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Numerical solution of 2D nonlinear equation Ask Question. Asked 2 years, 4 months ago. Active 2 years, 4 months ago.
Appendix 8. Numerical Methods for Solving. Nonlinear Equations. 1. An equation is said to be nonlinear when it involves terms of degree higher than 1 in. multivariable nonlinear equations, which involves using the Jacobian matrix. Numerical methods are used to approximate solutions of equations when exact.
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Extrapolation will be used. Which version are you on?
I am on version I am not sure how to code one point source. Sign up or log in Sign up using Google. Sign up using Facebook. For any nonlinear model, implicit timestepping techniques lead to a set of discrete nonlinear equations which must be solved at each timestep.
Several iterative methods for solving these equations are tested. In the cases of uncertain volatility and passport options, it is shown that the frozen coefficient method outperforms two different Newton-type methods. Further, it is proven that the frozen coefficient method is guaranteed to converge for a wide class of one factor problems. A major issue when solving nonlinear PDEs is the possibility of multiple solutions.
In a financial context, convergence to the viscosity solution is desired. Conditions under which the one factor uncertain volatility equations are guaranteed to converge to the viscosity solution are derived.
Unfortunately, the techniques used do not apply to passport options, primarily because a positive coefficient discretization is shown to not always be achievable. For both uncertain volatility and passport options, much work has already been done for one factor problems.
In this thesis, extensions are made for two factor problems. The importance of treating derivative estimates consistently between the discretization and an optimization procedure is discussed. For option pricing problems in general, non-smooth data can cause convergence difficulties for classical timestepping techniques.
In particular, quadratic convergence may not be achieved. Techniques for restoring quadratic convergence for linear problems are examined. Via numerical examples, these techniques are also shown to improve the stability of the nonlinear uncertain volatility and passport option problems.
Finally, two applications are briefly explored.