<|file_name|>one_D_helmholtz.py<|end_file_name|><｜fim▁begin｜>from numpy import array, zeros, ones, sqrt, ravel, mod, random, inner, conjugate
from scipy.sparse import csr_matrix, csc_matrix, coo_matrix, bmat, eye
from scipy import rand, mat, real, imag, linspace, hstack, vstack, exp, cos, sin, pi
from pyamg.util.linalg import norm
import pyamg
from scipy.optimize import fminbound, fmin

__all__ = ['one_D_helmholtz', 'min_wave']

def min_wave(A, omega, x, tol=1e-5, maxiter=25):
    '''

    parameters
    ----------
    A {matrix}
        1D Helmholtz Operator
    omega {scalar}
        Wavenumber used to discretize Helmholtz problem
    x {array}
        1D mesh for the problem
    tol {scalar}
        minimization tolerance
    maxit {integer}
        maximum iters for minimization algorithm

    returns
    -------
    Applies minimization algorithm to find numerically lowest energy wavenumber
    for the matrix A, i.e., the omega shift that minimizes <Ac, c> / <c, c>, 
    for c = cosine((omega+shift)x)
    
    '''

    x = ravel(x)

    # Define scalar objective function, ignoring the 
    # boundaries by only considering A*c at [1:-1]
    def <|fim_middle|>(alpha):
        c = cos((omega+alpha)*x)
        Ac = (A*c)[1:-1]
        return norm(Ac)/norm(c[1:-1])
    
    (xopt, fval, ierr, numfunc) = fminbound(obj_fcn, -0.99*omega, \
         0.99*omega, xtol=tol, maxfun=maxiter, full_output=True, disp=0)
    
    #print "Minimizer = %1.4f,  Function Value at Min = %1.4e\nError Flag = %d,\
    #        Number of function evals = %d" % (xopt, fval, ierr, numfunc)
    
    return xopt

def one_D_helmholtz(h, omega=1.0, nplane_waves=2):
    '''

    parameters
    ----------
    h {int}
        Number of grid spacings for 1-D Helmholtz
    omega {float}
        Defines Helmholtz wave number
    nplane_waves {int}
        Defines the number of planewaves used for the near null-space modes, B.
        1: B = [ exp(ikx) ]
        2: B = [ real(exp(ikx)), complex(exp(ikx)) ]

    returns
    -------
    dictionary containing: 
    
    A {matrix-like}
        LHS of linear system for Helmholtz problem,
        -laplace(u) - omega^2 u = f
    mesh_h {float}
        mesh size
    vertices {array-like}
        [X, Y] 
    elements {None}
        None, just using 1-D finite-differencing 

    '''
    
    # Ensure Repeatability of "random" initial guess
    random.seed(10)

    # Mesh Spacing
    mesh_h = 1.0/(float(h)-1.0)        
    
    # Construct Real Operator
    reA = pyamg.gallery.poisson( (h,), format='csr')
    reA = reA - mesh_h*mesh_h*omega*omega*\
          eye(reA.shape[0], reA.shape[1], format='csr')
    dimen = reA.shape[0]
    
    # Construct Imaginary Operator
    imA = csr_matrix( coo_matrix( (array([2.0*mesh_h*omega]), \
                    (array([0]), array([0]))), shape=reA.shape) )

    # Enforce Radiation Boundary Conditions at first grid point
    reA.data[1] = -2.0
    
    # In order to maintain symmetry scale the first equation by 1/2
    reA.data[0] = 0.5*reA.data[0]
    reA.data[1] = 0.5*reA.data[1]
    imA.data[0] = 0.5*imA.data[0]

    # Create complex-valued system
    complexA = reA + 1.0j*imA
    
    # For this case, the CG (continuous Galerkin) case is the default elements and vertices
    # because there is no DG mesh to speak of
    elements = None
    vertices = hstack((linspace(-1.0,1.0,h).reshape(-1,1), zeros((h,1))))
    
    # Near null-space modes are 1-D Plane waves: [exp(ikx), i exp(ikx)]
    B = zeros( (dimen, nplane_waves), dtype=complex )
    shift = min_wave(complexA, omega, vertices[:,0], tol=1e-9, maxiter=15)
    if nplane_waves == 1:
        B[:,0] = exp(1.0j*(omega+shift)*vertices[:,0])
    elif nplane_waves == 2:
        B[:,0] = cos((omega+shift)*vertices[:,0])
        B[:,1] = sin((omega+shift)*vertices[:,0])
        
    return {'A' : complexA, 'B' : B, 'mesh_h' : mesh_h, \
            'elements' : elements, 'vertices' : vertices}



<｜fim▁end｜>
obj_fcn