Unimportant and Advanced Features of Matrices and Solvers

This chapter introduces additional features of the PETSc matrices and solvers. Since most PETSc users should not need to use these features, we recommend skipping this chapter during an initial reading.

Extracting Submatrices

One can extract a (parallel) submatrix from a given (parallel) using

MatCreateSubMatrix(Mat A,IS rows,IS cols,MatReuse call,Mat *B);

This extracts the rows and columns of the matrix A into B. If call is MAT_INITIAL_MATRIX it will create the matrix B. If call is MAT_REUSE_MATRIX it will reuse the B created with a previous call.

Matrix Factorization

Normally, PETSc users will access the matrix solvers through the KSP interface, as discussed in KSP: Linear System Solvers, but the underlying factorization and triangular solve routines are also directly accessible to the user.

The LU and Cholesky matrix factorizations are split into two or three stages depending on the user’s needs. The first stage is to calculate an ordering for the matrix. The ordering generally is done to reduce fill in a sparse factorization; it does not make much sense for a dense matrix.

MatGetOrdering(Mat matrix,MatOrderingType type,IS* rowperm,IS* colperm);

The currently available alternatives for the ordering type are

• MATORDERINGNATURAL - Natural

• MATORDERINGND - Nested Dissection

• MATORDERING1WD - One-way Dissection

• MATORDERINGRCM - Reverse Cuthill-McKee

• MATORDERINGQMD - Quotient Minimum Degree

These orderings can also be set through the options database.

Certain matrix formats may support only a subset of these; more options may be added. Check the manual pages for up-to-date information. All of these orderings are symmetric at the moment; ordering routines that are not symmetric may be added. Currently we support orderings only for sequential matrices.

Users can add their own ordering routines by providing a function with the calling sequence

int reorder(Mat A,MatOrderingType type,IS* rowperm,IS* colperm);

Here A is the matrix for which we wish to generate a new ordering, type may be ignored and rowperm and colperm are the row and column permutations generated by the ordering routine. The user registers the ordering routine with the command

MatOrderingRegister(MatOrderingType ordname,char *path,char *sname,PetscErrorCode (*reorder)(Mat,MatOrderingType,IS*,IS*)));

The input argument ordname is a string of the user’s choice, either an ordering defined in petscmat.h or the name of a new ordering introduced by the user. See the code in src/mat/impls/order/sorder.c and other files in that directory for examples on how the reordering routines may be written.

Once the reordering routine has been registered, it can be selected for use at runtime with the command line option -pc_factor_mat_ordering_type ordname. If reordering from the API, the user should provide the ordname as the second input argument of MatGetOrdering().

The following routines perform complete, in-place, symbolic, and numerical factorizations for symmetric and nonsymmetric matrices, respectively:

MatCholeskyFactor(Mat matrix,IS permutation,const MatFactorInfo *info);
MatLUFactor(Mat matrix,IS rowpermutation,IS columnpermutation,const MatFactorInfo *info);

The argument info->fill > 1 is the predicted fill expected in the factored matrix, as a ratio of the original fill. For example, info->fill=2.0 would indicate that one expects the factored matrix to have twice as many nonzeros as the original.

For sparse matrices it is very unlikely that the factorization is actually done in-place. More likely, new space is allocated for the factored matrix and the old space deallocated, but to the user it appears in-place because the factored matrix replaces the unfactored matrix.

The two factorization stages can also be performed separately, by using the out-of-place mode, first one obtains that matrix object that will hold the factor

MatGetFactor(Mat matrix,MatSolverType package,MatFactorType ftype,Mat *factor);

and then performs the factorization

MatCholeskyFactorSymbolic(Mat factor,Mat matrix,IS perm,const MatFactorInfo *info);
MatLUFactorSymbolic(Mat factor,Mat matrix,IS rowperm,IS colperm,const MatFactorInfo *info);
MatCholeskyFactorNumeric(Mat factor,Mat matrix,const MatFactorInfo);
MatLUFactorNumeric(Mat factor,Mat matrix,const MatFactorInfo *info);

In this case, the contents of the matrix result is undefined between the symbolic and numeric factorization stages. It is possible to reuse the symbolic factorization. For the second and succeeding factorizations, one simply calls the numerical factorization with a new input matrix and the same factored result matrix. It is essential that the new input matrix have exactly the same nonzero structure as the original factored matrix. (The numerical factorization merely overwrites the numerical values in the factored matrix and does not disturb the symbolic portion, thus enabling reuse of the symbolic phase.) In general, calling XXXFactorSymbolic with a dense matrix will do nothing except allocate the new matrix; the XXXFactorNumeric routines will do all of the work.

Why provide the plain XXXfactor routines when one could simply call the two-stage routines? The answer is that if one desires in-place factorization of a sparse matrix, the intermediate stage between the symbolic and numeric phases cannot be stored in a result matrix, and it does not make sense to store the intermediate values inside the original matrix that is being transformed. We originally made the combined factor routines do either in-place or out-of-place factorization, but then decided that this approach was not needed and could easily lead to confusion.

We do not currently support sparse matrix factorization with pivoting for numerical stability. This is because trying to both reduce fill and do pivoting can become quite complicated. Instead, we provide a poor stepchild substitute. After one has obtained a reordering, with MatGetOrdering(Mat A,MatOrdering type,IS *row,IS *col) one may call

MatReorderForNonzeroDiagonal(Mat A,PetscReal tol,IS row, IS col);

which will try to reorder the columns to ensure that no values along the diagonal are smaller than tol in a absolute value. If small values are detected and corrected for, a nonsymmetric permutation of the rows and columns will result. This is not guaranteed to work, but may help if one was simply unlucky in the original ordering. When using the KSP solver interface the option -pc_factor_nonzeros_along_diagonal <tol> may be used. Here, tol is an optional tolerance to decide if a value is nonzero; by default it is 1.e-10.

Once a matrix has been factored, it is natural to solve linear systems. The following four routines enable this process:

MatSolve(Mat A,Vec x, Vec y);
MatSolveTranspose(Mat A, Vec x, Vec y);
MatSolveAdd(Mat A,Vec x, Vec y, Vec w);
MatSolveTransposeAdd(Mat A, Vec x, Vec y, Vec w);

matrix A of these routines must have been obtained from a factorization routine; otherwise, an error will be generated. In general, the user should use the KSP solvers introduced in the next chapter rather than using these factorization and solve routines directly.

Unimportant Details of KSP

PetscDrawAxisDraw(), are usually not used directly by the application programmer Again, virtually all users should use KSP through the KSP interface and, thus, will not need to know the details that follow.

It is possible to generate a Krylov subspace context with the command

KSPCreate(MPI_Comm comm,KSP *kps);

Before using the Krylov context, one must set the matrix-vector multiplication routine and the preconditioner with the commands

PCSetOperators(PC pc,Mat Amat,Mat Pmat);
KSPSetPC(KSP ksp,PC pc);

In addition, the KSP solver must be initialized with

KSPSetUp(KSP ksp);

Solving a linear system is done with the command

KSPSolve(KSP ksp,Vec b,Vec x);

Finally, the KSP context should be destroyed with

KSPDestroy(KSP *ksp);

It may seem strange to put the matrix in the preconditioner rather than directly in the KSP; this decision was the result of much agonizing. The reason is that for SSOR with Eisenstat’s trick, and certain other preconditioners, the preconditioner has to change the matrix-vector multiply. This procedure could not be done cleanly if the matrix were stashed in the KSP context that PC cannot access.

Any preconditioner can supply not only the preconditioner, but also a routine that essentially performs a complete Richardson step. The reason for this is mainly SOR. To use SOR in the Richardson framework, that is,

\[u^{n+1} = u^{n} + B(f - A u^{n}), \]

is much more expensive than just updating the values. With this addition it is reasonable to state that all our iterative methods are obtained by combining a preconditioner from the PC package with a Krylov method from the KSP package. This strategy makes things much simpler conceptually, so (we hope) clean code will result. Note: We had this idea already implicitly in older versions of KSP, but, for instance, just doing Gauss-Seidel with Richardson in old KSP was much more expensive than it had to be. With PETSc this should not be a problem.

Unimportant Details of PC

Most users will obtain their preconditioner contexts from the KSP context with the command KSPGetPC(). It is possible to create, manipulate, and destroy PC contexts directly, although this capability should rarely be needed. To create a PC context, one uses the command

PCCreate(MPI_Comm comm,PC *pc);

The routine

PCSetType(PC pc,PCType method);

sets the preconditioner method to be used. The routine

PCSetOperators(PC pc,Mat Amat,Mat Pmat);

set the matrices that are to be used with the preconditioner. The routine

PCGetOperators(PC pc,Mat *Amat,Mat *Pmat);

returns the values set with PCSetOperators().

The preconditioners in PETSc can be used in several ways. The two most basic routines simply apply the preconditioner or its transpose and are given, respectively, by

PCApply(PC pc,Vec x,Vec y);
PCApplyTranspose(PC pc,Vec x,Vec y);

In particular, for a preconditioner matrix, B, that has been set via PCSetOperators(pc,Amat,Pmat), the routine PCApply(pc,x,y) computes \(y = B^{-1} x\) by solving the linear system \(By = x\) with the specified preconditioner method.

Additional preconditioner routines are

PCApplyBAorAB(PC pc,PCSide right,Vec x,Vec y,Vec work);
PCApplyBAorABTranspose(PC pc,PCSide right,Vec x,Vec y,Vec work);
PCApplyRichardson(PC pc,Vec x,Vec y,Vec work,PetscReal rtol,PetscReal atol, PetscReal dtol,PetscInt maxits,PetscBool zeroguess,PetscInt *outits,PCRichardsonConvergedReason*);

The first two routines apply the action of the matrix followed by the preconditioner or the preconditioner followed by the matrix depending on whether the right is PC_LEFT or PC_RIGHT. The final routine applies its iterations of Richardson’s method. The last three routines are provided to improve efficiency for certain Krylov subspace methods.

A PC context that is no longer needed can be destroyed with the command

PCDestroy(PC *pc);