LinearSolversApplication

LinearSolversApplication

The LinearSolversApplication is a thin wrapper for the Eigen linear algebra library.

Direct sparse solvers

The application provides the following direct sparse solvers:

Python class	solver_type	Matrix kind	Domain	Dependencies
SparseLUSolver	sparse_lu	Square	Real	None
SparseQRSolver	sparse_qr	Rectangular	Real	None
SparseCGSolver	sparse_cg	SPD*	Real	None
PardisoLLTSolver	pardiso_llt	SPD*	Real	Intel® MKL
PardisoLDLTSolver	pardiso_ldlt	SPD*	Real	Intel® MKL
PardisoLUSolver	pardiso_lu	Square	Real	Intel® MKL
ComplexSparseLUSolver	sparse_lu_complex	Square	Complex	None
ComplexPardisoLLTSolver	pardiso_llt_complex	SPD*	Complex	Intel® MKL
ComplexPardisoLDLTSolver	pardiso_ldlt_complex	SPD*	Complex	Intel® MKL
ComplexPardisoLUSolver	pardiso_lu_complex	Square	Complex	Intel® MKL

*SPD = Symmetric Positive Definite

Example:

{
    "solver_type": "eigen_sparse_lu"
}

Direct dense solvers

The application provides the following direct solvers for dense systems of equations:

Python class	solver_type	Matrix requirements	Domain	Dependencies
DenseColPivHouseholderQRSolver**	dense_col_piv_householder_qr	None	Real	None
DenseHouseholderQRSolver**	dense_householder_qr	None	Real	None
DenseLLTSolver**	dense_llt	SPD*	Real	None
DensePartialPivLUSolver**	dense_partial_piv_lu	Invertible	Real	None
ComplexDenseColPivHouseholderQRSolver	complex_dense_col_piv_householder_qr	None	Complex	None
ComplexDenseHouseholderQRSolver	complex_dense_householder_qr	None	Complex	None
ComplexDensePartialPivLUSolver	complex_dense_partial_piv_lu	Invertible	Complex	None

*SPD = Symmetric Positive Definite

**Can also be used to solve equation systems with multiple right hand sides.

Generalized eigensystem solvers

The application provides the following generalized eigensystem Ax=λBx solver for sparse matrices.

Python class	solver_type	Matrix kind A	Matrix kind B	Domain	Dependencies
EigensystemSolver	eigen_eigensystem	Symmetric	SPD*	Real	None
SpectraSymGEigsShiftSolver	spectra_sym_g_eigs_shift	Symmetric	SPD*	Real	None
FEASTGeneralEigensystemSolver**	feast	General	General	Real	Intel® MKL
ComplexFEASTGeneralEigensystemSolver**	feast_complex	General	General	Complex	Intel® MKL

*SPD = Symmetric Positive Definite **A special version for symmetric matrices can be triggered in the solver settings.

EigensystemSolver and SpectraSymGEigsShiftSolver compute the smallest eigenvalues and corresponding eigenvectors of the system. MKL routines are used automatically if they are available.

SpectraSymGEigsShiftSolver interfaces a solver from the Spectra library, and has a shift mode that can be used to compute the smallest eigenvalues > shift.

Example:

{
    "solver_type": "spectra_sym_g_eigs_shift",
    "number_of_eigenvalues": 3,
    "max_iteration": 1000,
    "echo_level": 1
}

If the application is compiled with MKL, FEAST 4.0 can be used to solve the generalized eigenvalue problem for real and complex systems (symmetric or unsymmetric). The cmake switch USE_EIGEN_FEAST must be set to ON with

-DUSE_EIGEN_FEAST=ON \

Example:

{
    "solver_type": "feast",
    "symmetric": true,
    "number_of_eigenvalues": 3,
    "search_lowest_eigenvalues": true,
    "e_min" : 0.0,
    "e_max" : 0.2
}

Build instructions

1. Set the required definitions for cmake

   As any other app:

   Windows: in configure.bat

   set KRATOS_APPLICATIONS=%KRATOS_APPLICATIONS%%KRATOS_APP_DIR%\LinearSolversApplication;

   Linux: in configure.sh

   add_app ${KRATOS_APP_DIR}/LinearSolversApplication

2. Build Kratos

3. Setup the ProjectParameters.json

   "linear_solver_settings": {
       "solver_type" : "LinearSolversApplication.sparse_lu"
   }

4. Run the simulation

Enable MKL (optional)

In case you have installed MKL (see below), you can also use the Pardiso solvers.

1. Run the MKL setup script before building Kratos:

   Windows:

   call "C:\Program Files (x86)\Intel\oneAPI\mkl\latest\env\vars.bat" intel64 lp64

   Linux:

   source /opt/intel/oneapi/setvars.sh intel64

2. Add the following flag to CMake to your configure script:

   Windows:

   -DUSE_EIGEN_MKL=ON ^

   Linux:

   -DUSE_EIGEN_MKL=ON \

3. Build Kratos

4. Usage:

   Windows:

   call "C:\Program Files (x86)\Intel\oneAPI\mkl\latest\env\vars.bat" intel64 lp64

   Linux:

   Set the environment before using MKL

   source /opt/intel/oneapi/setvars.sh intel64

Install MKL on Ubuntu with apt

Intel MKL can be installed with apt on Ubuntu. A guide can be found in here. For example to install the MKL 2022 version

sudo bash
# <type your user password when prompted.  this will put you in a root shell>
# If they are not installed, you can install using the following command:
sudo apt update
sudo apt -y install cmake pkg-config build-essential
# use wget to fetch the Intel repository public key
wget https://apt.repos.intel.com/intel-gpg-keys/GPG-PUB-KEY-INTEL-SW-PRODUCTS.PUB
# add to your apt sources keyring so that archives signed with this key will be trusted.
sudo apt-key add GPG-PUB-KEY-INTEL-SW-PRODUCTS.PUB
# remove the public key
rm GPG-PUB-KEY-INTEL-SW-PRODUCTS.PUB
# Configure apt client to use Intel repository
sudo add-apt-repository "deb https://apt.repos.intel.com/oneapi all main"
# Install all MKL related dependencies. You can install full HPC with: sudo apt install intel-hpckit
sudo apt install intel-oneapi-mkl-devel
# Exit
exit

To enable the MKL environment (needs to be done before build/run) use

source /opt/intel/oneapi/setvars.sh intel64