LOBPCG

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is a matrix-free method for finding the largest (or smallest) eigenvalues and the corresponding eigenvectors of a symmetric positive definite generalized eigenvalue problem

$Ax=\lambda Bx,$

for a given pair $(A,B)$ of complex Hermitian or real symmetric matrices, where the matrix $B$ is also assumed positive-definite.

Algorithm

The method performs an iterative maximization (or minimization) of the generalized Rayleigh quotient

$\rho (x):=\rho (A,B;x):={\frac {x^{T}Ax}{x^{T}Bx}},$

which results in finding largest (or smallest) eigenpairs of $Ax=\lambda Bx.$

The direction of the steepest ascent, which is the gradient, of the generalized Rayleigh quotient is positively proportional to the vector

$r:=Ax-\rho (x)Bx,$

called the eigenvector residual. If a preconditioner $T$ is available, it is applied to the residual giving vector

$w:=Tr,$

called the preconditioned residual. Without preconditioning, we set $T:=I$ and so $w:=r,$. An iterative method

$x^{i+1}:=x^{i}+\alpha ^{i}T(Ax^{i}-\rho (x^{i})Bx^{i}),$

or, in short,

$x^{i+1}:=x^{i}+\alpha ^{i}w^{i},\,$

$w^{i}:=Tr^{i},\,$

$r^{i}:=Ax^{i}-\rho (x^{i})Bx^{i},$

is known as preconditioned steepest ascent (or descent), where the scalar $\alpha ^{i}$ is called the step size. The optimal step size can be determined by maximizing the Rayleigh quotient, i.e.,

$x^{i+1}:=\arg \max _{y\in span\{x^{i},w^{i}\}}\rho (y)$

(or $\arg \min$ in case of minimizing), in which case the method is called locally optimal. To further accelerate the convergence of the locally optimal preconditioned steepest ascent (or descent), one can add one extra vector to the two-term recurrence relation to make it three-term:

$x^{i+1}:=\arg \max _{y\in span\{x^{i},w^{i},x^{i-1}\}}\rho (y)$

(use $\arg \min$ in case of minimizing). The maximization/minimization of the Rayleigh quotient in a 3-dimensional subspace can be performed numerically by the Rayleigh–Ritz method.

This is a single-vector version of the LOBPCG method. It is one of possible generalization of the preconditioned conjugate gradient linear solvers to the case of symmetric eigenvalue problems. Even in the trivial case $T=I$ and $B=I$ the resulting approximation with $i>3$ will be different from that obtained by the Lanczos algorithm, although both approximations will belong to the same Krylov subspace.

Iterating several approximate eigenvectors together in a block in a similar locally optimal fashion, gives the full block version of the LOBPCG. It allows robust computation of eigenvectors corresponding to nearly-multiple eigenvalues.

Implementations

LOBPCG's inventor, Andrew Knyazev, published an implementation called Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX) with an interface to PETSc. Other implementations are available in ABINIT (including CUDA version), Octopus (software), PESCAN, Anasazi (Trilinos), SciPy, NGSolve, NVIDIA AmgX, and PYFEMax.

Applications

LOBPCG has been successfully used for multi-billion size matrices by Gordon Bell Prize finalists, on the Earth Simulator supercomputer in Japan.

External links

• Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX)
• LOBPCG code in MATLAB
• LOBPCG code in Octave
• LOBPCG code in SciPy
• LOBPCG code in Java