectors (sine-curves), starting with the lowest 4 frequencies, and then a few more frequencies up to the highest (number 31).

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- Let v(j) be the j-th eigenvector of T(i), and V the eigenvector matrix V=[v(1),v(2),...] whose j-th column is v(j). V is an orthogonal matrix, so any vector x can be written as a linear combination of the v(j).
- alpha(j) the j-th frequency component of x,
- On the upper half of the frequency domain, multigrid acting on P(m), damps the error in the solution by the solution operator S(i). On the next coarser grid, with half as many points, multigrid damps the upper half of the remaining frequency components in the error. On the next coarser grid, the upper half of the remaining frequency components are damped, and so on, until we solve the exact (1 unknown) problem P(1).

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- Here we consider a so-called weighted Jacobi smoothing operator S(i).
- Pure Jacobi would replace the j-th component of the approximate solution x(i) by the weighted average:
x(i)(j) = .5*( x(i)(j-1) + x(i)(j+1) + 4^i * b(i)(j) )

= x(i)(j) + .5*( x(i)(j-1) - 2*x(i)(j) + x(i)(j+1)

+ 4^i * b(i)(j) )

- Weighted Jacobi instead uses
x(i)(j) = x(i)(j) + w*.5*( x(i)(j-1) - 2*x(i)(j) + x(i)(j+1)

+ 4^i * b(i)(j) )

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- Choose the weight w to damp the highest frequencies; w=2/3 is a good choice
x(i)(j) = (1/3)*( x(i)(j-1) + x(i)(j) + x(i)(j+1)

+ 4^i * b(i)(j) )

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- Take i=6, so there are 2^6-1=63 unknowns. In the figures to follow, the true solution is a sine curve, shown as a dotted line in the leftmost plot in each row. The approximate solution is a solid line in the same plot. The middle plot shows the error alone. The rightmost plot shows the frequency components of the error.
- Initially, the norm of the vector decreases rapidly, from 1.78 to 1.11, but then decays more gradually, because there is little more error in the high frequencies to damp. Thus, it only makes sense to do 1 or perhaps 2 iterations of S(i) at a time.

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- The restriction operator R(i) takes a problem P(i) with right-hand-side b(i) and an approximate solution x(i), and maps it to a simpler problem P(i-1) with right-hand-side b(i-1) and approximate solution x(i-1).
- Let r(i) be the residual of the approximate solution x(i):
r(i) = T(i)*x(i) - b(i)

- If x(i) were the exact solution, r(i) would be zero.
- If we could solve the equation T(i) * d(i) = r(i) exactly for the correction d(i), then x(i)-d(i) would be the solution we seek, because
T(i) * ( x(i)-d(i) ) = T(i)*x(i) - T(i)*d(i)

= (r(i) + b(i)) - (r(i)) = b(i)

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Thus, to apply R(i) to ( b(i), x(i) ) and get P(i-1),

- compute the residual r(i),
- restrict it to the next coarser grid to get b(i-1) = r(i-1)
- letting the initial guess x(i-1) be all zeros.

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- Thus, P(i-1) will be the problem of solving T(i-1)*d(i-1) = r(i-1), for the correction d(i-1), starting with the initial guess x(i-1) of all zeros.
- The restriction will be accomplished by averaging nearest neighbors of r(i) on the fine grid (with 2^i-1 unknowns) to get an approximation on the coarse grid (with 2^(i-1)-1 unknowns): The value at a coarse grid point is .5 times the value at the corresponding fine grid point, plus .25 times each of the fine grid point neighbors.

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- The interpolation operator In(i-1) takes the solution of the residual equation
T(i-1)*x(i-1) = r(i-1)

and uses it to correct the fine grid approximation x(i).

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- The next graphs summarize the convergence of Multigrid in two ways. The graph on the right shows the residual
residual(k) = norm(T(i)x(i)-b(i))

after k iterations of Full Multigrid (FMG). The graph on the left shows the ratio of consecutive residuals, showing how the error (residual) is decreased by a factor bounded away from 1 at each step (for less smooth solutions, the residual will tend to decrease by a factor closer to .5 at each step.)

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- Look more carefully at a parallel version of multigrid in 2D, including communication costs.
- Multigrid requires each grid point to be updated depending on as many as 8 neighbors (those to the N, E, S, W, NW, SW, SE and NE). This will determine communication costs.
- Assume an n=(2^m+1)-by-n grid of data, and that p=s^2 is a perfect square.
- Assume the data is laid out on an s-by-s grid of processors, with each processor owning an ((n-1)/s)-by-((n-1)/s) subgrid.

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- This is illustrated in the following diagram by the pink dotted line, which encircles the grid points owned by the corresponding processor. The grid points in the top processor row have been labeled (and color coded) by the grid number i of the problem P(i) in which they participate. There is exactly one point labeled 2 per processor. The only grid point in P(1) with a nonboundary value is owned by the processor above the one with the pink dotted line. In general, if p>=16, there will be .5*log_2(p)-1 grids P(i) in which fewer than all processor own participating grid points.

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- The processor indicated by the pink dotted line requires certain grid point data from its neighbors to execute the algorithm. For example, to update its own (blue) grid points for P(5), the processor requires 8 blue grid point values from its N, S, E, and W neighbors, as well as single blue grid point values from its NW, SW, SE and NE neighbors. Similarly, updating the (red) grid points for P(4) requires 4 red grid point values from the N, W, E and W neighbors, and one each from the NW, SW, SE and NE neighbors. This pattern continues until each processor has only one grid point. After this, only some processors participate in the computation, requiring one value each from 8 other processors.

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- From this, we can do a simple complexity analysis of the communication costs. For simplicity, let p=2^(2*k), so each processor owns a 2^(m-k)-by-2^(m-k) subgrid. Consider a V-cycle starting at level m. Then in levels m through k of the V-cycle, each processor will own at least one grid point. After that, in levels k-1 through 1, fewer than all processors will own an active grid point, and so some processors will remain idle. Let f be the time per flop, alpha the time for a message startup, and beta the time per word to send a message.

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At level m >= i >= k, the time in the V-cycle will be

Time at level i = O( 4^(i-k) ) * f ... number of

flops, proportional to the

number of grid points

per processor

+ O( 1 ) * alpha ... send a constant

number of messages

to neighbors

+ O( 2^(i-k) ) * ... number of words sent

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Summing all these terms from i=k to m yields Time at levels k to m =

O( 4^(m-k) )*f + O( m-k )*alpha

+ O( 2^(m-k) )*beta

= O( n^2/p )*f + O( log(n/p) )*alpha

+ O( n/sqrt(p) )*beta

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- On levels k >= i >= 1, this effect disappears, because each processor needs to communicate about as many words as it does floating point operations:
Time at level i = O( 1 ) * f ... number of flops, proportional

to # of grid points per processor

+ O( 1 ) * alpha ... send a constant number of

messages to neighbors

+ O( 1 ) * beta ... number of words sent

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Summing all these terms yields

Time at levels 1 to k-1 = O( k-1 )*f

+ O( k-1 )*alpha + O( k-1 )*beta

= O( log(p) )*f + O( log(p) )*alpha

+ O( log(p) )*beta

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The total time for a V-cycle starting at the finest level is therefore

Time = O( n^2/p + log(p) )*f

+ O( log(n) )*alpha

+ O( n/sqrt(p) + log(p) )*beta

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For Full Multigrid, assuming n^2 >= p, so that each processor has at least 1 grid point:

Time = O( N/p + log(p)*log(N) )*f

+ O( (log(N))^2 )*alpha

+ O( sqrt(N/p) + log(p)*log(N) )*beta

where N=n^2 is the number of unknowns. The speedup over the serial floating point work O( N ), is nearly perfect, O( N/p + log(p)*log(N) ), when N>>p, but reduces to log(N)^2 when p=N (the PRAM model).

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- In practice, the time spent on the coarsest grids, when each processor has one or zero active grid points, while negligible on a serial machine, is significant on a parallel machine, enough so to make the efficiency noticeably lower.

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