scholarly journals ESERK Methods to Numerically Solve Nonlinear Parabolic PDEs in Complex Geometries: Using Right Triangles

2020 ◽  
Vol 8 ◽  
Author(s):  
Jesús Martín-Vaquero
Keyword(s):  
Complex Geometries ◽  
Parabolic Pdes ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

scholarly journals A probabilistic numerical method for fully nonlinear parabolic PDEs

2011 ◽  
Vol 21 (4) ◽  
pp. 1322-1364 ◽  
Author(s):  
Arash Fahim ◽  
Nizar Touzi ◽  
Xavier Warin
Keyword(s):  
Numerical Method ◽  
Parabolic Pdes ◽  
Fully Nonlinear ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

Simulations of nonlinear parabolic PDEs with forcing function without linearization

2021 ◽  
Vol 71 (4) ◽  
pp. 1005-1018
Author(s):  
Shko Ali Tahir ◽  
Murat Sari
Keyword(s):  
Nonlinear Term ◽  
Numerical Solutions ◽  
Physical Processes ◽  
Parabolic Pdes ◽  
Differentiation Formula ◽  
Numerical Tests ◽  
Natural Behaviour ◽  
Forcing Function ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

Abstract This paper aims at producing numerical solutions of nonlinear parabolic PDEs with forcing term without any linearization. Since the linearization of nonlinear term leads to lose real features, without doing linearization, this paper focuses on capturing natural behaviour of the mechanism. Therefore we concentrate on analysis of the physical processes without losing their properties. To carry out this study, a backward differentiation formula in time and a spline method in space have been combined in leading to the discretized equation. This method leads to a very reliable alternative in solving the problem by conserving the physical properties of the nature. The efficiency of the present method are proved theoretically and illustrated by various numerical tests.

scholarly journals Dividends Sharing Convertible Bonds Pricing and Numerical Evaluation

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xu Guo ◽  
Haiyang Wang
Keyword(s):  
New York ◽  
Convertible Bonds ◽  
Obstacle Problems ◽  
Convertible Bond ◽  
Bond Price ◽  
Parabolic Pdes ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Underlying Stocks ◽  
Discrete Dividends

The convertible bond is becoming one of the most important financial instruments for the company to raise capital fund since it was first issued by American New York Erie Company in 1843. In this paper, it is the first time to study the pricing problem for convertible bond whose underlying stocks pay dividends via the reflected backward stochastic differential equations. Associating the solutions of reflected BSDEs with the obstacle problems for nonlinear parabolic PDEs, we establish the pricing formulas for convertible bonds with continuous and discrete dividends by means of the viscosity solutions for some PDEs. Besides, we also derive the price of convertible bonds with higher borrowing rate which is realistic in the financial market. Then the numerical evaluations are provided by the radial basis functions method. Moreover, we discuss the influence of dividends paying as well as higher borrowing rate on the convertible bond price at last.

Solving nonlinear parabolic PDEs via extended hybrid BDF methods

2014 ◽  
Vol 45 (3) ◽  
pp. 395-412
Author(s):  
Moosa Ebadi ◽  
M. Y. Gokhale
Keyword(s):  
Parabolic Pdes ◽  
Bdf Methods ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

scholarly journals A collage-based approach to solving inverse problems for second-order nonlinear parabolic PDEs

2013 ◽  
Vol 406 (1) ◽  
pp. 120-133 ◽  
Author(s):  
K.M. Levere ◽  
H. Kunze ◽  
D. La Torre
Keyword(s):  
Inverse Problems ◽  
Second Order ◽  
Parabolic Pdes ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic

scholarly journals Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

Nonlinearity ◽  
2021 ◽  
Vol 35 (1) ◽  
pp. 278-310
Author(s):  
Weinan E ◽  
Jiequn Han ◽  
Arnulf Jentzen
Keyword(s):  
Monte Carlo ◽  
Differential Equations ◽  
Numerical Algorithms ◽  
Variational Problems ◽  
Least Square ◽  
Picard Iteration ◽  
Parabolic Pdes ◽  
Nonlinear Parabolic Pdes ◽  
Nonlinear Parabolic ◽  
Very High

Abstract In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the deep backward stochastic differential equations method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.

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