Locally optimal block preconditioned conjugate gradient (LOBPCG)

Solves the generalized eigenproblem $Ax = λBx$ approximately where $A$ and $B$ are Hermitian linear maps, and $B$ is positive definite. $B$ is taken to be the identity by default. It can find the smallest (or largest) k eigenvalues and their corresponding eigenvectors which are B-orthonormal. It also admits a preconditioner and a "constraints" matrix C, such that the algorithm returns the smallest (or largest) eigenvalues associated with the eigenvectors in the nullspace of C'B.

Usage

IterativeSolvers.lobpcgFunction

The Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)

Finds the nev extremal eigenvalues and their corresponding eigenvectors satisfying AX = λBX.

A and B may be generic types but Base.mul!(C, AorB, X) must be defined for vectors and strided matrices X and C. size(A, i::Int) and eltype(A) must also be defined for A.

lobpcg(A, [B,] largest, nev; kwargs...) -> results

Arguments

• A: linear operator;
• B: linear operator;
• largest: true if largest eigenvalues are desired and false if smallest;
• nev: number of eigenvalues desired.

Keywords

• log::Bool: default is false; if true, results.trace will store iterations states; if false only results.trace will be empty;

• P: preconditioner of residual vectors, must overload ldiv!;

• C: constraint to deflate the residual and solution vectors orthogonal to a subspace; must overload mul!;

• maxiter: maximum number of iterations; default is 200;

• tol::Real: tolerance to which residual vector norms must be under.

Output

• results: a LOBPCGResults struct. r.λ and r.X store the eigenvalues and eigenvectors.
source
lobpcg(A, [B,] largest, X0; kwargs...) -> results

Arguments

• A: linear operator;
• B: linear operator;
• largest: true if largest eigenvalues are desired and false if smallest;
• X0: Initial guess, will not be modified. The number of columns is the number of eigenvectors desired.

Keywords

• not_zeros: default is false. If true, X0 will be assumed to not have any all-zeros column.

• log::Bool: default is false; if true, results.trace will store iterations states; if false only results.trace will be empty;

• P: preconditioner of residual vectors, must overload ldiv!;

• C: constraint to deflate the residual and solution vectors orthogonal to a subspace; must overload mul!;

• maxiter: maximum number of iterations; default is 200;

• tol::Real: tolerance to which residual vector norms must be under.

Output

• results: a LOBPCGResults struct. r.λ and r.X store the eigenvalues and eigenvectors.
source

lobpcg(A, [B,] largest, X0, nev; kwargs...) -> results

Arguments

• A: linear operator;
• B: linear operator;
• largest: true if largest eigenvalues are desired and false if smallest;
• X0: block vectors such that the eigenvalues will be found size(X0, 2) at a time; the columns are also used to initialize the first batch of Ritz vectors;
• nev: number of eigenvalues desired.

Keywords

• log::Bool: default is false; if true, results.trace will store iterations states; if false only results.trace will be empty;

• P: preconditioner of residual vectors, must overload ldiv!;

• C: constraint to deflate the residual and solution vectors orthogonal to a subspace; must overload mul!;

• maxiter: maximum number of iterations; default is 200;

• tol::Real: tolerance to which residual vector norms must be under.

Output

• results: a LOBPCGResults struct. r.λ and r.X store the eigenvalues and eigenvectors.
source
IterativeSolvers.lobpcg!Function
lobpcg!(iterator::LOBPCGIterator; kwargs...) -> results

Arguments

• iterator::LOBPCGIterator: a struct having all the variables required for the LOBPCG algorithm.

Keywords

• not_zeros: default is false. If true, the initial Ritz vectors will be assumed to not have any all-zeros column.

• log::Bool: default is false; if true, results.trace will store iterations states; if false only results.trace will be empty;

• maxiter: maximum number of iterations; default is 200;

• tol::Real: tolerance to which residual vector norms must be under.

Output

• results: a LOBPCGResults struct. r.λ and r.X store the eigenvalues and eigenvectors.
source

Implementation Details

A LOBPCGIterator is created to pre-allocate all the memory required by the method using the constructor LOBPCGIterator(A, B, largest, X, P, C) where A and B are the matrices from the generalized eigenvalue problem, largest indicates if the problem is a maximum or minimum eigenvalue problem, X is the initial eigenbasis, randomly sampled if not input, where size(X, 2) is the block size bs. P is the preconditioner, nothing by default, and C is the constraints matrix. The desired k eigenvalues are found bs at a time.

References

Implementation is based on [Knyazev1993] and [Scipy].

• Knyazev1993Andrew V. Knyazev. "Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method" SIAM Journal on Scientific Computing, 23(2):517–541 2001.
• ScipySee Scipy LOBPCG implementation