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Targeting large-scale linear algebra applications where the matrix matvec subroutine), this package provides lightweight Fortran implementations of certain of the most useful Krylov methods to solve a variety of problems, among which:
-
Eigenvalue Decomposition
$$\mathbf{A} \mathbf{x} = \lambda \mathbf{x}$$ -
Singular Value Decomposition
$$\mathbf{A} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^T$$ -
Linear system of equations
$$\mathbf{Ax} = \mathbf{b}$$
Krylov methods are particularly appropriate in situations where such problems must be solved but factorizing the matrix
-
$\mathbf{A}$ is not available explicitly but only implicitly through amatvecsubroutine computing the matrix-vector product$\mathbf{Ax}$ . -
$\mathbf{A}$ or its factors (e.g.LUorCholesky) are dense and would consume an excessive amount of memory.
Krylov methods are iterative methods, i.e. they iteratively refine the solution of the problem until a desired accuracy is reached. While they are not recommended when a machine-precision solution is needed, they can nonetheless provide highly accurate approximations of the solution after a relatively small number of iterations. Krylov methods form the workhorses of large-scale numerical linear algebra.
LightKrylov leverages Fortran's abstract type feature to provide generic implementations of the various Krylov methods.
The only requirement from the user to benefit from the capabilities of LightKrylov is to extend the abstract_vector and abstract_linop types to define their notion of vectors and linear operators. LightKrylov then provides the following functionalities:
- Krylov factorizations :
arnoldi_factorization,lanczos_tridiagonalization,lanczos_bidiagonalization. - Spectral analysis :
eigs,eighs,svds. - Linear systems :
gmres,cg.
For the sake of simplicity, LightKrylov only works with real or double precision data. While this might seem restrictive at first, consider that a complex-valued LightKrylov is to couple it with high-performance solvers in computational mechanics which often use exclusively real-valued data types. As such, we do not have any plan to natively support complex-valued linear operators or vectors.
Several examples can be found in the example folder. These include:
- Ginzburg-Landau : Serial computation of the leading eigenpairs of a complex-valued linear operator via time-stepping.
- Laplace operator : Parallel computation of the leading eigenpairs of the Laplace operator defined on the unit-square.
Alternatively, you can also look at neklab, a bifurcation and stability analysis toolbox based on LightKrylov and designed to augment the functionalities of the massively parallel spectral element solver Nek5000.
| Ginzburg-Landau | Laplace operator |
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| ADD FIGURE | ADD FIGURE |
Provided you have git installed, getting the code is as simple as:
git clone https://github.com/nekStab/LightKrylov
Alternatively, using gh-cli, you can type
gh repo clone nekStab/LightKrylov
LightKrylov has a very minimal set of dependencies. These only include:
And that's all of it. To date, the tested compilers include:
gfortran 12.0.3
Provided you have cloned the repo, installing LightKrylov with fpm is as simple as
fpm build --profile release
N/A
To see if the library has been compiled correctly, a set of unit tests are provided in test. If you use fpm, running these tests is as simple as
fpm test
If everything went fine, you should see
All tests successfully passed!
If not, please feel free to open an Issue.
To run the examples:
fpm run --example
This command will run all of the examples sequentially. You can alternatively run a specific example using e.g.
fpm run --example Ginzburg-Landau
For more details, please refer to each of the examples.
LightKrylov is currently developed and maintained by a team of three:
- Jean-Christophe Loiseau : Assistant Professor of Applied maths and Fluid dynamics at DynFluid, Arts et Métiers Institute of Technology, Paris, France.
- Ricardo Frantz : PhD in Fluid dynamics (Arts et Métiers, France, 2022) and currently postdoctoral researcher at DynFluid.
- Simon Kern : PhD in Fluid dynamics (KTH, Sweden, 2023) and currently postdoctoral researcher at DynFluid.
Anyone else interested in contributing is obviously most welcomed!
The development of LightKrylov is part of an on-going research project funded by Agence Nationale pour la Recherche (ANR) under the grant agreement ANR-22-CE46-0008. The project started in January 2023 and will run until December 2026.
LightKrylov is the base package of our ecosystem. If you like it, you may also be interested in :
LightROM: a lightweight Fortran package providing a set of functions for reduced-order modeling, control and estimation of large-scale linear time invariant dynamical systems.neklab: a bifurcation and stability analysis toolbox based onLightKrylovfor the massively parallel spectral element solverNek5000.