finite_difference/src/approx/approx2D.py

"""
Approximation of functions by linear combination of basis functions in
function spaces and the least squares method (or the Galerkin method).
2D version.
"""
import sympy as sym
import numpy as np

def least_squares(f, psi, Omega, symbolic=True, print_latex=False):
    """
    Given a function f(x,y) on a rectangular domain
    Omega=[[xmin,xmax],[ymin,ymax]],
    return the best approximation to f(x,y) in the space V
    spanned by the functions in the list psi.
    """
    N = len(psi) - 1
    A = sym.zeros((N+1, N+1))
    b = sym.zeros((N+1, 1))
    x, y = sym.symbols('x y')
    print ('...evaluating matrix...')
    for i in range(N+1):
        for j in range(i, N+1):
            print ('(%d,%d)' % (i, j))

            integrand = psi[i]*psi[j]
            if symbolic:
                I = sym.integrate(integrand,
                                 (x, Omega[0][0], Omega[0][1]),
                                 (y, Omega[1][0], Omega[1][1]))
            if not symbolic or isinstance(I, sym.Integral):
                # Could not integrate symbolically, use numerical int.
                print ('numerical integration of', integrand)
                integrand = sym.lambdify([x,y], integrand)
                I = sym.mpmath.quad(integrand,
                                   [Omega[0][0], Omega[0][1]],
                                   [Omega[1][0], Omega[1][1]])
            A[i,j] = A[j,i] = I
        integrand = psi[i]*f
        if symbolic:
            I = sym.integrate(integrand,
                             (x, Omega[0][0], Omega[0][1]),
                             (y, Omega[1][0], Omega[1][1]))
        if not symbolic or isinstance(I, sym.Integral):
            # Could not integrate symbolically, use numerical int.
            print ('numerical integration of', integrand)
            integrand = sym.lambdify([x,y], integrand)
            I = sym.mpmath.quad(integrand,
                               [Omega[0][0], Omega[0][1]],
                               [Omega[1][0], Omega[1][1]])
        b[i,0] = I
    print ('A:\n', A, '\nb:\n', b)
    if symbolic:
        c = A.LUsolve(b)  # symbolic solve
        # c is a sympy Matrix object, numbers are in c[i,0]
        c = [c[i,0] for i in range(c.shape[0])]
    else:
        c = sym.mpmath.lu_solve(A, b)  # numerical solve
    print ('coeff:', c)

    u = sum(c[i]*psi[i] for i in range(len(psi)))
    print ('approximation:', u)
    print ('f:', sym.expand(f))
    if print_latex:
        print (sym.latex(A, mode='plain'))
        print (sym.latex(b, mode='plain'))
        print (sym.latex(c, mode='plain'))
    return u, c


def comparison_plot(f, u, Omega, plotfile='tmp', title=''):
    """Compare f(x,y) and u(x,y) for x,y in Omega in a plot."""
    x, y = sym.symbols('x y')

    f = sym.lambdify([x,y], f, modules="numpy")
    u = sym.lambdify([x,y], u, modules="numpy")
    # When doing symbolics, Omega can easily contain symbolic expressions,
    # assume .evalf() will work in that case to obtain numerical
    # expressions, which then must be converted to float before calling
    # linspace below
    for r in range(2):
        for s in range(2):
            if not isinstance(Omega[r][s], (int,float)):
                Omega[r][s] = float(Omega[r][s].evalf())

    resolution = 41  # no of points in plot
    xcoor = np.linspace(Omega[0][0], Omega[0][1], resolution)
    ycoor = np.linspace(Omega[1][0], Omega[1][1], resolution)
    xv, yv = np.meshgrid(xcoor, ycoor)
    # Vectorized functions expressions does not work with
    # lambdify'ed functions without the modules="numpy"
    exact  = f(xv, yv)
    approx = u(xv, yv)
    error = exact - approx

    import matplotlib.pyplot as plt
    plt.figure()
    contours = plt.contour(xv, yv, error, 8)     # 8 contour lines
    plt.clabel(contours, inline=1, fontsize=10)  # labels
    if title: plt.title(title)
    if plotfile:
        plt.savefig('%s_error_c.pdf' % plotfile)
        plt.savefig('%s_error_c.png' % plotfile)
    fig = plt.figure()
    from mpl_toolkits.mplot3d import Axes3D
    from matplotlib import cm
    #ax = fig.add_subplot(111) #, projection='3d')
    ax = fig.gca(projection='3d')
    surf = ax.plot_surface(xv, yv, error, rstride=1, cstride=1,
                           cmap=cm.coolwarm, linewidth=0,
                           antialiased=False)
    fig.colorbar(surf, shrink=0.5, aspect=5)
    if title: plt.title(title)
    if plotfile:
        plt.savefig('%s_error_s.pdf' % plotfile)
        plt.savefig('%s_error_s.png' % plotfile)
    plt.show()

if __name__ == '__main__':
    print ('Module file not meant for execution.')