Preconditioners

GaussianMarkovRandomFields.AbstractPreconditioner Type

AbstractPreconditioner

Abstract type for preconditioners. Should implement the following methods:

• ldiv!(y, P::AbstractPreconditioner, x::AbstractVector)

• ldiv!(P::AbstractPreconditioner, x::AbstractVector)

• (P::AbstractPreconditioner, x::AbstractVector)

• size(P::AbstractPreconditioner)

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GaussianMarkovRandomFields.BlockJacobiPreconditioner Type

BlockJacobiPreconditioner

A preconditioner that uses a block Jacobi preconditioner, i.e. P = diag(A₁, A₂, ...), where each Aᵢ is a preconditioner for a block of the matrix.

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GaussianMarkovRandomFields.FullCholeskyPreconditioner Type

FullCholeskyPreconditioner

A preconditioner that uses a full Cholesky factorization of the matrix, i.e. P = A, so P⁻¹ = A⁻¹. Does not make sense to use on its own, but can be used as a building block for more complex preconditioners.

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GaussianMarkovRandomFields.temporal_block_gauss_seidel Function

temporal_block_gauss_seidel(A, block_size)

Construct a temporal block Gauss-Seidel preconditioner for a spatiotemporal matrix with constant spatial mesh size (and thus constant spatial block size).

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GaussianMarkovRandomFields.TridiagonalBlockGaussSeidelPreconditioner Type

TridiagonalBlockGaussSeidelPreconditioner{T}(D_blocks, L_blocks)
TridiagonalBlockGaussSeidelPreconditioner{T}(D⁻¹_blocks, L_blocks)

Block Gauss-Seidel preconditioner for block tridiagonal matrices. For a matrix given by

$$A=\left[\begin{array}{ccccc}D₁& L₁ᵀ& 0& \cdots & 0\\ L₁& D₂& L₂ᵀ& 0& \cdots \\ 0& L₂& D₃& L₃ᵀ& \cdots \\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & 0& Lₙ₋₁& Lₙ\end{array}\right]$$

this preconditioner is given by

$$P=\left[\begin{array}{cccc}D₁& 0& \cdots & 0\\ L₁& D₂& 0& \cdots \\ 0& L₂& D₃& \cdots \\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & 0& Lₙ₋₁& Dₙ\end{array}\right]$$

Solving linear systems with the preconditioner is made efficient through block forward / backward substitution. The diagonal blocks must be inverted. As such, they may be specified

1. directly as matrices: in this case they will be transformed into FullCholeskyPreconditioners.

2. in terms of their invertible preconditioners

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GaussianMarkovRandomFields.TridiagSymmetricBlockGaussSeidelPreconditioner Type

TridiagSymmetricBlockGaussSeidelPreconditioner{T}(D_blocks, L_blocks)
TridiagSymmetricBlockGaussSeidelPreconditioner{T}(D⁻¹_blocks, L_blocks)

Symmetric Block Gauss-Seidel preconditioner for symmetric block tridiagonal matrices. For a symmetric matrix given by the block decomposition A = L + D + Lᵀ, where L is strictly lower triangular and D is diagonal, this preconditioner is given by P = (L + D) D⁻¹ (L + D)ᵀ ≈ A.

Solving linear systems with the preconditioner is made efficient through block forward / backward substitution. The diagonal blocks must be inverted. As such, they may be specified

1. directly as matrices: in this case they will be transformed into FullCholeskyPreconditioners.

2. in terms of their invertible preconditioners

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