Error estimates for approximate solutions to Bellman equations associated with controlled jump-diffusions

We derive error estimates for approximate (viscosity) solutions of Bellman equations associated to controlled jump-diffusion processes, which are fully nonlinear integro-partial differential equations. Two main results are obtained: (i) error bounds for a class of monotone approximation schemes, which under some assumptions includes finite difference schemes, and (ii) bounds on the error induced when the original Lévy measure is replaced by a finite measure with compact support, an approximation process that is commonly used when designing numerical schemes for integro-partial differential equations. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen. This work is supported by the European network HYKE, contract HPRN-CT-2002-00282. The research of E. R. Jakobsen is supported by the Research Council of Norway through grant no 151608/432. The research of K. H. Karlsen is supported by an Outstanding Young Investigators Award from the Research Council of Norway. This work was done while C. La Chioma visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway.     

On the discretization of differential and Volterra integral equations with variable delay

In this paper we analyze the attainable order ofm-stage implicit (collocation-based) Runge-Kutta methods for differential equations and Volterra integral equations of the second kind with variable delay of the formqt (0<q<1). It will be shown that, in contrast to equations without delay, or equations with constant delay, collocation at the Gauss (-Legendre) points will no longer yield the optimal (local) orderO(h ^2m ). This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Research Grant OGP0009406).     

A smoothing projected Newton-type algorithm for semi-infinite programming

This paper presents a smoothing projected Newton-type method for solving the semi-infinite programming (SIP) problem. We first reformulate the KKT system of the SIP problem into a system of constrained nonsmooth equations. Then we solve this system by a smoothing projected Newton-type algorithm. At each iteration only a system of linear equations needs to be solved. The feasibility is ensured via the aggregated constraint under some conditions. Global and local superlinear convergence of this method is established under some standard assumptions. Preliminary numerical results are reported. Qi’s work is supported by the Hong Kong Research Grant Council. Ling’s work was supported by the Zhejiang Provincial National Science Foundation of China (Y606168). Tong’s work was done during her visit to The Hong Kong Polytechnic University. Her work is supported by the NSF of China (60474070) and the Technology Grant of Hunan (06FJ3038). Zhou’s work is supported by Australian Research Council.     

The Wazewski topological method for contingent equations

Summary A Wazewski — type theorem for contingent equations is obtained using the fundamental theory of contingent equations. This work was done at the Istituto Matematico dell'Università di Firenze under the auspices of the Italian Research Council (C.N.R.). The resident addresses of the authors are, respectively, the University of Colorado, Boulder, Colo., U.S.A. and Michigan State University, East Lansing, Mich., U.S.A. Entrata in Redazione il 20 febbraio 1970.     

Nonlinear Anisotropic Elliptic and Parabolic Equations in R<Superscript>N</Superscript> with Advection and Lower Order Terms and Locally Integrable Data

We prove existence and regularity results for distributional solutions in R^N for nonlinear elliptic and parabolic equations with general anisotropic diffusivities as well as advection and lower-order terms that satisfy appropriate growth conditions. The data are assumed to be merely locally integrable. Mathematics Subject Classifications (2000) 35J60, 35K55.This work was supported by the BeMatA program of the Research Council of Norway and the European network HYKE, funded by the EC as contract HPRN-CT-2002-00282. This work was done while M. Bendahmane visited the Centre of Mathematics for Applications (CMA) at the University of Oslo, Norway.     

The duality of option investment strategies for hedge funds

This paper explores the structure of optimal investment strategies using stochastic programming and duality theory in investment portfolios containing options for a hedge fund manager who attempts to beat a benchmark. Explicit optimal conditions for option investments are obtained for several models. This research was supported by Inquire, the Social Sciences and Humanities Research Council of Canada, the Natural Sciences and Engineering Research Council of Canada, the National Center of Competence in Research FINRISK, a research instrument of the Swiss National Science Foundation, and MIT’s Sloan School of Management. J. R. Rodríguez-Mancilla thanks Gabriel Casillas-Olvera, Deputy Manager of Risk Control at Banco de Mexico, who read several drafts of this paper and made many helpful comments. The opinions in this paper do not necessarily represent those of Banco de México.     

An elementary and direct proof of the Painlevé property for the Painlevé equations I,II and IV

We present a direct and elementary proof that all the solutions of the Painlevé Equations I, II and IV are meromorphic functions on the whole complex plane. The proof uses some ideas from the existing proofs but applies the ideas in a different setting. Research was supported by Hong Kong Research Grant Council Earmark Grant HKUST6123/00P. Research was supported by Hong Kong Research Grant Council Earmark Grant HKUST6107/02P.     

Additional necessary conditions for optimal control with state-variable inequality constraints

It is shown that, when the set of necessary conditions for an optimal control problem with state-variable inequality constraints given by Bryson, Denham, and Dreyfus is appropriately augmented, it is equivalent to the (different) set of conditions given by Jacobson, Lele, and Speyer. Relationships among the various multipliers are given.This work was done at NASA Ames Research Center, Moffett Field, California, under a National Research Council Associateship.     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

The Lagrangian Globalization Method for Nonsmooth Constrained Equations

The difficulty suffered in optimization-based algorithms for the solution of nonlinear equations lies in that the traditional methods for solving the optimization problem have been mainly concerned with finding a stationary point or a local minimizer of the underlying optimization problem, which is not necessarily a solution of the equations. One method to overcome this difficulty is the Lagrangian globalization (LG for simplicity) method. This paper extends the LG method to nonsmooth equations with bound constraints. The absolute system of equations is introduced. A so-called Projected Generalized-Gradient Direction (PGGD) is constructed and proved to be a descent direction of the reformulated nonsmooth optimization problem. This projected approach keeps the feasibility of the iterates. The convergence of the new algorithm is established by specializing the PGGD. Numerical tests are given. This author's work was done when she was visiting The Hong Kong Polytechnic University. His work is also supported by the Research Grant Council of Hong Kong.     

A parameterized Newton method and a quasi-Newton method for nonsmooth equations

This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-defined even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial differential equations and nonlinear complementarity problems. The approximation function method generalizes the splitting function method for nonsmooth equations. Locally superlinear convergence results are proved for the two methods. Numerical examples are given to compare the two methods with some other methods.This work is supported by the Australian Research Council.     

Solution of complex matrix equations with changing phase

A method is presented for solving a succession of complex matrix equations in which the phase of the real and imaginary components changes. The method is more efficient than the technique obtained by using complex Gaussian elimination on each of the matrix equations separately. In addition, some interesting theoretical relationships are presented for the solution of complex matrix equations in general, using only real-valued arithmetic operations.This work was performed while the author was with the Electrical Engineering Department, McGill University, Montreal, Canada. Financial support was provided by the National Research Council of Canada.     

Optimal load frequency control with governor backlash

A new technique is described for designing an optimal controller for a system whose dynamical equations contain a backlash element. The approach is applied to the problem of load frequency control (LFC) of a single area steam power system.This work was supported in part by the National Research Council of Canada, Grant No. A4146.     

Boundary optimal control of MHD flows

This paper deals with some optimal control problems associated with the equations of steady-state, incompressible magnetohydrodynamics. These problems have direct applications to nuclear reactor technology, magnetic propulsion devices, and design of electromagnetic pumps. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations is derived. The theory is applied to an example.The work of L. S. Hou was supported in part by the Natural Science and Engineering Research Council of Canada under Grant Number OGP-0137436 and by a Simon Fraser University President's Research grant.     

Second order parallel algorithms for piecewise smooth displacement kernels

Second order parallel algorithms for Fredholm integral equations with piecewise smooth displacement kernels are derived. One is based on a difference scheme of Runge-Kutta type for an unusual partial differential equations for continuous functions of two variables. The other is based on the trapezoidal quadrature rule applied to a modified integral equations. It is found that the Runge-Kutta type algorithm exhibits certain advantages.The work of these authors was supported in part by the NSF Grant DMS-9007030The work of this author was supported in part by a grant from the National Science and Engineering Research Council of Canada     

Elementary equivalent pairs of algebras associated with sets

The elementary equivalence of two full relation algebras, partition lattices or function monoids are shown to be equivalent to the second order equivalence of the cardinalities of the corresponding sets. This is shown to be related to elementary equivalence of permutation groups and ordinals. Infinite function monoids are shown to be ultrauniversal.Presented by Walter Taylor.The work of the second author was supported by a grant from the University of Cape Town Research Committee, and by the Topology Research Group from the University of Cape Town and the South African Council for Scientific and Industrial Research.     

Midpoint collocation for Cauchy singular integral equations

Summary A Cauchy singular integral equation on a smooth closed curve may be solved numerically using continuous piecewise linear functions and collocation at the midpoints of the underlying grid. Even if the grid is non-uniform, suboptimal rates of convergence are proved using a discrete maximum principle for a modified form of the collocation equations. The same techniques prove negative norm estimates when midpoint collocation is used to determine piecewise constant approximations to the solution of first kind equations with the logarithmic potential.This work was supported by the Australian Research Council through the program grant Numerical analysis for integrals, integral equations and boundary value problems     

The L��vy�CKhintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

Ito??s construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy?CKhintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein?CKantorovich metrics. This is a non-trivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.     

Efficient algorithms for the matrix cosine and sine

Several improvements are made to an algorithm of Higham and Smith for computing the matrix cosine. The original algorithm scales the matrix by a power of 2 to bring the ∞-norm to 1 or less, evaluates the [8/8] Padé approximant, then uses the double-angle formula cos (2A)=2cos ²A−I to recover the cosine of the original matrix. The first improvement is to phrase truncation error bounds in terms of ‖A²‖^1/2 instead of the (no smaller and potentially much larger quantity) ‖A‖. The second is to choose the degree of the Padé approximant to minimize the computational cost subject to achieving a desired truncation error. A third improvement is to use an absolute, rather than relative, error criterion in the choice of Padé approximant; this allows the use of higher degree approximants without worsening an a priori error bound. Our theory and experiments show that each of these modifications brings a reduction in computational cost. Moreover, because the modifications tend to reduce the number of double-angle steps they usually result in a more accurate computed cosine in floating point arithmetic. We also derive an algorithm for computing both cos (A) and sin (A), by adapting the ideas developed for the cosine and intertwining the cosine and sine double angle recurrences. AMS subject classification 65F30 Numerical Analysis Report 461, Manchester Centre for Computational Mathematics, February 2005. Gareth I. Hargreaves: This work was supported by an Engineering and Physical Sciences Research Council Ph.D. Studentship. Nicholas J. Higham: This work was supported by Engineering and Physical Sciences Research Council grant GR/T08739 and by a Royal Society–Wolfson Research Merit Award.     

Attractive cycles in the iteration of meromorphic functions

Summary The existence of attractive cycles constitutes a serious impediment to the solution of nonlinear equations by iterative methods. This problem is illustrated in the case of the solution of the equationz tanz=c, for complex values ofc, by Newton's method. Relevant results from the theory of the iteration of rational functions are cited and extended to the analysis of this case, in which a meromorphic function is iterated. Extensive numerical results, including many attractive cycles, are summarized.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grants A3028 and A7691