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

Škulj, Damjan (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: imprecise Markov chain in continuous time, nonlinear Kolmogorov backward equation, imprecise transition operator, normal cone
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FDV - Fakulteta za družbene vede
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2023
• Št. strani: 21 str.
• Številčenje: Vol. 158, art. 108919
• PID: 20.500.12556/RUL-145138
• UDK: 519.217
• ISSN pri članku: 0888-613X
• DOI: 10.1016/j.ijar.2023.03.005
• COBISS.SI-ID: 148229123
• Datum objave v RUL: 07.04.2023

Gradivo je del revije

• Naslov: International journal of approximate reasoning
• Skrajšan naslov: Int. j. approx. reason.
• Založnik: North-Holland
• ISSN: 0888-613X
• COBISS.SI-ID: 14231301

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: markovski procesi, nelinearne diferencialne enačbe

Projekti

• Financer: ARRS - Agencija za raziskovalno dejavnost Republike Slovenije
• Številka projekta: P5-0168
• Naslov: Družboslovna metodologija, statistika in informatika