Heat equation inversion
The heat equation is another classic example of an inverse problem. The following example shows how you can estimate time dependent temperature distribution on a 1d domain using several point measurements when heating is localized to a few fixed points. This is done with the help of sparse matrices and regularized least squares. Regularization of the second derivative used used to ensure smoothness of the reconstructed heating.
A generalization of this to 2d and 3d (and with advection added to the model) has applications for example in oceanography, where the inverse problem is to estimate the methane release from localized sources in the sea floor.
#!/usr/bin/env python # # Inverse problems with the 1d heat equation # (c) 2017 Juha Vierinen # import numpy as n import matplotlib.pyplot as plt from scipy.sparse import coo_matrix import scipy.sparse.linalg as s import lsqlin def create_theory_matrix(n_t=200,n_x=200, dt=2.0,dx=1.0,C=0.5, …
A generalization of this to 2d and 3d (and with advection added to the model) has applications for example in oceanography, where the inverse problem is to estimate the methane release from localized sources in the sea floor.
#!/usr/bin/env python # # Inverse problems with the 1d heat equation # (c) 2017 Juha Vierinen # import numpy as n import matplotlib.pyplot as plt from scipy.sparse import coo_matrix import scipy.sparse.linalg as s import lsqlin def create_theory_matrix(n_t=200,n_x=200, dt=2.0,dx=1.0,C=0.5, …
