80 research outputs found
Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory
In the present paper, we consider large scale nonsymmetric differential
matrix Riccati equations with low rank right hand sides. These matrix equations
appear in many applications such as control theory, transport theory, applied
probability and others. We show how to apply Krylov-type methods such as the
extended block Arnoldi algorithm to get low rank approximate solutions. The
initial problem is projected onto small subspaces to get low dimensional
nonsymmetric differential equations that are solved using the exponential
approximation or via other integration schemes such as Backward Differentiation
Formula (BDF) or Rosenbrok method. We also show how these technique could be
easily used to solve some problems from the well known transport equation. Some
numerical experiments are given to illustrate the application of the proposed
methods to large-scale problem
Solving differential matrix Riccati equations by a piecewise-linearized method based on diagonal Padé approximants
Differential Matrix Riccati Equations (DMREs) appear in several branches of science such as applied physics and engineering. For example, these equations play a fundamental role in control theory, optimal control, filtering and estimation, decoupling and order reduction, etc. In this paper a new method based on a theorem proved in this paper is described for solving DMREs by a piecewise-linearized approach. This method is applied for developing two block-oriented algorithms based on diagonal Padé approximants. MATLAB versions of the above algorithms are developed, comparing, under equal conditions, accuracy and computational costs with other piecewise-linearized algorithms implemented by the authors. Experimental results show the advantages of solving stiff or non-stiff DMREs by the implemented algorithms.Ibáñez González, JJ.; Hernández García, V. (2011). Solving differential matrix Riccati equations by a piecewise-linearized method based on diagonal Padé approximants. Computer Physics Communications. 182(3):669-678. doi:10.1016/j.cpc.2010.11.024S669678182
Adaptive high-order splitting schemes for large-scale differential Riccati equations
We consider high-order splitting schemes for large-scale differential Riccati
equations. Such equations arise in many different areas and are especially
important within the field of optimal control. In the large-scale case, it is
critical to employ structural properties of the matrix-valued solution, or the
computational cost and storage requirements become infeasible. Our main
contribution is therefore to formulate these high-order splitting schemes in a
efficient way by utilizing a low-rank factorization. Previous results indicated
that this was impossible for methods of order higher than 2, but our new
approach overcomes these difficulties. In addition, we demonstrate that the
proposed methods contain natural embedded error estimates. These may be used
e.g. for time step adaptivity, and our numerical experiments in this direction
show promising results.Comment: 23 pages, 7 figure
Spline approximations for systems of ordinary differential equations
El objetivo de esta tesis doctoral es desarrollar nuevos métodos basados en splines para la resolución de sistemas de ecuaciones diferenciales del tipo
Y'(x)=f(x,Y(x)) , a<x<b
Y(a)=Y_a (1)
donde Y_a, Y(x) son matrices rxq, comenzando con splines de tipo cúbico y con un algoritmo similar al propuesto por Loscalzo y Talbot en el caso escalar [20], intentando poder aumentar el orden del spline, lo que con el método dado en [20] no puede hacerse de forma convergente. Trataremos también de aplicar dicho método al problema
Y''(x)=f(x,Y(x),Y'(x)) , a<x<b
Y(a)=Y_a
Y'(a)=Y_b (2)
sin aumentar la dimensión del problema para evitar el sobrecoste computacional. Los métodos presentados se compararán con los existentes en la literatura y serán implementados en algoritmos para ponerlos, debidamente documentados, a disposición de la comunidad científica.Tung, MM. (2013). Spline approximations for systems of ordinary differential equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/31658TESISPremios Extraordinarios de tesis doctorale
Differential-Algebraic Equations
Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed
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