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Some special cases of nonlinear programming have specialized solution methods: If the objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and general methods from convex optimization can be used in most cases.
[6] It is generally used in solving non-linear equations like Euler's equations in computational fluid dynamics. Matrix-free conjugate gradient method has been applied in the non-linear elasto-plastic finite element solver. [7] Solving these equations requires the calculation of the Jacobian which is costly in terms of CPU time and storage. To ...
Newton–Krylov methods are numerical methods for solving non-linear problems using Krylov subspace linear solvers. [1] [2] Generalising the Newton method to systems of multiple variables, the iteration formula includes a Jacobian matrix. Solving this directly would involve calculation of the Jacobian's inverse, when the Jacobian matrix itself ...
Powell's dog leg method, also called Powell's hybrid method, is an iterative optimisation algorithm for the solution of non-linear least squares problems, introduced in 1970 by Michael J. D. Powell. [1] Similarly to the Levenberg–Marquardt algorithm, it combines the Gauss–Newton algorithm with gradient descent, but it uses an explicit trust ...
[5] [6] [7] They have also been developed for solving nonlinear systems of equations. [1] Relaxation methods are important especially in the solution of linear systems used to model elliptic partial differential equations, such as Laplace's equation and its generalization, Poisson's equation. These equations describe boundary-value problems, in ...
In mathematics, quasilinearization is a technique which replaces a nonlinear differential equation or operator equation (or system of such equations) with a sequence of linear problems, which are presumed to be easier, and whose solutions approximate the solution of the original nonlinear problem with increasing accuracy.
The LMA is used in many software applications for solving generic curve-fitting problems. By using the Gauss–Newton algorithm it often converges faster than first-order methods. [6] However, like other iterative optimization algorithms, the LMA finds only a local minimum, which is not necessarily the global minimum.
In the last twenty years, the HAM has been applied to solve a growing number of nonlinear ordinary/partial differential equations in science, finance, and engineering. [8] [9] For example, multiple steady-state resonant waves in deep and finite water depth [10] were found with the wave resonance criterion of arbitrary number of traveling gravity waves; this agreed with Phillips' criterion for ...
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