The Landscape of Ax=b SolversThe Landscape of Ax=b Solvers

Pivoting

LU

GMRES, QMR, …

Cholesky

Conjugate gradient

DirectA = LU

Iterativey’ = Ay

Non-symmetric

Symmetricpositivedefinite

More Robust Less Storage (if sparse)

More Robust

More General

D

Conjugate gradient iterationConjugate gradient iteration

• One matrix-vector multiplication per iteration• Two vector dot products per iteration• Four n-vectors of working storage

x0 = 0, r0 = b, p0 = r0

for k = 1, 2, 3, . . .

αk = (rTk-1rk-1) / (pT

k-1Apk-1) step length

xk = xk-1 + αk pk-1 approx solution

rk = rk-1 – αk Apk-1 residual

βk = (rTk rk) / (rT

k-1rk-1) improvement

pk = rk + βk pk-1 search direction

Conjugate gradient: Krylov subspacesConjugate gradient: Krylov subspaces

• Eigenvalues: Au = λu { λ1, λ2 , . . ., λn}

• Cayley-Hamilton theorem:

(A – λ1I)·(A – λ2I) · · · (A – λnI) = 0

Therefore Σ ciAi = 0 for some ci

so A-1 = Σ (–ci/c0) Ai–1

• Krylov subspace:

Therefore if Ax = b, then x = A-1 b and

x span (b, Ab, A2b, . . ., An-1b) = Kn (A, b)

0 i n

1 i n

Conjugate gradient: Orthogonal sequencesConjugate gradient: Orthogonal sequences

• Krylov subspace: Ki (A, b) = span (b, Ab, A2b, . . ., Ai-1b)

• Conjugate gradient algorithm:for i = 1, 2, 3, . . .

find xi Ki (A, b)

such that ri = (Axi – b) Ki (A, b)

• Notice ri Ki+1 (A, b), so ri rj for all j < i

• Similarly, the “directions” are A-orthogonal:(xi – xi-1 )T·A· (xj – xj-1 ) = 0

• The magic: Short recurrences. . .A is symmetric => can get next residual and direction from the previous one, without saving them all.

Conjugate gradient: ConvergenceConjugate gradient: Convergence

• In exact arithmetic, CG converges in n steps (completely unrealistic!!)

• Accuracy after k steps of CG is related to:• consider polynomials of degree k that are equal to 1 at 0.• how small can such a polynomial be at all the eigenvalues of A?

• Thus, eigenvalues close together are good.

• Condition number: κ(A) = ||A||2 ||A-1||2 = λmax(A) / λmin(A)

• Residual is reduced by a constant factor by O(κ1/2(A)) iterations of CG.

(Matlab demo)(Matlab demo)

• CG on grid5(15) and bcsstk08• n steps of CG on bcsstk08

• Lay out matrix and vectors by rows

• Hard part is matrix-vector product y = A*x

• AlgorithmEach processor j: Broadcast x(j) Compute y(j) = A(j,:)*x

• May send more of x than needed

• Partition / reorder matrix to reduce communication

x

y

P0

P1

P2

P3

P0 P1 P2 P3

Conjugate gradient: Parallel implementationConjugate gradient: Parallel implementation

(Matlab demo)(Matlab demo)

• 2-way partition of eppstein mesh • 8-way dice of eppstein mesh

PreconditionersPreconditioners

• Suppose you had a matrix B such that:

1. condition number κ(B-1A) is small2. By = z is easy to solve

• Then you could solve (B-1A)x = B-1b instead of Ax = b

• B = A is great for (1), not for (2)• B = I is great for (2), not for (1)• Domain-specific approximations sometimes work• B = diagonal of A sometimes works

• Or, bring back the direct methods technology. . .

(Matlab demo)(Matlab demo)

• bcsstk08 with diagonal precond

Incomplete Cholesky factorization (IC, ILU)Incomplete Cholesky factorization (IC, ILU)

• Compute factors of A by Gaussian elimination, but ignore fill

• Preconditioner B = RTR A, not formed explicitly

• Compute B-1z by triangular solves (in time nnz(A))

• Total storage is O(nnz(A)), static data structure

• Either symmetric (IC) or nonsymmetric (ILU)

x

A RT R

(Matlab demo)(Matlab demo)

• bcsstk08 with ic precond

Incomplete Cholesky and ILU: VariantsIncomplete Cholesky and ILU: Variants

• Allow one or more “levels of fill”• unpredictable storage requirements

• Allow fill whose magnitude exceeds a “drop tolerance”• may get better approximate factors than levels of fill• unpredictable storage requirements• choice of tolerance is ad hoc

• Partial pivoting (for nonsymmetric A)

• “Modified ILU” (MIC): Add dropped fill to diagonal of U or R• A and RTR have same row sums• good in some PDE contexts

2 3

1 4

2 3

1 4

Incomplete Cholesky and ILU: IssuesIncomplete Cholesky and ILU: Issues

• Choice of parameters• good: smooth transition from iterative to direct methods• bad: very ad hoc, problem-dependent• tradeoff: time per iteration (more fill => more time)

vs # of iterations (more fill => fewer iters)

• Effectiveness• condition number usually improves (only) by constant factor

(except MIC for some problems from PDEs)• still, often good when tuned for a particular class of problems

• Parallelism• Triangular solves are not very parallel• Reordering for parallel triangular solve by graph coloring

(Matlab demo)(Matlab demo)

• 2-coloring of grid5(15)

Sparse approximate inversesSparse approximate inverses

• Compute B-1 A explicitly

• Minimize || B-1A – I ||F (in parallel, by columns)

• Variants: factored form of B-1, more fill, . .

• Good: very parallel

• Bad: effectiveness varies widely

A B-1

Support graph preconditioners: example Support graph preconditioners: example [Vaidya]

• A is symmetric positive definite with negative off-diagonal nzs

• B is a maximum-weight spanning tree for A (with diagonal modified to preserve row sums)

• factor B in O(n) space and O(n) time

• applying the preconditioner costs O(n) time per iteration

G(A) G(B)

• support each edge of A by a path in B

• dilation(A edge) = length of supporting path in B

• congestion(B edge) = # of supported A edges

• p = max congestion, q = max dilation

• condition number κ(B-1A) bounded by p·q (at most O(n2))

G(A) G(B)

Support graph preconditioners: example Support graph preconditioners: example

• can improve congestion and dilation by adding a few

strategically chosen edges to B

• cost of factor+solve is O(n1.75), or O(n1.2) if A is planar

• in recent experiments [Chen & Toledo], often better than

drop-tolerance MIC for 2D problems, but not for 3D.

G(A) G(B)

Support graph preconditioners: example Support graph preconditioners: example

Domain decomposition Domain decomposition (introduction)(introduction)

• Partition the problem (e.g. the mesh) into subdomains

• Use solvers for the subdomains B-1 and C-1 to precondition an iterative solver on the interface

• Interface system is the Schur complement:

S = G – ET B-1E – FT C-1F

• Parallelizes naturally by subdomains

B 0 E

0 C F

ET FT G

A =

(Matlab demo)(Matlab demo)

• grid and matrix structure for overlapping 2-way partition of eppstein

Multigrid Multigrid (introduction)(introduction)

• For a PDE on a fine mesh, precondition using a solution on a coarser mesh

• Use idea recursively on hierarchy of meshes• Solves the model problem (Poisson’s eqn) in linear time!• Often useful when hierarchy of meshes can be built• Hard to parallelize coarse meshes well

• This is just the intuition – lots of theory and technology

Other Krylov subspace methodsOther Krylov subspace methods

• Nonsymmetric linear systems:• GMRES:

for i = 1, 2, 3, . . . find xi Ki (A, b) such that ri = (Axi – b) Ki (A, b)

But, no short recurrence => save old vectors => lots more space

• BiCGStab, QMR, etc.:Two spaces Ki (A, b) and Ki (AT, b) w/ mutually orthogonal bases

Short recurrences => O(n) space, but less robust

• Convergence and preconditioning more delicate than CG• Active area of current research

• Eigenvalues: Lanczos (symmetric), Arnoldi (nonsymmetric)

The Landscape of Sparse Ax=b SolversThe Landscape of Sparse Ax=b Solvers

Pivoting

LU

GMRES, QMR, …

Cholesky

Conjugate gradient

DirectA = LU

Iterativey’ = Ay

Non-symmetric

Symmetricpositivedefinite

More Robust Less Storage

More Robust

More General

Complexity of direct methodsComplexity of direct methods

n1/2 n1/3

2D 3D

Space (fill): O(n log n) O(n 4/3 )

Time (flops): O(n 3/2 ) O(n 2 )

Time and space to solve any problem on any well-shaped finite element mesh

Complexity of linear solversComplexity of linear solvers

2D 3DSparse Cholesky: O(n1.5 ) O(n2 )

CG, exact arithmetic: O(n2 ) O(n2 )

CG, no precond: O(n1.5 ) O(n1.33 )

CG, modified IC: O(n1.25 ) O(n1.17 )

CG, support trees: O(n1.20 ) O(n1.75 )

Multigrid: O(n) O(n)

n1/2 n1/3

Time to solve model problem (Poisson’s equation) on regular mesh