scholarly journals The Sum and Difference of Two Lognormal Random Variables

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
C. F. Lo
Keyword(s):  
Stochastic Process ◽  
Probability Distributions ◽  
Operator Splitting ◽  
Splitting Method ◽  
Unified Approach ◽  
New Approach ◽  
Numerical Examples ◽  
Operator Splitting Method ◽  
Approximate Probability ◽  
Analytical Series

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum ofNlognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

scholarly journals Modified Jacobian Newton Iterative Method: Theory and Applications

2009 ◽  
Vol 2009 ◽  
pp. 1-24 ◽  
Author(s):  
Jürgen Geiser
Keyword(s):  
Convergence Rates ◽  
Nonlinear Differential Equations ◽  
Operator Splitting ◽  
Splitting Method ◽  
Linearization Method ◽  
Newton Methods ◽  
Convergence Properties ◽  
New Approach ◽  
Newton Iterative Method ◽  
Operator Splitting Method

This article proposes a new approach to the construction of a linearization method based on the iterative operator-splitting method for nonlinear differential equations. The convergence properties of such a method are studied. The main features of the proposed idea are the linearization of nonlinear equations and the application of iterative splitting methods. We present an iterative operator-splitting method with embedded Newton methods to solve nonlinearity. We confirm with numerical applications the effectiveness of the proposed iterative operator-splitting method in comparison with the classical Newton methods. We provide improved results and convergence rates.

scholarly journals A Fast and Efficient Numerical Algorithm for the Nonlocal Conservative Swift–Hohenberg Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Jingying Wang ◽  
Shuying Zhai
Keyword(s):  
Spectral Method ◽  
Numerical Algorithm ◽  
Total Mass ◽  
Operator Splitting ◽  
Splitting Method ◽  
Three Dimensions ◽  
Simulation Tool ◽  
Numerical Examples ◽  
Efficient Simulation ◽  
Operator Splitting Method

In this paper, we consider a new Swift–Hohenberg equation, where the total mass of this model is conserved through a nonlocal Lagrange multiplier. Based on the operator splitting method and spectral method, a fast and efficient numerical algorithm is proposed. Three numerical examples in both two and three dimensions are provided to illustrate that the proposed algorithm is a practical, accurate, and efficient simulation tool for the nonlocal Swift–Hohenberg equation.

scholarly journals COSMO: A Conic Operator Splitting Method for Convex Conic Problems

2021 ◽  
Vol 190 (3) ◽  
pp. 779-810
Author(s):  
Michael Garstka ◽  
Mark Cannon ◽  
Paul Goulart
Keyword(s):  
Convex Sets ◽  
Operator Splitting ◽  
Splitting Method ◽  
Computational Cost ◽  
Coefficient Matrix ◽  
Benchmark Problems ◽  
Portfolio Optimisation ◽  
Splitting Algorithm ◽  
Semidefinite Programs ◽  
Operator Splitting Method

AbstractThis paper describes the conic operator splitting method (COSMO) solver, an operator splitting algorithm and associated software package for convex optimisation problems with quadratic objective function and conic constraints. At each step, the algorithm alternates between solving a quasi-definite linear system with a constant coefficient matrix and a projection onto convex sets. The low per-iteration computational cost makes the method particularly efficient for large problems, e.g. semidefinite programs that arise in portfolio optimisation, graph theory, and robust control. Moreover, the solver uses chordal decomposition techniques and a new clique merging algorithm to effectively exploit sparsity in large, structured semidefinite programs. Numerical comparisons with other state-of-the-art solvers for a variety of benchmark problems show the effectiveness of our approach. Our Julia implementation is open source, designed to be extended and customised by the user, and is integrated into the Julia optimisation ecosystem.

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