Minimization of Sample Clauses

Minimization of. J This section addresses the problem of the minimization of J, i.e., find- ing the solution xˆ. This is an area where many techniques and methods can be employed and where the tuning of the adopted scheme can turn out to be as important as the scheme itself. Essentially any method of finding the minimum is acceptable in a sense, with the caveat that in an operational context it must be robust and fast. The particular characteristics of this problem are that:‌ • First and second derivatives of J (with respect to x) are available and continuous, which implies that descent algorithms that makes use of the local gradient are possible and these are generally faster than methods that do not. • Multiple minima are unlikely This condition is however not met in the present case, and the solution is searched for different first guess values (see Section 4.3.2.2) • J is likely to be approximately quadratic in the region of the solution, far from quadratic elsewhere. This characteristics is a result of the reasonably strongly non-linear nature of the forward (radiative transfer) problem. It means that quick convergence from a poor starting position is unlikely. The cost function J(xˆ) is minimized, for example using a steepest descent method, or the Xxxxxxxxx- Xxxxxxxxx method (combined steepest-descent method / Xxxxx-Xxxxxx method, as described in Sec- tion (4.3.2.3)). Aerosol-CCI+ ATBD XXXXX-SLSTR REF. : Aerosol_cci+_ATBD_C ISSUE : V1.2 DATE : December 2021 PAGE : 21 x y m ΩΛ˜ x b a a a L L L The first and second derivatives of J with respect to x are given by: ∂x Jr = ∂J = KF S—1(y (x, b, Ω) − y ) + S—1 (x − x ) + HF S—1 H x + HF S—1 H x (58) rr ∂2J