31. F. B. Hanson, Vector multiprocessor implementation for stochastic dynamic programming, IEEE Distributed Processing Technical Committee Newsletter, vol. 10, pp. 44-48, 1988.

Brief Abstract: This paper presents some preliminary optimization results for stochastic dynamic programming on the Alliant FX/8 vector multiprocessor.

33. F. B. Hanson, Parallel computation for stochastic dynamic programming: Row versus column code orientation, Proc. 1988 IEEE Conf. on Parallel Processing, Vol. III Algorithms and Applications, pp. 117-119, August 1988.

Brief Abstract: In this paper, presented at the 1988 IEEE Conf. on Parallel Processing, St. Charles, IL, August 1988, it is demonstrated that column oriented multiplications for dynamic programming is not the best performer in a Fortran environment of the Alliant FX/8 vector multiprocessor for a 2 state, 2 control resource application, due to the nonstandard nature of the multiplications in the general, variable coefficient, dynamic programming problem.

36. F. B. Hanson, Stochastic Dynamic Programming: advanced Computing Constructs , in Proc. 28th IEEE Conf. on Decision and Control, vol. 1, pp. 901-903, December 1989.

Brief Abstract: In this paper, presented at the 28th CDC, Tampa, Fl, December 1989, a vector data structure is introduced that paved the way for general state space implementation and for facilitating supercomputing optimizations.

37. F. B. Hanson, Parallel and Vector Computation for Optimal Control Applications , in Proc. 3rd Annual Conference on Aerospace Computational Control, JPL Publication 85-45, Vol. 1, D. E. Bernard and G. K. Man, Editors, Jet Propulsion Laboratory, Pasadena, pp. 184-197, December 1989.

Brief Abstract: In this paper, presented at the 3rd Annual Conf. Aerospace Computational Control, Williamsburg, VA, July 1989, the block decomposition of vector data structure loops is introduced to further enhance performance for 3 state space resource applications.

40. S.-L. Chung and F. B. Hanson, Parallel Optimizations for Computational Stochastic Dynamic Programming , in Proc. 1990 Int. Conf. on Parallel Processing, Vol. III Algorithms and Applications, pp. 254-260, August 1990.

Brief Abstract: In this paper, presented at the 1990 Int. Conf. on Parallel Processing, St. Charles, IL, Aug. 1990, more performance is attained by modified blocking, with more work transformed to inner loops and even greater performance by collapsing loops into vector-concurrent form.

42. S.-L. Chung and F. B. Hanson, Optimization Techniques for Stochastic Dynamic Programming , in Proc. 29th IEEE Conf. on Decision and Control, vol. 4, pp. 2450-2455, December 1990.

Brief Abstract: In this paper was presented at the 29th CDC, Honolulu, HI, December, 1990. More improvements are obtained by more efficient handling of boundary nodes on the Cray X/MP/48 vector supercomputer, while use of the massive memory Cray 2S/4-128 supercomputer permitted the solution of the 4 state space problem.

44. S.-L. Chung, F. B. Hanson and H. Xu, Supercomputer Optimizations for Stochastic Optimal Control Applications , in Proc. 4th NASA Workshop on Computational Control of Flexible Aerospace Systems, NASA Conf. Publ. 10065, Part 1, L. W. Taylor, Editor, NASA Langley Research Center, pp. 57-70, March 1991.

Brief Abstract: In this paper, presented during July 1990 in Williamsberg, multitasking (Cray parallelization) is used on the Cray 2S/4-128 to gain additional performance for the 4 state problem, and the massively data parallel Connection Machine CM-2 is applied to show that the performance stochastic dynamic programming code, in spite of significant conditional dependencies, scales up almost linearly with the number of state dimensions or mesh points per state.

45. F. B. Hanson, Computational Dynamic Programming on a Vector Multiprocessor , IEEE Trans. Automatic Control, vol. 36(4), pp. 507-511, April 1991.

Brief Abstract: In this paper, presented at the First Int. Conf. on Industrial and Applied Math., Paris, July 1987, the fundamental numerical method, with its parallel-vector optimizations, is described and high performance is demonstrated.

46. H. H. Xu, F. B. Hanson and S.-L. Chung, Optimal Data Parallel Methods for Stochastic Dynamic Programming , in Proc. 1991 Int. Conf. on Parallel Processing, Vol. III Algorithms and Applications, pp. 142-146, August 1991.

Brief Abstract: In this paper, presented at the 1991 ICPP meeting, operator decomposition and 1-to-many broadcasting techniques were implemented, helping to significantly improve the mapping of the dynamic programming problem to the CM-2.

47. H. H. Xu, F. B. Hanson and S.-L. Chung, Data Parallel Solutions of Dimensionality Problems in Stochastic Dynamic Programming , in Proc. 30th IEEE Conf. on Decision and Control, vol. 2, pp. 1717-1723, December 1991.

Brief Abstract: In this invited paper, presented at the 1991 CDC meeting, Brighton, UK, December 1991. The broadcasting and decomposition data structures are refined to make additional gains on the Bellman dimensionality problem.

48. F. B. Hanson, D. J. Jarvis and H. H. Xu, Applications of FORALL-Formed Computations in Large Scale Stochastic Dynamic Programming , Proc. Scalable High Perf. Comp. Conf.: SHPCC-92, IEEE Computer Society, pp. 182-185, April 1992.

Brief Abstract: In this paper, for the 1992 Scalable High Performance Computing Conference, the definition and structure of forall-formed computations is introduced to promote the transformation of the computations into optimizing data parallel forall loops.

49. H. H. Xu, D. J. Jarvis and F. B. Hanson, Parallel Data Vault Methods for Larger Scale Stochastic Dynamic Programming , in Proc. 1992 Amer. Control Conf., vol. 1, pp. 142-146, June 1992.

Brief Abstract: In this invited paper, presented at the 1992 ACC meeting, the preliminary development using the operator decomposition to map the problem onto the CM-2 mass storage data vaults was discussed, with the projection that this would enable the computation of six state problems.

50. S.-L. Chung, F. B. Hanson and H. H. Xu, Parallel Stochastic Dynamic Programming: Finite Element Methods , Lin. Alg. Applic., vol. 172, pp. 197-218, July 1992.

Brief Abstract: In this paper, presented in an invited talk at the Second NIU Conference on Linear Algebra, Numerical Linear Algebra and Applications, DeKalb, IL, May 1991, the finite element version of the stochastic dynamic programming code is developed and the storage efficiency advantages over the finite difference version are calculated.

51. C. J. Pratico, F. B. Hanson, H. H. Xu, D. J. Jarvis and M. S. Vetter, Visualization for the Management of Renewable Resources in an Uncertain Environment , Proc. Supercomputing '92 , pp. 258-266, color plates p. 843, November 1992.

Brief Abstract: This paper was presented at the Supercomputing '92 meeting in Minneapolis, MN, November 1992 with actual workstation demonstration was at SIGGRAPH '92 Showcase. Our multidimensional inner-outer world visualization of a three state resource management application in uncertain environments is described, along with state and vector control law views, as well as parameter sensitivity simulated animation.

52. F. B. Hanson, C. J. Pratico, M. S. Vetter, and H. H. Xu, Multidimensional Visualization Applied to Renewable Resource Management, Proc. Sixth SIAM Conference on Parallel Processing for Scientific Computing, vol. 2, pp. 1033-1036, March 1993.

Brief Abstract: This paper was presented at the Sixth SIAM Conference on Parallel Processing for Scientific Computing, Norfork, VA, March 1993, with a more advanced version presented at the 3rd SIAM Conference on Linear Algebra, Signals, Systems, and Control, Seattle, WA, August 1993. This paper describes the port of our graphical visualization system to the NeXT computer and compares the results to the original Silicon Graphics implementation.

53 [D&C93]. K. Naimipour and F. B. Hanson, Convergence of a Numerical Method for the Bellman Equation of Stochastic Optimal Control with Quadratic Costs and Constrained Control , Dynamics and Control, vol. 3(3), pp.237-259, July 1993.

Brief Abstract: In this paper, the uniform numerical convergence mesh ratio criteria, that is so critical to the robust computation with the stochastic dynamic programming approach, is presented.

54 [IFAC93]. F. B. Hanson and K. Naimipour, " Convergence of Numerical Method for Multistate Stochastic Dynamic Programming ," Proc. Int. Federation of Automatic Control 12th World Congress, vol. 9, pp. 501-504, July 1993.

Brief Abstract: This invited paper was presented at the International Federation of Automatic Control 12th World Congress, Sydney, Australia, July 1993. This presents additional results for the multidimensional stochastic dynamics programming corrected convergence problem and robust mesh ratio criterion.

56 [CDC94]. F. B. Hanson and J. J. Westman, " Computational Stochastic Dynamic Programming Problems: Groundwater Quality Remediation, " in Proc. 33th IEEE Conf. on Decision and Control, vol. 1, pp. 455-460, December 1994.

Brief Abstract: This paper will be presented at the 33rd CDC in December at Buena Vista, FL. This is an introduction to the stochastic control model for ground water modeling using stochastic dynamic programming with both state and space dependence.

57 [ICCI95]. F. B. Hanson and J. J. Westman, " Parallel Stochastic Dynamic Programming Applications: Spatial and State Finite Elements, " Proceedings of International Conference on Control and Information, The Chinese University Press, Chinese University of Hong Kong, Sha Tin, Hong Kong, pp. 145-150, June 1995.

Brief Abstract: This paper covers more of the finite element techniques used in the computational stochastic dynamic programming for groundwater applications.

58 [CADS96]. F. B. Hanson, " Techniques in Computational Stochastic Dynamic Programming, " in Stochastic Digital Control System Techniques, within series Control and Dynamical Systems: Advances in Theory and Applications, vol. 76, (C. T. Leondes, Editor), Academic Press, New York, NY, pp. 103-162, Invited Book Chapter, April 1996.

Brief Abstract: This invited book chapter summarizes much of the stochastic dynamic programming accomplishments of my scientific supercomputing group. Our methods are compared to the Markov Chain Approximation methods of Kushner and the Differential Dynamic Programming Method for deterministic problems. ( Click for Technical Abstract and Table of Contents?)

61 [ACC97]. John J. Westman and Floyd B. Hanson, The LQGP problem: a manufacturing application, in Proceedings of 1997 American Control Conference, vol. 1, American Automatic Control Council, pp. 566-570, June 1997. (paper presented in Session WM06 on 04 June 1997; Westman won a Finalist Award of $700 in the Best Student Paper Competition; for color copy of preprint or slides click here).

Abstract: The Linear Quadratic Gaussian Poisson (LQGP) problem denotes an optimal control problem with linear dynamics and quadratic costs with both Gaussian and Poisson noise disturbances. The LQGP problem provides a benchmark model with sufficient complexity while permitting formal solutions for testing both theoretical and computational methods. A numerical example for a multistage manufacturing system application of the LQGP problem is presented for the regular or unconstrained control problem.

62 [D&C97] R. M. Abu-Saris and F. B. Hanson, " Computational Efficient Suboptimal Filter for a Class of Wiener-Poisson Driven Stochastic Processes, " Dynamics and Control, vol. 7, no. 3, pp. 279-292, July 1997.

Abstract: The minimum mean square estimate (MMSE) for a stochastic process driven simultaneously by Wiener and Poisson processes is characterized by an infinite number of stochastic differential equations (even in the simplest linear case), and so is not practically implementable. In this article, a practical approximation to the solution is developed in terms of a computationally suboptimal filter for the estimation problem. Basically, it detects and estimates the Poisson driving process using a Maximum A Posteriori (MAP) criterion, and then reconstructs the entire system state using MMSE applied to a system approximating the original one.

65 [SIAMCCIA98poster]. J. J. Westman and Floyd B. Hanson, "Stochastic Optimal Control for Palliative Treatment of Cancer Using Chemotherapy and Radiotherepy", awarded the Best Poster Presentation at the Fourth SIAM Conference on Control and Its Applications, held May 7-9, 1998 in Jacksonville, Florida; Poster Presentation by Westman.

65 [ACC98]. J. J. Westman and Floyd B. Hanson, " The NLQGP Problem: Application to a Multistage Manufacturing System ," in Proceedings of 1998 American Control Conference, American Automatic Control Council, vol. 2, pp. 1104-1108, June 1998. (paper presented in Session W08 on 24 June 1998; again Westman won a Finalist Award of $700 in the Best Student Paper Competition).

Abstract: The Nonlinear Quadratic Gaussian Poisson (NLQGP) problem denotes an optimal control problem with nonlinear dynamics and quadratic costs with both Gaussian and Poisson noise disturbances. The NLQGP problem provides a comprehensive model for many applications since the noises considered are quite robust and add extra realism to physical models. The problem is examined and is illustrated with an application to a multistage manufacturing system (MMS) in an uncertain environment.

66 [CIMASI98]. Floyd B. Hanson and J. J. Westman, " Nonlinear State Dynamics, Computational Approximations and Multistage Manufacturing System Application ," in Proc. Deuxième Conférence Internationale Sur les Mathématiques Appliquées et les Sciences de l'Ingénieur (CIMASI'98), Ecole Supérierure de Technologie de Casablanca, Morocco, vol. 2, pp. 736-739, 27 October 1998.

Abstract: In this paper, we treat stochastic optimal control problems that are nonlinear in the state, but otherwise are a Linear-Quadratic-Gaussian-Poisson problem in the control (LQGP/U), such that the dynamics are linear in the control with quadratic costs in control. The uncertainty in the environment is modeled by Gaussian noise for continuous background fluctuations and discrete random jumps by Poisson noise. Computational approximations in the dynamic programming approach include least squares equivalent LQGP problems in the state for a state independent approximation to solutions, modified Gauss quadrature to handle arbitrary jump distributions, and multidimensional linear interpolation for Poisson functional terms. The approximations are applied to a multistage manufacturing system (MMS) in an uncertain environment, along with implementation procedures needed to modify the formal theory for hybrid systems.

67 [CDC98]. Daniel L. Kern and Floyd B. Hanson, "Systematic Perturbations of Discrete-Time Stochastic Dynamical Systems," in Proc. 37th IEEE Conference on Decision and Control, pp. 1871-1876, December 1998.

Abstract: The discrete-time stochastic optimal control problem is approximated by a variation of differential dynamic programming with systematic calculations of the perturbations due to small stochastic noise. This problem is related to the dual control aspects of stochastic optimal control problems. The motivation is to correct prior calculations for missing terms and to examine the foundations of the method. The state vector is properly expanded asymptotically, in addition to the control vector, in contrast to previous solutions. Corrections are given for the small noise expansions of the solution.

68 [ACC99k]. Daniel L. Kern and Floyd B. Hanson, " Filtering Approximation Using Systematic Perturbations of a Discrete-Time Stochastic Dynamical System," Proceedings of 1999 American Control Conference, Paper no. ACC99-IEEE0116, pp. 445-449, June 1999.

Abstract: The standard problem of groundwater pollutant remediation by well pumping is modeled as a discrete-time LQG stochastic optimal control problem. The control is approximated by using a variation of differential dynamic programming (DDP) that includes systematic perturbations. Kalman filtering is used to estimate the partially observed state variables in a tractable format. This is a filtering application of the DDP method used by the authors in an earlier perturbation paper.

69 [ACC99w]. J. J. Westman and Floyd B. Hanson, " Computational Method for Nonlinear Stochastic Optimal Control, " Proceedings of 1999 American Control Conference, Paper no. ACC99-IEEE0038, pp. 2798-2802, June 1999.

Abstract: Nonlinear stochastic optimal control problems are treated that are nonlinear in the state dynamics, but are linear in the control. The cost functional is a general function of the state, but the costs are quadratic in the control. The system is subject random fluctuations due to discontinuous Poisson noise that depends on both the state and control, as well as due to discontinuous Gaussian noise. This general framework provides a comprehensive model for numerous applications that are subject to random environments. A stochastic dynamic programming approach is used and the theory for an iterative algorithm is formulated utilizing a least squares equivalent of a genuine LQGP problem to approximate the nonlinear state space dependence of the LQGP problem in control only in order to accelerate the convergence of the nonlinear control problem. A particular contribution of this paper is the treatment of a Poisson jump process that is linear in the control vector, within the context of a nonlinear control problem.

70 [CDC99]. J. J. Westman and Floyd B. Hanson, " State Dependent Jump Models in Optimal Control," in Proceedings of 38th IEEE Conference on Decision and Control, pp. 2378-2384, 07 December 1999.

Abstract: Models of linear and nonlinear optimal control applications are considered in which random discrete jumps in the system are state dependent in both rate and amplitude. These discrete jumps are treated as a Poisson processes in continuous time. This type of random noise allows for greater realism while modeling industrial and natural phenomena in which important changes occur with jumps. Modeling concerns are described and the appropriate modifications are indicated for numerically solving the resulting optimal control problems. Applications to a multistage manufacturing system and to the management of a natural resource under stochastic price fluctuations are used to illustrate this type of dynamical formulation.

71 [IJC00]. J. J. Westman and F. B. Hanson, " Nonlinear State Dynamics: Computational Methods and Manufacturing Application," International Journal of Control, pp. 464-480, April 2000.

Abstract: Stochastic optimal control problems are considered that are nonlinear in the state dynamics, but otherwise are an LQGP problem in the control, i.e., the dynamics are linear in the control vector and the costs are quadratic in the control. In addition the system is randomly perturbed by both continuous Gaussian (G) and discontinuous Poisson (P) noise. The approach to the solution is by way of computational stochastic dynamic programming using a least squares equivalent LQGP problem in the state to accelerate the iterative convergence. Methods are discussed for numerically handling the Poisson integral terms. The methods are illustrated for a multistage manufacturing system (MMS) in an uncertain environment, together with implementation procedures needed to modify the formal general theory.

72 [MTNS00]. F. B. Hanson and J. J. Westman, " Computational Stochastic Multistage Manufacturing Systems with Strikes and Other Adverse Random Events," Proceedings of Mathematical Theory of Networks and Systems, 2000, 10 pages, 19 June 2000.

Abstract: Multistage manufacturing systems (MMS) are models for the assembly of consumable goods. In the simple case, a linear assembly line of workstations, components, or value, are added to the product. Some examples assembly line products are automobiles or printed circuit boards. Production scheduling typically takes in to account workstation repair, failure, and defective pieces as stochastic events, effecting the workstation production rates. The supply routing problem of raw materials is not usually taken into account. However, in this treatment, the effects of strikes and natural disasters, which may affect the routing of raw materials, are considered for the MMS. Numerical results illustrate the optimal control of MMS undergoing strikes, as well as workstation repair and failure.

73 [ACC00]. J. J. Westman and F. B. Hanson, " MMS Production Scheduling Subject to Strikes in Random Environments ," Proceedings of 2000 American Control Conference, Paper no. ACC00-IEEE1327, pp. 2194-2198, 28 June 2000.

Abstract: A multistage manufacturing system (MMS) is the normal paradigm used for the final assembly of a consumable good, for example a car or printed circuit board. A simple MMS consists normally of a linear sequence of workstations at which components or value is added to a product, this is essentially an assembly line. When planning the production scheduling, normally only workstation repair, failure, and defective pieces are considered as stochastic events that can affect the production rates for the various workstations. Additionally, it is assumed that all of the raw materials necessary to assemble the finished product are available, and therefore the supply routing problem of raw materials is not considered. In this treatment of the problem, we consider the effects of strikes and of natural disasters. A strike or natural disaster can affect the MMS itself and/or the way in which raw materials are introduced into the MMS that is the supply routing problem is treated as well. A numerical example is presented to illustrate the model that includes a strike as well as workstation repair and failure.

74 [CCA00]. J. J. Westman and Floyd B. Hanson, " Manufacturing Production Scheduling with Preventive Maintenance in Random Environments ," Proceedings of IEEE International Conference on Control and Applications, pp. 582-587, 25 Septmber 2000.

Abstract: Consider the optimal control of a manufacturing system consisting of k stages in which a single consumable good is produced in a random jump environment. At each stage of the manufacturing process there are n workstations that can fail and be repaired. The workstations are assumed to have different operating parameters for a given stage. The mean time to failure for a given workstation, on a given stage, is modeled as a function of the uptime of the workstation. The uptime of the workstations is a monotone increasing function, which can be reset to a lower level by preventive maintenance. This formulation combines features of flexible and multistage manufacturing systems. The goal is to schedule the production of the consumable good subject to random effects and preventive maintenance.

75 [ACC01mfg]. J. J. Westman, F. B. Hanson and E. K. Boukas, " Optimal Production Scheduling for Manufacturing Systems with Preventive Maintenance in an Uncertain Environment ," Proceedings of 2001 American Control Conference, pp. 1375-1380, 25 June 2001.

Abstract: Consider a manufacturing system in which a single consumable good is fabricated in a process that consists of $k$ stages in an uncertain environment. On each stage, there are a number of workstations that are assumed to have different operating parameters that are subject to failure, repair, and preventive maintenance which generate discrete jumps in the value of the state. A Just-In-Time manufacturing discipline is assumed for the workstations with running costs that include penalties for shortfall and surplus production. The formulation presented here for the optimal production scheduling for the manufacturing system requires extensions to the results of the LQGP problem with state dependent Poisson processes (SDPP) by the inclusion of coefficients for the dynamics and the costs that are parameterized by the value of the state. The cost functional used is fully quadratic which is an enhancement for the LQGP problem. The functionality of this canonical model is demonstrated with a numerical example.

76 [ACC01fin]. F. B. Hanson and J. J. Westman, " Optimal Consumption and Portfolio Policies for Important Jump Events: Modeling and Computational Considerations ," Proceedings of 2001 American Control Conference, pp.4456-4661, 25 June 2001.

Abstract: While the volatility of portfolios are often modeled by continuous Brownian motion processes, discontinuous jump processes are more appropriate for modeling important external events that significantly affect the prices of financial assets. Here the discontinuous jump processes are modeled by state and control dependent compound Poisson processes, such that the random jumps come at the times of a pure Poisson process with jump amplitudes that are randomly distributed. The optimal consumption and investment portfolio policy formulation is in terms of stochastic differential equations with optimal discounted utility objectives. This paper was motivated by a recent paper of Rishel (1999) concerning portfolio optimization when prices are dependent on external events. However, the model has been significantly generalized for realistic computational considerations.

77 [CDC01]. J. J. Westman, E. K. Boukas and F. B. Hanson, " Optimal Production Scheduling for Manufacturing Systems with Preventive Maintenance and Finite Buffers," Proceedings of 2001 Conference on Decision and Control, pp. 2343-2348, 04 December 2001.

Abstract: Consider the production of a single consumable product that is fabricated in a process of $k$ stages that is subject to an uncertain environment. There are a number of workstations on each stage that have different operating parameters. The workstations are subject to the discrete events of repair, failure, and preventive maintenance that generate a jump in the state of the system. Between each stage of the manufacturing process is a finite buffer that holds pieces before they can be processed by the next stage. If a buffer is full, then the preceding stage cannot produce pieces since there will be no place for them to go. This formulation of a manufacturing system is a hybrid system consisting of the meeting the global production demand while locally managing the operational status of the workstations. This formulation of the optimal scheduling of production is a quasi-LQGP problem whose jumps are generated by State Dependent Poisson Processes (SDPP). A numerical example is presented to illustrate the model.

78 [ACC02]. F. B. Hanson and J. J. Westman, " Optimal Consumption and Portfolio Control for Jump-Diffusion Stock Process with Log-Normal Jumps (corrected), " Proceedings of 2002 American Control Conference, pp. 4256-4261, 8 May 2002.

Abstract: A jump-diffusion with log-normal jumps is presented. The distribution of the changes in the logarithm of the price of the financial instrument has heavier tails and a higher mode than the usual geometric Brownian motion. Such a distribution is called leptokurtic and is more characteristic of real market distributions than the corresponding log-normal distribution. In addition, this jump-diffusion return distribution can be negatively skewed as is real market returns distribution. The geometric Gaussian mark jump-diffusion process is modeled, basic properties are derived and applied to the problem of portfolio and consumption control optimization.

79 [CMS02]. Daniel L. Kern, J. J. Westman and Floyd B. Hanson, " Locally Optimal Pumping and Treatment Rates in Uncertain Environments ," in Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment, AMS Contempory Mathematics, vol. 295, edited by Zhangxin Chen and Richard E. Ewing, pp. 305-315, July 2002.

Abstract: Stochastic control theory can be applied to the problem of groundwater remediation in order to determine optimal pumping and treatment rates in an uncertain environment. The approach utilizes stochastic dynamic programming that optimizes a cost functional for treatment subject to the constraints of the contaminant transport equations via a search over the state-control space. However, this approach for even small problems requires huge amounts of memory to solve numerically and is exceptionally prohibitive even using finite elements for the state-control space and time large jumps present. The approach here is to reduce the computational complexity by casting the contaminant flow equations to be linear locally while allowing for jumps and small background fluctuations in the dynamical system. The quasi-LQGP problem can then be used to determine the local optimal rates for pumping and treating, which does not require prohibitive amounts of memory. This linearized model can then be used as the starting iterate to hopefully accelerate the solution to the true nonlinear system for the contaminant transport equations.

80 [KU02FM]. Floyd B. Hanson and J. J. Westman, " Stochastic Analysis of Jump-Diffusions for Financial Log-Return Processes (corrected)," in Stochastic Theory and Control, Proceedings of a Workshop held in Lawrence, Kansas, October 18-20, 2001, Lecture Notes in Control and Information Sciences, B.Pasik-Duncan (Editor), Springer-Verlag, New York, pp. 169-184, 24 July 2002 (invited paper).

Abstract: A jump-diffusion log-return process with log-normal jump amplitudes is presented. The probability density and other properties of the theoretical model are rigorously derived. This theoretical density is fit to empirical log-returns of Standard & Poor's 500 stock index data. The model repairs some failures found from the log-normal distribution of geometric Brownian motion to model features of realistic financial instruments: (1) No large jumps or extreme outliers, (2) Not negatively skewed such that the negative tail is thicker than the positive tail, and (3) Non-leptokurtic due to the lack of thicker tails and higher mode.

83 [MTNS02FMB]. F. B. Hanson and J. J. Westman, " Computational Methods for Portfolio and Consumption Optimization in Log-Normal Diffusion, Log-Uniform Jump Environments," Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, 9 pages, 12 August 2002 (invited paper).

Abstract: Computational methods for a jump-diffusion portfolio optimization application using a log-uniform jump distribution are considered. In contrast to the usual geometric Brownian motion problem based upon two parameters, mean appreciation and diffusive volatility, the jump-diffusion model will have at least five, since jump process needs at least a rate, a mean and a variance, depending on the jump-amplitude distribution. As the number number of parameters increases, the computational complexity of the problem of determining the parameter set of the underlying model becomes greater. In a companion stochastic parameter estimation paper, real market data, here a decade of log-returns for Standard and Poor's 500 index closings, is used to fit the jump-diffusion parameters, with constraints based on matching the data mean and variance to keep the unconstrained parameter space to 3 dimensions. A weighted least squares method has been used. The jump-diffusion theoretical distribution and weights has been derived. In this computational paper, the computational features of a new multidimensional, derivative-less global search method used in the companion paper are discussed. The main part of this paper is to discuss the computational solution of an optimal portfolio and consumption finance application with these more realistic parameter results. The constant relative risk aversion (CRRA) canonical model is used to reduce the high dimensionality of the PDE of stochastic dynamic programming problem to something more reasonable. Many computational issues arise due to the jump process part of the model, since several jump integrals arise which are not present in the pure diffusion with drift model. The log-uniformly distributed jumps allow a wider range of portfolio policies than does previous work with normally distributed jumps.

84 [MTNS02FMT]. F. B. Hanson and J. J. Westman, " Jump-Diffusion Stock Return Models in Finance: Stochastic Process Density with Uniform-Jump Amplitude," Proceedings of the 15th International Symposium on Mathematical Theory of Networks and Systems, 7 pages, 12 August 2002 (invited paper).

Abstract: The stochastic analysis is presented for the parameter estimation problem for fitting a theoretical jump-diffusion model to the log-returns from closing data of the Standard and Poor's 500 (S\&P500) stock index during the prior decade 1992-2001. The jump-diffusion model combines a the usual geometric Brownian motion for the diffusion and a space-time Poisson process for the jumps such that the jump amplitudes are uniformly distributed. The uniform jump distribution accounts for the rare large outlying log-returns, both negative and positive in magnitude. The log-normal, log-uniform jump-diffusion density is derived, leading to a jump-diffusion simulator approximation for the case the the log-return time is a small fraction of a year. There are five jump-diffusion parameters that need to be determined, the means and variances for both diffusion and jumps, as well as the jump rate, given the average log-return time. A weighted least squares is used to fit the theoretical jump-diffusion model to the S\&P500 data optimizing with respect to three free parameters, with the two other parameters constrained by the mean and variance of the S\&P500 data. The weight distribution derives from stochastic methods. The ideal fitted model determines the three free parameters, but the corresponding simulated results resemble the original S\&P500 data better. This stochastic analysis paper is a companion to a computational methods and portfolio optimization paper at this conference

85 [CDC02]. F. B. Hanson and J. J. Westman, " Portfolio Optimization with Jump--Diffusions: Estimation of Time-Dependent Parameters and Application," Proceedings of 2002 Conference on Decision and Control, pp. 377-382, 09-13 December 2002, invited paper CDC02-INV0302.

Abstract: This paper treats jump--diffusion processes in continuous time, with emphasis on the jump--amplitude distributions, developing more appropriate models using parameter estimation for the market in one phase and then applying the resulting model to a stochastic optimal portfolio application in a second phase. The new developments are the use of uniform jump--amplitude distributions and time--varying market parameters, introducing more realism into the application model, a Log--Normal--Diffusion, Log--Uniform--Jump model.

86 [ACC03]. F. B. Hanson and J. J. Westman, " Jump--Diffusion Stock-Return Model with Weighted Fitting of Time-Dependent Parameters," Proceedings of 2003 American Control Conference, pp. 4869-4874, 04 June 2003.

Abstract: This paper treats jump-diffusion processes in continuous time, with emphasis on the jump-amplitude distributions, developing more appropriate models using parameter estimation for the market. The proposed method of parameter estimation is weighted least squares of the difference between theoretical and experimental bin frequencies, where the weights or reciprocal variances are chosen as by the theory of jump-diffusion simulation applied to bin frequencies. The empirical data is taken from a decade of Standard & Poor 500 index of stock closings and are viewed as one moderately large simulation. The new developments are the combined use of uniform jump-amplitude distributions, least squares weights and time-varying market parameters, introducing more realism into the model, a Log-Normal-Diffusion, Log-Uniform-Jump financial market model. The optimal parameter estimation is highly nonlinear, computationally intensive, and the optimization is with respect to the three parameters of the log-uniform jump distribution, while the diffusion parameters are constrained by the first two moments of the Standard & Poor's data.

88 [CDC03]. Floyd B. Hanson, Computational Stochastic Control: Basic Foundations, Complexity and Techniques, Proceedings of 2003 Conference on Decision and Control, Invited Poster/Interactive Paper in a Control Education Session, pp. 3024-3029, December 2003.

Longer 9 page Submitted Version.

Abstract: Much research in control systems is purely mathematical, but advances in stochastic control problem solving can be used beyond the limits of where theoretical mathematics can help. Theoretical and computational mathematics are complementary. Computation is important where the problem is mathematically intractable, of high dimension as in stochastic dynamic programming or solving the problem is urgent as in competitive financial engineering predictions. Many advances in solving large scale control problems have been gained through technical improvements in computing hardware, but as many advances have been made in the development of new and better algorithms, the theoretical side of computation. Both analysis and computation are important in solving problems. Both rely on mathematics, but rely on them in different ways. An important part of educational training is general preparation for problem solving since the postgraduate job is uncertain in the current world.

In this expository paper, a selection of basic computational considerations, high performance computers and some useful algorithms are surveyed. Some of the computational methodology in both algorithms and advanced computers arose from the author's own research. Much of the knowledge has been transferred to classes in computation and control, so that student instruction is at the leading edge, a buffer against obsolescence. An important general lesson in computational education is that the computation, if done properly, forms the other half of mathematics, beyond the topics of regular or traditional mathematics courses. Computation has its own algebra, oriented to finite precision arithmetic, and its own analysis that is numerically oriented. The view is that of an applied analyst and computational control scientist explaining the field to those with a regular mathematics background.

89 [TAC04]. Floyd B. Hanson and John J. Westman, "Optimal Portfolio and Consumption Policies Subject to Rishel's Important Jump Events Model: Computational Methods," Trans. Automatic Control, vol. 49, no. 3, Special Issue on Stochastic Control Methods in Financial Engineering, pp. 326-337, March 2004.

Abstract: At important events or announcements, there can be large changes in the value of financial portfolios. Events and their corresponding jumps can occur at random or scheduled times. However, the amplitude of the response in either case can be unpredictable or random. While the volatility of portfolios are often modeled by continuous Brownian motion processes, discontinuous jump processes are more appropriate for modeling the response to important external events that significantly affect the prices of financial assets. Discontinuous jump processes are modeled by compound Poisson processes for random events or by quasi-deterministic jump processes for scheduled events. In both cases, the responses are randomly distributed and are modeled in a stochastic differential equation formulation. The objective is the maximal, expected total discounted utility of terminal wealth and instantaneous consumption. This paper was motivated by a paper of Rishel (1999) concerning portfolio optimization when prices are dependent on external events. However, the model has been significantly generalized for more realistic computational considerations with constraints and parameter values. The problem is illustrated for a canonical risk-adverse power utility model, but the usual explicit canonical solution is not strictly valid. Fortunately, iterations about the canonical solution result in computationally feasible approximations.

90 [CSM04]. Molly H. Shor and Floyd B. Hanson, " Bringing Control to Students and Teachers," Control Systems Magazine, vol. 24, no. 3, pp. 20-30, June 2004.

Abstract: CDC03 Conference Report on ``Ideas and Technology of Control Systems: NSF Workshop for Middle School and High School Students and Teachers'' in Maui, Hawaii in December 2003. See CSM issue for figures and sidebars.

91 [SB03/CM04]. Floyd B. Hanson, John J. Westman and Zongwu Zhu, " Maximum Multinomial Likelihood Estimation of Market Parameters for Stock Jump-Diffusion Models," in Proc. 2003 AMS/IMS/SIAM Summer Research Conference on Mathematics of Finance, AMS Contemporary Mathematics, pp.~1-15, December 2003, invited paper to appear 2004.

Abstract: The estimated parameters of the log-return density for log-normal-diffusion, log-uniform jump process are found for an observed financial market distribution. When the observed data is collected into bins, it is shown that the appropriate parameter estimation method is the multinomial maximum likelihood estimation. This result is independent of the theoretical distribution, since it is only assumed that the observed distribution is the simulation of independent, identically distributed random variables. For the application to the theoretical jump-diffusion distribution, the estimation procedure is constrained by forcing the first two moments of the theoretical distribution to be the same as that for observed market distribution. The Standard and Poor's 500 stock index for the 1992-2001 decade is used as the observed market data. Numerically, the classical Nelder-Mead and our own direct search method are used to find the maximum likelihood estimation of the parameters. The results and performance of these numerical methods are compared along with the our older weighted least squares estimation method. The results of the two numerical approximations for the multinomial estimation methods were similar, but the weighted least squares results are not as good. In the severe test on the third and fourth moment measures, the multinomial based methods differed significantly from the same measures on the observed data, but did much better than the normal distribution based weighted least squares.

92 [CDC04]. Floyd B. Hanson and Zongwu Zhu, " Comparison of Market Parameters for Jump-Diffusion Distributions Using Multinomial Maximum Likelihood Estimation," Proceedings of 43th IEEE Conference on Decision and Control, pp. 1-6, invited paper, 13 December 2004.

Abstract: Previously, we have shown that the proper method for estimating parameters from discrete, binned stock log returns is the multinomial maximum likelihood estimation, and its performance is superior to the method of least squares. Useful formulas have been derived for the jump-diffusion distributions. Numerically, the parameter estimation can be a large scale nonlinear optimization, but we have used techniques to reduce the computation demands of multi-dimensional direct search. In this paper, three jump-diffusion models using different jump-amplitude distributions are compared. These are the normal, uniform and double-exponential. The parameters of all three models are fit to the Standard and Poor's 500 log-return market data, constrained by the data first and second moments. While the results for the skew and kurtosis moments are mixed, the uniform jump distribution has superior qualitative performance since it produces genuine fat tails that are typical of market data, whereas the others have exponentially thin tails. However, the log-normal model has a big advantage in computational time of parameter estimation compared with the others, while the double-exponential is most costly due to having one more model parameter.

93 [ACC05] Siddhartha P. Chakrabarty and Floyd B. Hanson, " Optimal Control of Drug Delivery to Brain Tumors for a Distributed Parameters Model," Proceedings of 2005 American Control Conference, pp. 973-978, June 2004.

Abstract: The growth and treatment of brain tumors is mathematically examined using a distributed parameters model. The model is a system of three coupled reaction diffusion equations involving the tumor cells, normal tissue and the drug concentration. An optimal control problem is designed, with the drug delivery rate as the control and solved to obtain the state and co-state equations as well as the regular control using a modified double shot forward-backward method. This gives rise to a coupled system of equations with a forward state equation and a backward co-state equation, which is solved using a double shot method. A numerical procedure based upon the Crank-Nicolson method is used to solve the coupled system of two three-dimension partial differential equations.

94 [CDC05hc] Floyd B. Hanson and Siddhartha P. Chakrabarty, " Optimal Control of Drug Delivery to Brain Tumors for a PDE Driven Model Using the Galerkin Finite Element Method ," Proceedings of Joint 44th IEEE Conference on Decision and Control and European Control Confence, pp. 1613-1618, 12 December 2005.

Abstract: The Galerkin finite element method is used to examine the optimal drug delivery to brain tumors. The PDE driven mathematical model is a system of three coupled reaction diffusion equations involving the tumor cells, the normal tissue and the drug concentration. An optimal control problem is formulated keeping in mind the primary goals of the treatment, i.e., minimizing the tumor cell density and reducing the side effects of drugs. A distributed parameter method based on application of variational calculus to a pseudo-Hamiltonian, is used to obtain a coupled system of forward state equations and backward co-state equations. The Galerkin form of the finite element method is used due to its greater facility in numerically representing complex structures such as those in the brain. Finally, a two-dimensional circular disk test case is considered and partitioned into a set of rectangular finite elements in polar coordinates, with bilinear basis functions in the interior, but linear-quadratic basis function for elements adjacent to the boundary to exactly satisfy the no-flux boundary conditions.

95 [CDC05zh]. Zongwu Zhu and Floyd B. Hanson, " A Monte-Carlo Option-Pricing Algorithm for Log-Uniform Jump-Diffusion Model , Proceedings of Joint 44th IEEE Conference on Decision and Control and European Control Conference, pp. 5221-5226, 12 December 2005.

Abstract: A reduced European call option pricing formula by risk-neutral valuation is given. It is shown that the European call and put options for jump-diffusion models are worth more than that for the Black-Scholes (diffusion) model with the common parameters. Due to the complexity of the jump-diffusion models, obtaining a closed option pricing formula like that of Black-Scholes is not viable. Instead, a Monte Carlo algorithm is used to compute European option prices. Monte Carlo variance reduction techniques such as both antithetic and control variates are used. The numerical results show that this is a practical, efficient and easily implementable algorithm.

97 [Sethi06] Zongwu Zhu and Floyd B. Hanson, " Optimal Portfolio Application with Double-Uniform Jump Model," Stochastic Processes, Optimization, and Control Theory:Applications in Financial Engineering, Queueing Networks and Manufacturing Systems/A Volume in Honor of Suresh Sethi, International Series in Operations Research & Management Sciences, Vol. 94, H. Yan, G. Yin, Q. Zhang (Eds.), Springer Verlag, New York, pp. 331-358, Invited chapter, June 2006.

Abstract: This paper treats jump-diffusion processes in continuous time, with emphasis on the jump-amplitude distributions, developing more appropriate models using parameter estimation for the market in one phase and then applying the resulting model to a stochastic optimal portfolio application in a second phase. The new developments are the use of double-uniform jump-amplitude distributions and time-varying market parameters, introducing more realism into the application model -- a log-normal diffusion, log-double-uniform jump-amplitude model. Although unlimited borrowing and short-selling play an important role in pure diffusion models, it is shown that borrowing and shorting is limited for jump-diffusions, but finite jump-amplitude models can allow very large limits in contrast to infinite range models which severely restrict the instant stock fraction to [0,1]. Among all the time-dependent parameters modeled, it appears that the interest and discount rate have the strongest effects.

98 [CDC06] Siddhartha P. Chakrabarty and Floyd B. Hanson, " Cancer Drug Delivery in Three Dimensions For a Distributed Parameter Control Model Using Finite Elements ," Proceedings of 45th Conference on Decision and Control , pp.~1-6, San Diego, CA, 13-15 December 2006.

Abstract: Optimal distributed parameter control and a Galerkin finite element method are used to develop procedures for an ideal model of optimal drug delivery to brain tumors. The mathematical model comprises of a system of three coupled reaction diffusion models, involving the density of tumor cells, normal tissue and also the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameter method based on a simple application of variational calculus is used on a pseudo-Hamiltonian, which is then used to obtain a coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure. Finally, an ideal three dimensional test case is considered and partitioned into a set of brick finite elements in spherical coordinates, with tri-linear basis functions. Non-uniqueness of nodes in spherical coordinates is removed by combining like nodes, such as at the origin, at the poles and at the polar angle discontinuity. The Galerkin ODEs are solved by a combination of Crank-Nicolson and predictor-corrector methods.

100 [SIAMbook]. Floyd B. Hanson, Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis and Computation, SIAM Books: Advances in Design and Control Series, Order Code DC13, 28 + 441 pages, published 03 October 2007.

101 [MBS09]. Siddhartha P. Chakrabarty and Floyd B. Hanson, "Distributed Parameters Deterministic Model for Treatment of Brain Tumors Using Galerkin Finite Element Method ," Mathematical Biosciences, pp. 1-54, accepted for publication, 27 March 2009.

Abstract: In this paper we present a distributed parameters deterministic model for treatment of brain tumors using Galerkin finite element method. The dynamic model comprises of system of three coupled reaction diffusion models, involving the tumor cells, the normal tissues and the drug concentration. An optimal control problem is formulated with the goal of minimizing the tumor cell density and reducing the side effects of the drug. A distributed parameters method based on the application of variational calculus is used on an integral-Hamiltonian, which is then used to obtain an optimal coupled system of forward state equations and backward co-state equations. The Galerkin finite element method is used to realistically represent the brain structure as well as to facilitate computation. Finally a three dimensional test case is considered and partitioned into a set of spherical finite elements, using tri-linear basis functions, except for the elements effected by singularities of polar and azimuthal angles, as well the origin.