Application of normal cones to the computation of solutions of the nonlinear Kolmogorov backward equation

Škulj, Damjan (author)

A numerical approach to computing solutions of a generalized Kolmogorov backward equation is proposed, in which the stochastic matrix is replaced by a nonlinear operator obtained as the lower bound of a set of stochastic matrices. The equation is central to the theory of imprecise Markov chains in continuous time, which has made rapid progress in recent years. One of the obstacles to its implementation remains the high computational complexity, with the prevailing existing approaches relying on a discretization of the time interval. In order to achieve sufficient accuracy of the approximations, the grid must typically contain a large number of points on which optimization steps are performed, usually using linear programming. The main goal of this work is to develop a new, more efficient approach by significantly reducing the number of optimization steps required to achieve the prescribed accuracy of the solutions. Our approach is based on the Lipschitz continuity of the solutions of the equation with respect to time, which results in the optimization problems occurring at nearby points of the time interval having similar optimal solutions. This property is exploited using the theory of normal cones of convex polytopes. If the solution vectors remain within the same normal cone of a polytope corresponding to the nonlinear operator in a given interval, the optimization problem to be solved becomes linear, which allows much faster computations. This paper is primarily concerned with providing the theoretical basis for the new technique. However, initial tests show that it significantly outperforms existing methods in most cases.

• Language: English
• Keywords: imprecise Markov chain in continuous time, nonlinear Kolmogorov backward equation, imprecise transition operator, normal cone
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FDV - Faculty of Social Sciences
• Publication status: Published
• Publication version: Version of Record
• Year: 2023
• Number of pages: 21 str.
• Numbering: Vol. 158, art. 108919
• PID: 20.500.12556/RUL-145138
• UDC: 519.217
• ISSN on article: 0888-613X
• DOI: 10.1016/j.ijar.2023.03.005
• COBISS.SI-ID: 148229123
• Publication date in RUL: 07.04.2023

Record is a part of a journal

• Title: International journal of approximate reasoning
• Shortened title: Int. j. approx. reason.
• Publisher: North-Holland
• ISSN: 0888-613X
• COBISS.SI-ID: 14231301

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: markovski procesi, nelinearne diferencialne enačbe

Projects

• Funder: ARRS - Slovenian Research Agency
• Project number: P5-0168
• Name: Družboslovna metodologija, statistika in informatika