(6) |

where ( ). This is coded in

(7) |

The Hessian matrix (see Eq. (45) in Reference [11]) in three dimensions () is:

(8) |

where is the expectation operator, which for a scalar-valued function is:

(9) |

The Hessian matrix and its inverse are computed in

Let
denote the converged solution
for the Lagrange multipliers and the corresponding
basis function solution for the th node.
Since
(-), the Hessian
(`hessian(.true.)` is the call) is

(10) |

The expressions for the derivatives of the basis functions are given below. We adopt the notations and approach presented in Arroyo and Ortiz [12]; in the interest of space, just the final results are indicated. We can write Eq. (4) as

where is implicitly dependent on . On using Eq. (11), we have

and therefore the gradient of is

If the prior is a Gaussian radial basis function
(see Reference [13]), then
and Eq. (13) reduces to
!
This result appears in the Appendix of Reference [12]. In
general,
depends on
through
the expression given in Eq. (13).
The gradient of the basis functions is computed in
`function dphimaxent()`.

On taking the gradient of Eq. (13), we obtain the following expression for the second derivatives of the max-ent basis functions:

where on using the identity , the gradient of the inverse of the Hessian can be written as

(15) |

and therefore

(16) |

The term in Eq. (14) is given by

(17) |

and the term is:

The Hessian of the basis functions is computed in

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