ex14.py: Lyapunov equation with the shifted 2-D Laplacian — slepc4py 3.25.0 documentation

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# ex14.py: Lyapunov equation with the shifted 2-D Laplacian

This example illustrates the use of slepc4py linear matrix equations solver object. In particular, a Lyapunov equation is solved, where the coefficient matrix is the shifted 2-D Laplacian.

The full source code for this demo can be downloaded here.

Initialization is similar to previous examples.

    import sys, slepc4py
    slepc4py.init(sys.argv)

    from petsc4py import PETSc
    from slepc4py import SLEPc

    Print = PETSc.Sys.Print

Once again, a function to build the discretized Laplacian operator in two dimensions. In this case, we build the shifted negative Laplacian because the Lyapunov equation requires the coefficient matrix being stable.

    def Laplacian2D(m, n, sigma):
        # Create matrix for 2D Laplacian operator
        A = PETSc.Mat().create()
        A.setSizes([m*n, m*n])
        A.setFromOptions()
        # Fill matrix
        Istart, Iend = A.getOwnershipRange()
        for I in range(Istart, Iend):
            A[I,I] = -4.0-sigma
            i = I//n    # map row number to
            j = I - i*n # grid coordinates
            if i> 0  : J = I-n; A[I,J] = 1.0
            if i< m-1: J = I+n; A[I,J] = 1.0
            if j> 0  : J = I-1; A[I,J] = 1.0
            if j< n-1: J = I+1; A[I,J] = 1.0
        A.assemble()
        return A

The following function solves the continuous-time Lyapunov equation

\[AX + XA^* = -C\]

where \(C\) is supposed to have low rank. The solution \(X\) also has low rank, which will be restricted to `rk` if it was provided as an argument of the function. After solving the equation, this function computes and prints a residual norm.

    def solve_lyap(A, C, rk=0):
        # Setup the solver
        L = SLEPc.LME().create()
        L.setProblemType(SLEPc.LME.ProblemType.LYAPUNOV)
        L.setCoefficients(A)
        L.setRHS(C)

        N = C.size[0]
        if rk>0:
            X1 = PETSc.Mat().createDense((N,rk))
            X1.assemble()
            X = PETSc.Mat().createLRC(None, X1, None, None)
            L.setSolution(X)

        L.setTolerances(1e-7)
        L.setFromOptions()

        # Solve the problem
        L.solve()

        if rk==0:
            X = L.getSolution()

        its = L.getIterationNumber()
        Print(f'Number of iterations of the method: {its}')
        sol_type = L.getType()
        Print(f'Solution method: {sol_type}')
        tol, maxit = L.getTolerances()
        Print(f'Stopping condition: tol={tol}, maxit={maxit}')
        Print(f'Error estimate reported by the solver: {L.getErrorEstimate()}')
        if N<500:
            Print(f'Computed residual norm: {L.computeError()}')
        Print('')

        return X

The main function processes several command-line arguments such as the grid dimension (`n`, `m`), the shift `sigma` and the rank of the solution, `rank`.

The coefficient matrix `A` is built with the function above, while the right-hand side matrix `C` must be built as a low-rank matrix using `Mat.createLRC()`.

    if __name__ == '__main__':
        opts = PETSc.Options()
        n = opts.getInt('n', 15)
        m = opts.getInt('m', n)
        N = m*n
        sigma = opts.getScalar('sigma', 0.0)
        rk = opts.getInt('rank', 0)

        # Coefficient matrix A
        A = Laplacian2D(m, n, sigma)

        # Create a low-rank Mat to store the right-hand side C = C1*C1'
        C1 = PETSc.Mat().createDense((N,2))
        rstart, rend = C1.getOwnershipRange()
        v = C1.getDenseArray()
        for i in range(rstart,rend):
            if i<N//2: v[i-rstart,0] = 1.0
            if i==0: v[i-rstart,1] = -2.0
            if i==1 or i==2: v[i-rstart,1] = -1.0
        C1.assemble()
        C = PETSc.Mat().createLRC(None, C1, None, None)

        X = solve_lyap(A, C, rk)