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.

A stochastically weighted operator splitting Monte Carlo (SWOSMC) method for the numerical simulation of complex aerosol dynamic processes

2017 ◽  
Vol 27 (1) ◽  
pp. 263-278 ◽  
Author(s):  
Shuyuan Liu ◽  
Tat L. Chan
Keyword(s):  
Numerical Simulation ◽  
Monte Carlo ◽  
Operator Splitting ◽  
Splitting Method ◽  
Computational Cost ◽  
Particle Method ◽  
Dynamic Processes ◽  
Content Type ◽  
Aerosol Dynamics ◽  
Operator Splitting Method

Purpose The purpose of this paper is to study the complex aerosol dynamic processes by using this newly developed stochastically weighted operator splitting Monte Carlo (SWOSMC) method. Design/methodology/approach Stochastically weighted particle method and operator splitting method are coupled to formulate the SWOSMC method for the numerical simulation of particle-fluid systems undergoing the complex simultaneous processes. Findings This SWOSMC method is first validated by comparing its numerical simulation results of constant rate coagulation and linear rate condensation with the corresponding analytical solutions. Coagulation and nucleation cases are further studied whose results are compared with the sectional method in excellent agreement. This SWOSMC method has also demonstrated its high numerical simulation capability when used to deal with simultaneous aerosol dynamic processes including coagulation, nucleation and condensation. Originality/value There always exists conflict and tradeoffs between computational cost and accuracy for Monte Carlo-based methods for the numerical simulation of aerosol dynamics. The operator splitting method has been widely used in solving complex partial differential equations, while the stochastic-weighted particle method has been commonly used in numerical simulation of aerosol dynamics. However, the integration of these two methods has not been well investigated.

A New Fast Algorithm Based on Half-Step Discretization for 3D Quasilinear Hyperbolic Partial Differential Equations

2018 ◽  
Vol 16 (01) ◽  
pp. 1850090
Author(s):  
R. K. Mohanty ◽  
Gunjan Khurana
Keyword(s):  
Hyperbolic Equations ◽  
Operator Splitting ◽  
Splitting Method ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Hyperbolic Partial Differential Equations ◽  
Benchmark Problems ◽  
Operator Splitting Method ◽  
Singular Coefficients ◽  
Grid Points

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function [Formula: see text] For the derivation of the method, we use two inner half-step points each in [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.

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.

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