文档介绍：Assume that you have a guess U(n)of the solution. IfU(n) is close enough to the exact solution, an improved approximation U(n+1) is obtained by solving the linearized problem where isa positive number. (It isnot necessary that have a solution . even if has. In this case, the Gauss-Newton iteration tends tobe the minimizer of the residual, ., the solution of minU It is well known that for sufficiently small And is called a descent direction for, where | is the l2-norm. The iteration is where is chosen as large as possible such that the step has a reasonable Gauss-Newton method is local, and convergence is assured only when U (0)is close enough to the solution. In general, the first guess may be outside thergion of convergence. To improve convergence from bad initial guesses, a damping strategy is implemented for choosing , the Armijo-Goldstein line search . It chooses the largest damping coefficient out of the sequence 1, 1/2,1/4, ... such that the following inequality holds: | which guarantees a reduction of the residual norm by at least Note that each step of the line-search algorithm requires an evaluation of the residual An important point of this strategy is that when U(n) approaches the solution, then and thus the convergence rate increases. If there isa solution to the scheme ultimately recovers the quadratic convergence rate of the standard Newton iteration. Closely related to the above problem is the choice of the initial guess U (0). By default, the solver sets U (0) and then assembles the FEM matrices K and F putes The damped Gauss-Newton iteration is then started with U (1), which should bea better guess than U (0). If the boundary conditions donot depend on the solution u, then U (1) satisfies them even ifU (0) does not. Furthermore, if the equation is linear, then U (1) is the exact FEM solution and the solver does not enter the Gauss-Newton loop. There are situations where U (0) =0 makes no sense or convergence is impossible. In some sit