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KrylovKit.jl

A Julia package collecting a number of Krylov-based algorithms for linear problems, singular value and eigenvalue problems and the application of functions of linear maps or operators to vectors.

Overview

KrylovKit.jl accepts general functions or callable objects as linear maps, and general Julia objects with vector like behavior (see below) as vectors.

The high level interface of KrylovKit is provided by the following functions:

  • linsolve: solve linear systems A*x = b
  • eigsolve: find a few eigenvalues and corresponding eigenvectors of an eigenvalue problem A*x = λ x
  • geneigsolve: find a few eigenvalues and corresponding vectors of a generalized eigenvalue problem A*x = λ*B*x
  • svdsolve: find a few singular values and corresponding left and right singular vectors A*x = σ * y and A'*y = σ*x
  • exponentiate: apply the exponential of a linear map to a vector x=exp(t*A)*x₀
  • expintegrator: exponential integrator for a linear non-homogeneous ODE (generalization of exponentiate)

Furthermore, for specialised use cases, there are functions that can deal with so-called "real linear maps", which arise e.g. in the context of differentiable programming:

Package features and alternatives

This section could also be titled "Why did I create KrylovKit.jl"?

There are already a fair number of packages with Krylov-based or other iterative methods, such as

  • IterativeSolvers.jl: part of the JuliaMath organisation, solves linear systems and least square problems, eigenvalue and singular value problems
  • Krylov.jl: part of the JuliaSmoothOptimizers organisation, solves linear systems and least square problems on CPU or GPU for any data type that supports mul!(), including dense and sparse matrices, and abstract operators such as those defined from LinearOperators.jl or LinearMaps.jl.
  • KrylovMethods.jl: specific for sparse matrices
  • Expokit.jl: application of the matrix exponential to a vector
  • ArnoldiMethod.jl: Implicitly restarted Arnoldi method for eigenvalues of a general matrix
  • JacobiDavidson.jl: Jacobi-Davidson method for eigenvalues of a general matrix
  • ExponentialUtilities.jl: Krylov subspace methods for matrix exponentials and phiv exponential integrator products. It has specialized methods for subspace caching, time stepping, and error testing which are essential for use in high order exponential integrators.
  • OrdinaryDiffEq.jl: contains implementations of high order exponential integrators with adaptive Krylov-subspace calculations for solving semilinear and nonlinear ODEs.

Some of these packages have certainly inspired and influenced the development of KrylovKit.jl. However, KrylovKit.jl distinguishes itself from the previous packages in the following ways:

  1. KrylovKit accepts general functions to represent the linear map or operator that defines the problem, without having to wrap them in a LinearMap or LinearOperator type. Of course, subtypes of AbstractMatrix are also supported. If the linear map (always the first argument) is a subtype of AbstractMatrix, matrix vector multiplication is used, otherwise it is applied as a function call.

  2. KrylovKit does not assume that the vectors involved in the problem are actual subtypes of AbstractVector. Any Julia object that behaves as a vector is supported, so in particular higher-dimensional arrays or any custom user type that supports the interface as defined in VectorInterface.jl. Aside from arrays filled with scalar entries, this includes tuples, named tuples, and arbitrarily nested combinations of tuples and arrays. Furthermore, CuArray objects are fully supported as vectors, so that the application of the linear operator on the vector can be executed on a GPU. The computations performed within the Krylov subspace, such as diagonalising the projected matrix, are however always performed on the CPU.

  3. Since version 0.8, KrylovKit.jl supports reverse-mode AD by defining ChainRulesCore.rrule definitions for the most common functionality (linsolve, eigsolve, svdsolve). Hence, reverse mode AD engines that are compatible with the ChainRules ecosystem will be able to benefit from an optimized implementation of the adjoint of these functions. The rrule definitions for the remaining functionality (geneigsolve and expintegrator, of which exponentiate is a special case) will be added at a later stage. There is a dedicated documentation page on how to configure these rrules, as they also require to solve large-scale linear or eigenvalue problems.

Current functionality

The following algorithms are currently implemented

  • linsolve: CG, GMRES, BiCGStab
  • eigsolve: a Krylov-Schur algorithm (i.e. with tick restarts) for extremal eigenvalues of normal (i.e. not generalized) eigenvalue problems, corresponding to Lanczos for real symmetric or complex hermitian linear maps, and to Arnoldi for general linear maps.
  • geneigsolve: an customized implementation of the inverse-free algorithm of Golub and Ye for symmetric / hermitian generalized eigenvalue problems with positive definite matrix B in the right hand side of the generalized eigenvalue problem $A v = B v λ$. The Matlab implementation was described by Money and Ye and is known as EIGIFP; in particular it extends the Krylov subspace with a vector corresponding to the step between the current and previous estimate, analogous to the locally optimal preconditioned conjugate gradient method (LOPCG). In particular, with Krylov dimension 2, it becomes equivalent to the latter.
  • svdsolve: finding largest singular values based on Golub-Kahan-Lanczos bidiagonalization (see GKL)
  • exponentiate: a Lanczos or Arnoldi based algorithm for the action of the exponential of linear map.
  • expintegrator: exponential integrator for a linear non-homogeneous ODE, computes a linear combination of the ϕⱼ functions which generalize ϕ₀(z) = exp(z).

Future functionality?

Here follows a wish list / to-do list for the future. Any help is welcomed and appreciated.

  • More algorithms, including biorthogonal methods:
    • for linsolve: L-GMRES, MINRES, BiCG, IDR(s), ...
    • for eigsolve: BiLanczos, Jacobi-Davidson JDQR/JDQZ, subspace iteration (?), ...
    • for geneigsolve: trace minimization, ...
  • Support both in-place / mutating and out-of-place functions as linear maps
  • Reuse memory for storing vectors when restarting algorithms (related to previous)
  • Support non-BLAS scalar types using GeneralLinearAlgebra.jl and GeneralSchur.jl
  • Least square problems
  • Nonlinear eigenvalue problems
  • Preconditioners
  • Refined Ritz vectors, Harmonic Ritz values and vectors
  • Block versions of the algorithms
  • More relevant matrix functions

Partially done:

  • Improved efficiency for the specific case where x is Vector (i.e. BLAS level 2 operations): any vector v::AbstractArray which has IndexStyle(v) == IndexLinear() now benefits from a multithreaded (use export JULIA_NUM_THREADS = x with x the number of threads you want to use) implementation that resembles BLAS level 2 style for the vector operations, provided ClassicalGramSchmidt(), ClassicalGramSchmidt2() or ClassicalGramSchmidtIR() is chosen as orthogonalization routine.