We validate our numerical solution against the solution of the depth-averaged equation, which has been used to model intercalation in the limit of highly orthotropic diffusivity and gradient penalty tensors.
Figure 11 Predicted larva density (Eq.8.51Duchon) using 1 st order derivatives in the penalty along a northing gradient through the Hessian Ried for a fixed easting coordinate = 3469000 and constant values of DWT = 4.0 m and CTH = 0.
By automatically estimating the distribution, which nearly becomes the Laplace distribution, the penalty of the intensity gradient becomes weaker than that defined by using the Gaussian distribution, where the details of its estimation are shown in Section 3.2.
Algorithm 1 (Penalty Algorithm Based on Conjugate Gradient Method).
Then, in Section 3, we are going to develop a penalty algorithm based on conjugate gradient methods.
Based on the discussion above, we now come to develop a penalty algorithm based on conjugate gradient method in the last of this section.
So far, we presented very general derivations, starting from Maxwell's equations and ending up with the gradient of the penalty function, which is required in gradient-type inversion schemes.
PSO, as a type of simple and effective random searching algorithm, may lead to better results than the gradient descent method, penalty function method, and genetic algorithm, when solving some non-linear optimization problems [34, 35].
Existence and uniqueness of solutions were proved in [8] for a simplified model, and in [9] they treated a one-dimensional penalty problem and studied the gradient flow of the associated type Ginzburg-Landau functional.
These methods require a computation of the gradient of the penalty function ϕ with respect to the model parameters, i.e. (18)While the gradient of the regularization term is easily calculated analytically and given by (19 the calculation of the data misfit gradient is more challenging.
