linalg.kron#

Functions#

kronecker_solve(solvers, rhs[, out])

Solve linear system Ax=b with A=kron( A_n, A_{n-1}, ..., A_2, A_1 ), given \(n\) separate linear solvers \(L_n\) for the 1D problems \(A_n x_n = b_n\):

Classes#

Inheritance diagram of psydac.linalg.kron

`KroneckerDenseMatrix`(V, W, *args[, with_pads])	Kronecker product of 1D dense matrices.
`KroneckerLinearSolver`(V, W, solvers)	A solver for Ax=b, where A is a Kronecker matrix from arbirary dimension d, defined by d solvers.
`KroneckerStencilMatrix`(V, W, *args)	Kronecker product of 1D stencil matrices.

Details#

class KroneckerStencilMatrix(V, W, *args)[source]#

Bases: LinearOperator

Kronecker product of 1D stencil matrices.

Parameters:

VStencilVectorSpace: The domain.
WStencilVectorSpace: The codomain.
argslist of StencilMatrix: Factors of the Kronecker product (one for each dimension).

property domain#: The domain of the linear operator - an element of Vectorspace

property codomain#: The codomain of the linear operator - an element of Vectorspace

property dtype#

property ndim#

property mats#

dot(x, out=None)[source]#: Apply linear operator to Vector v. Result is written to Vector out, if provided.

copy()[source]#

tostencil()[source]#

tosparse()[source]#

toarray()[source]#: Convert to Numpy 2D array.

transpose(conjugate=False)[source]#

Transpose the LinearOperator .

If conjugate is True, return the Hermitian transpose.

class KroneckerLinearSolver(V, W, solvers)[source]#

Bases: LinearOperator

A solver for Ax=b, where A is a Kronecker matrix from arbirary dimension d, defined by d solvers. We also need information about the space of b.

Parameters:

VStencilVectorSpace: The space b will live in; i.e. which gives us information about the distribution of the right-hand side b.
WStencilVectorSpace: The space x will live in; i.e. which gives us information about the distribution of the unknown vector x.
solverslist of LinearSolver: The components of A in each dimension.

Attributes:

domainStencilVectorSpace: The domain of the linear operator - an element of Vectorspace
codomainStencilVectorSpace: The codomain of the linear operator - an element of Vectorspace

property domain#: The domain of the linear operator - an element of Vectorspace

property codomain#: The codomain of the linear operator - an element of Vectorspace

property dtype#

toarray()[source]#: Convert to Numpy 2D array.

tosparse()[source]#

transpose(conjugate=False)[source]#

Transpose the LinearOperator .

If conjugate is True, return the Hermitian transpose.

dot(v, out=None)[source]#: Apply linear operator to Vector v. Result is written to Vector out, if provided.

property solvers#: Returns an immutable view onto references to the one-dimensional solvers.

solve(rhs, out=None)[source]#: Solves Ax=b where A is a Kronecker product matrix (and represented as such), and b is a suitable vector.

class KroneckerSolverSerialPass(solver, nglobal, mglobal)[source]#

Bases: object

Solves a linear equation for several right-hand sides at the same time, given that the data is already in memory.

Parameters:

solverBandedSolver or SparseSolver: The internally used solver class.
nglobalint: The length of the dimension which we want to solve for.
mglobalint: The number of right-hand sides we want to solve. Equals the product of the number of dimensions which we do NOT want to solve for (when squashing all these dimensions into a single one). I.e. mglobal*nglobal is the total data size.

required_memory()[source]#: Returns the required memory for this operation. Minimum size for the workmem and tempmem parameters.

solve_pass(workmem, tempmem)[source]#

Solves the data available in workmem, assuming that all data is available locally.

Parameters:

workmemndarray: The data which is to be solved. It is a one-dimensional ndarray which contains all columns contiguously ordered in memory one after another. Its minimum size is also given by self.required_mem().
tempmemndarray: Ignored, it exists for compatibility with the parallel solver.

class KroneckerSolverParallelPass(solver, mpi_type, i, cart, mglobal, nglobal, nlocal, localsize)[source]#

Bases: object

Solves a linear equation for several right-hand sides at the same time, using an Alltoallv operation to distribute the data.

The parameters use the form of n and m; here n denotes the length of the dimension we want to solve for, and m is the length of all other dimensions, multiplied with each other. These n and m are then suffixed with local and global, denoting how much of them we have (or want to have) locally. So, nglobal is the dimension of the columns we want to solve, nlocal is the part we have on our local processor. mglobal is the number of right-hand sides to solve in the whole communicator, and mlocal is the number of right-hand sides we will solve on our local processor.

Parameters:

solverBandedSolver or SparseSolver: The internally used solver class.
mpi_typeMPI type: The MPI type of the space. Used for the Alltoallv.
iint: The index of the dimension.
cartCartDecomposition: The cartesian decomposition we use.
mglobalint: The number of right-hand sides we want to solve. Equals the product of the number of dimensions which we do NOT want to solve for (when squashing all these dimensions into a single one). I.e. mglobal*nglobal is the total data size in our communicator (not on the whole grid though).
nglobalint: The length of the dimension which we want to solve. (the total length, not the one we have on this process)
nlocalint: The length of the part of the dimension to solve which is located on this process already.
localsizeint: The size of data on our local process. Equals mlocal * nlocal (given that we know the former).

required_memory()[source]#: Returns the required memory for this operation. Minimum size for the workmem and tempmem parameters.

solve_pass(workmem, tempmem)[source]#

Solves the data available in workmem in a distributed manner, using MPI_Alltoallv.

Parameters:

workmemndarray: The data which is used for solving. All columns to be solved are ordered contiguously. Its minimum size is given by self.required_mem()
tempmemndarray: Temporary array of the same minimum size as workmem.

class KroneckerDenseMatrix(V, W, *args, with_pads=False)[source]#

Bases: LinearOperator

Kronecker product of 1D dense matrices.

Parameters:

VStencilVectorSpace: The domain.
WStencilVectorSpace: The codomain.
argslist of ndarray: Factors of the Kronecker product (one for each dimension).

property domain#: The domain of the linear operator - an element of Vectorspace

property codomain#: The codomain of the linear operator - an element of Vectorspace

property dtype#

property ndim#

property mats#

dot(x, out=None)[source]#: Apply linear operator to Vector v. Result is written to Vector out, if provided.

copy()[source]#

tosparse(**kwargs)[source]#

toarray()[source]#: Convert to Numpy 2D array.

transpose(conjugate=False)[source]#

Transpose the LinearOperator .

If conjugate is True, return the Hermitian transpose.

exchange_assembly_data()[source]#

set_backend(backend)[source]#

kronecker_solve(solvers, rhs, out=None)[source]#

Solve linear system Ax=b with A=kron( A_n, A_{n-1}, …, A_2, A_1 ), given \(n\) separate linear solvers \(L_n\) for the 1D problems \(A_n x_n = b_n\):

x_n = L_n.solve( b_n )

Parameters:

solverslist( LinearSolver ): List of linear solvers along each direction: [L_1, L_2, …, L_n].
rhsStencilVector: Right hand side vector of linear system Ax=b.