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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