Special Issue: Optimization and Stochastic Control in Finance,Journal of Optimization Theory and Applications

In this paper, we study the near-optimal control for systems governed by forward–backward stochastic differential equations via dynamic programming principle. Since the nonsmoothness is inherent in this field, the viscosity solution approach is employed to investigate the relationships among the value function, the adjoint equations along near-optimal trajectories. Unlike the classical case, the definition of viscosity solution contains a perturbation factor, through which the illusory differentiability conditions on the value function are dispensed properly. Moreover, we establish new relationships between variational equations and adjoint equations. As an application, a kind of stochastic recursive near-optimal control problem is given to illustrate our theoretical results.     

Sensitivities Calculation for Unsteady Flow Problems

In this work we present an efficient approach for the calculation of the sensitivities in the case of unsteady problems like the compressible high-speed flows. The sensitivities are of a known response with respect to the initial conditions. If the number of parameters is large, the adjoint method is preferred since the cost of computing the sensitivities is independent of the number of design parameters. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Issues in the numerical solution of evolutionary delay differential equations

Delay differential equations are of sufficient importance in modelling real-life phenomena to merit the attention of numerical analysts. In this paper, we discuss key features of delay differential equations (DDEs) and consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide an introduction to the existing literature and numerical codes, and in particular we indicate the approaches adopted by the authors. We also indicate some of the unresolved issues in the numerical solution of DDEs. Communicated by J.C. Mason     

Issues in the numerical solution of evolutionary delay differential equations

Delay differential equations are of sufficient importance in modelling real-life phenomena to merit the attention of numerical analysts. In this paper, we discuss key features of delay differential equations (DDEs) and consider the main issues to be addressed when constructing robust numerical codes for their solution. We provide an introduction to the existing literature and numerical codes, and in particular we indicate the approaches adopted by the authors. We also indicate some of the unresolved issues in the numerical solution of DDEs. Communicated by J.C. Mason     

The Adjoint Newton Algorithm for Large-Scale Unconstrained Optimization in Meteorology Applications

A new algorithm is presented for carrying out large-scale unconstrained optimization required in variational data assimilation using the Newton method. The algorithm is referred to as the adjoint Newton algorithm. The adjoint Newton algorithm is based on the first- and second-order adjoint techniques allowing us to obtain the Newton line search direction by integrating a tangent linear equations model backwards in time (starting from a final condition with negative time steps). The error present in approximating the Hessian (the matrix of second-order derivatives) of the cost function with respect to the control variables in the quasi-Newton type algorithm is thus completely eliminated, while the storage problem related to the Hessian no longer exists since the explicit Hessian is not required in this algorithm. The adjoint Newton algorithm is applied to three one-dimensional models and to a two-dimensional limited-area shallow water equations model with both model generated and First Global Geophysical Experiment data. We compare the performance of the adjoint Newton algorithm with that of truncated Newton, adjoint truncated Newton, and LBFGS methods. Our numerical tests indicate that the adjoint Newton algorithm is very efficient and could find the minima within three or four iterations for problems tested here. In the case of the two-dimensional shallow water equations model, the adjoint Newton algorithm improves upon the efficiencies of the truncated Newton and LBFGS methods by a factor of at least 14 in terms of the CPU time required to satisfy the same convergence criterion.The Newton, truncated Newton and LBFGS methods are general purpose unconstrained minimization methods. The adjoint Newton algorithm is only useful for optimal control problems where the model equations serve as strong constraints and their corresponding tangent linear model may be integrated backwards in time. When the backwards integration of the tangent linear model is ill-posed in the sense of Hadamard, the adjoint Newton algorithm may not work. Thus, the adjoint Newton algorithm must be used with some caution. A possible solution to avoid the current weakness of the adjoint Newton algorithm is proposed.     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.     

Hybrid evolutionary algorithm with Hermite radial basis function interpolants for computationally expensive adjoint solvers

In this paper, we present an evolutionary algorithm hybridized with a gradient-based optimization technique in the spirit of Lamarckian learning for efficient design optimization. In order to expedite gradient search, we employ local surrogate models that approximate the outputs of a computationally expensive Euler solver. Our focus is on the case when an adjoint Euler solver is available for efficiently computing the sensitivities of the outputs with respect to the design variables. We propose the idea of using Hermite interpolation to construct gradient-enhanced radial basis function networks that incorporate sensitivity data provided by the adjoint Euler solver. Further, we conduct local search using a trust-region framework that interleaves gradient-enhanced surrogate models with the computationally expensive adjoint Euler solver. This ensures that the present hybrid evolutionary algorithm inherits the convergence properties of the classical trust-region approach. We present numerical results for airfoil aerodynamic design optimization problems to show that the proposed algorithm converges to good designs on a limited computational budget.     

Linear stability of general linear methods for systems of neutral delay differential equations

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the stability of general linear methods for systems of neutral DDEs with multiple delays. A type of interpolation procedure is considered for general linear methods. Linear stability properties of general linear methods with this interpolation procedure are investigated. Many extant results are unified.     

D-CONVERGENCE OF RUNGE-KUTTA METHODS FOR STIFF DELAY DIFFERENTIAL EQUATIONS

1. IntroductionWhen considering the applicability of numerical methods for the solution of the delay differential equation (DDE) y'(t) = f(t, y(t), y(t - T)), it is necessary to analyze the error behaviourof the methods. In fact, many papers have investigated the local and global error behaviour ofDDE solvers (cL[1,2,14]). These error analyses are based on the assumption that the fUnctionf(t,y,z) satisfies Lipschitz conditions in both the last two variables. They are suitable fornonstiff …     

Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations

The characteristic equation of a system of delay differential equations (DDEs) is a nonlinear equation with infinitely many zeros. The stability of a steady state solution of such a DDE system is determined by the number of zeros of this equation with positive real part. We present a numerical algorithm to compute the rightmost, i.e., stability determining, zeros of the characteristic equation. The algorithm is based on the application of subspace iteration on the time integration operator of the system or its variational equations. The computed zeros provide insight into the system’s behaviour, can be used for robust bifurcation detection and for efficient indirect calculation of bifurcation points. This revised version was published online in June 2006 with corrections to the Cover Date.     

Shape Optimal Design Problem with Convective and Radiative Heat Transfer: Analysis and Implementation

We present a study of an optimal design problem for a coupled system, governed by a steady-state potential flow equation and a thermal equation taking into account radiative phenomena with multiple reflections. The state equation is a nonlinear integro-differential system. We seek to minimize a cost function, depending on the temperature, with respect to the domain of the equations. First, we consider an optimal design problem in an abstract framework and, with the help of the classical adjoint state method, give an expression of the cost function differential. Then, we apply this result in the two-dimensional case to the nonlinear integro-differential system considered. We prove the differentiability of the cost function, introduce the adjoint state equation, and give an expression of its exact differential. Then, we discretize the equations by a finite-element method and use a gradient-type algorithm to decrease the cost function. We present numerical results as applied to the automotive industry.     

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

We shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations. We devise a Petrov-Galerkin finite element (FE) interpretation of the BDF method and its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the FE approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation. We also show convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over techniques on global error estimation from FE methods to BDF methods.     

Software for ordinary and delay differential equations: Accurate discrete approximate solutions are not enough

Numerical methods for both ordinary differential equations (ODEs) and delay differential equations (DDEs) are traditionally developed and assessed on the basis of how well the accuracy of the approximate solution is related to the specified error tolerance on an adaptively-chosen, discrete mesh. This may not be appropriate in numerical investigations that require visualization of an approximate solution on a continuous interval of interest (rather than at a small set of discrete points) or in investigations that require the determination of the ‘average’ values or the ‘extreme’ values of some solution components.In this paper we will identify modest changes in the standard error-control and stepsize-selection strategies that make it easier to develop, assess and use methods which effectively deliver approximations to differential equations (both ODEs and DDEs) that are more appropriate for these type of investigations. The required changes will typically increase the cost per step by up to 40%, but the improvements and advantages gained will be significant. Numerical results will be presented for these modified methods applied to two example investigations (one ODE and one DDE).     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

Delay independent stability of linear switching systems with time delay

We consider a switching system with time delay composed of a finite number of linear delay differential equations (DDEs). Each DDE consists of a sum of a linear ODE part and a linear DDE part. We study two particular cases: (a) all the ODE parts are stable and (b) all the ODE parts are unstable and determine conditions for delay independent stability. For case (a), we extend a standard result of linear DDEs via the multiple Lyapunov function and functional methods. For case (b) the standard DDE result is not directly applicable, however, we are able to obtain uniform asymptotic stability using the single Lyapunov function and functional methods.     

A review of theoretical and numerical analysis for nonlinear stiff Volterra functional differential equations

In this review, we present the recent work of the author in comparison with various related results obtained by other authors in literature. We first recall the stability, contractivity and asymptotic stability results of the true solution to nonlinear stiff Volterra functional differential equations (VFDEs), then a series of stability, contractivity, asymptotic stability and B-convergence results of Runge-Kutta methods for VFDEs is presented in detail. This work provides a unified theoretical foundation for the theoretical and numerical analysis of nonlinear stiff problems in delay differential equations (DDEs), integro-differential equations (IDEs), delayintegro-differential equations (DIDEs) and VFDEs of other type which appear in practice.      

Solution operator approximations for characteristic roots of delay differential equations

In this paper a new method for the numerical computation of characteristic roots for linear autonomous systems of Delay Differential Equations (DDEs) is proposed. The new approach enlarges the class of methods recently developed (see [SIAM J. Numer. Anal. 40 (2002) 629; D. Breda, Methods for numerical computation of characteristic roots for delay differential equations: experimental comparison, in: BIOCOMP2002: Topics in Biomathematics and Related Computational Problems at the Beginning of the Third Millennium, Vietri, Italy, 2002, Sci. Math. Jpn. 58 (2) pp. 377–388; D. Breda, The infinitesimal generator approach for the computation of characteristic roots for delay differential equations using BDF methods, Research Report RR2/2002, Department of Mathematics and Computer Science, Università di Udine, Italy, 2002; IMA J. Numer. Anal. 24 (2004) 1; SIAM J. Sci. Comput. (2004), in press]) and in particular it is based on a Runge–Kutta (RK) time discretization of the solution operator associated with the system. Hence this paper revisits the Linear Multistep (LMS) approach presented in [SIAM J. Numer. Anal. 40 (2002) 629] for the multiple discrete delay case and moreover extends it to the distributed delay case. We prove that the method converges with the same order as the underlying RK scheme and illustrate this with some numerical tests that are also used to compare the method with other existing techniques.     

DISSIPATIVITY AND EXPONENTIAL STABILITY OF θ-METHODS FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH A BOUNDED LAG

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small ε > 0. We will study the numerical solution defined by the linear θ-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small ε > 0 if and only if θ = 1.     

D-CONVERGENCE OF ONE-LEG METHODS FOR STIFF DELAY DIFFERENTIAL EQUATIONS

1. IntroductionIn recent yeaJrs, many paPers discussed numerical methods for the solution of delay deential equation (DDE)y,(t) = f(t,y(t),y(t -- T)). (1.1)For linear stability of ntunerical methods, a sedcant nUIner of results have aiready beenfound for both Rase--Kutta methods and linear mchistev mehods (cf[4] [7] [8]).Recently wefurther established the relationship between G-stability and llonhnear stability (cf[3]). Erroranalysis of DDE sobors is another imPortant issue. In faCt, ma…