Discrete Green's Functions and Interaction Rank

Copyright (C) 2026 Andreas Kloeckner

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import numpy as np
import scipy.linalg as la

import matplotlib.pyplot as pt

Let's solve $u''=-30x^2$ with $u(0)=1$ and $u(1)=-1$.

n = 50

mesh = np.linspace(0, 1, n)
h = mesh[1] - mesh[0]

Set up the system matrix A to carry out centered finite differences

$$ u''(x)\approx \frac{u(x+h) - 2u(x) + u(x-h)}{h^2}. $$

Use np.eye(n, k=...). What needs to be in the first and last row?

A = (np.eye(n, k=1) + -2*np.eye(n) + np.eye(n, k=-1))/h**2
A[0] = 0
A[-1] = 0
A[0,0] = 1
A[-1,-1] = 1

Next, fix the right hand side:

b = -30*mesh**2
b[0] = 1
b[-1] = -1

Compute a reference solution x_true to the linear system:

x_true = la.solve(A, b)
pt.plot(mesh, x_true)

[<matplotlib.lines.Line2D at 0x7f723ca66a50>]

Next, let's consider the influence of each individual RHS component:

b = np.zeros_like(mesh)

# experiment with these set to zero/nonzero
b[17] = 1
b[0] = 0
b[-1] = 0

x_true = la.solve(A, b)
pt.plot(mesh, x_true)

[<matplotlib.lines.Line2D at 0x7fd7b75e27b0>]

pt.imshow(np.log10(1e-15+np.abs(la.inv(A))))

<matplotlib.image.AxesImage at 0x7fd7b7494a50>