Parameter Estimation in Chaotic Systems

This report examines how to estimate the parameters of a chaotic system given noisy observations of the state behavior of the system. Investigating parameter estimation for chaotic systems is interesting because of possible applications for high-precision measurement and for use in other signal processing, communication, and control applications involving chaotic systems. In this report, we examine theoretical issues regarding parameter estimation in chaotic systems and develop an efficient algorithm to perform parameter estimation. We discover two properties that are helpful for performing parameter estimation on non-structurally stable systems. First, it turns out that most data in a time series of state observations contribute very little information about the underlying parameters of a system, while a few sections of data may be extraordinarily sensitive to parameter changes. Second, for one-parameter families of systems, we demonstrate that there is often a preferred direction in parameter space governing how easily trajectories of one system can "shadow'" trajectories of nearby systems. This asymmetry of shadowing behavior in parameter space is proved for certain families of maps of the interval. Numerical evidence indicates that similar results may be true for a wide variety of other systems. Using the two properties cited above, we devise an algorithm for performing parameter estimation. Standard parameter estimation techniques such as the extended Kalman filter perform poorly on chaotic systems because of divergence problems. The proposed algorithm achieves accuracies several orders of magnitude better than the Kalman filter and has good convergence properties for large data sets.

Preliminary Observations on Program Instability

This white paper reports emerging findings at the end of Phase I of the Lean Aircraft Initiative in the Policy focus group area. Specifically, it provides details about research on program instability. Its objective is to discuss high-level findings detailing: 1) the relative contribution of different factors to a program’s overall instability; 2) the cost impact of program instability on acquisition programs; and 3) some strategies recommended by program managers for overcoming and/or mitigating the negative effects of program instability on their programs. Because this report comes as this research is underway, this is not meant to be a definitive document on the subject. Rather, is it anticipated that this research may potentially produce a number of reports on program instability-related topics. The government managers of military acquisition programs rated annual budget or production rate changes, changes in requirements, and technical difficulties as the three top contributors, respectively, to program instability. When asked to partition actual variance in their program’s planned cost and schedule to each of these factors, it was found that the combined effects of unplanned budget and requirement changes accounted for 5.2% annual cost growth and 20% total program schedule slip. At a rate of approximately 5% annual cost growth from these factors, it is easy to see that even conservative estimates of the cost benefits to be gained from acquisition reforms and process improvements can quickly be eclipsed by the added cost associated with program instability. Program management practices involving the integration of stakeholders from throughout the value chain into the decision making process were rated the most effective at avoiding program instability. The use of advanced information technologies was rated the most effective at mitigating the negative impact of program instability.

Certified Rapid Solution of Parametrized Linear Elliptic Equations: Application to Parameter Estimation

We present a technique for the rapid and reliable evaluation of linear-functional output of elliptic partial differential equations with affine parameter dependence. The essential components are (i) rapidly uniformly convergent reduced-basis approximations — Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N (optimally) selected points in parameter space; (ii) a posteriori error estimation — relaxations of the residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs; and (iii) offline/online computational procedures — stratagems that exploit affine parameter dependence to de-couple the generation and projection stages of the approximation process. The operation count for the online stage — in which, given a new parameter value, we calculate the output and associated error bound — depends only on N (typically small) and the parametric complexity of the problem. The method is thus ideally suited to the many-query and real-time contexts. In this paper, based on the technique we develop a robust inverse computational method for very fast solution of inverse problems characterized by parametrized partial differential equations. The essential ideas are in three-fold: first, we apply the technique to the forward problem for the rapid certified evaluation of PDE input-output relations and associated rigorous error bounds; second, we incorporate the reduced-basis approximation and error bounds into the inverse problem formulation; and third, rather than regularize the goodness-of-fit objective, we may instead identify all (or almost all, in the probabilistic sense) system configurations consistent with the available experimental data — well-posedness is reflected in a bounded "possibility region" that furthermore shrinks as the experimental error is decreased.

Preliminary Observations on Program Instability

This white paper reports emerging findings at the end of Phase I of the Lean Aircraft Initiative in the Policy focus group area. Specifically, it provides details about research on program instability. Its objective is to discuss high-level findings detailing: 1) the relative contribution of different factors to a program’s overall instability; 2) the cost impact of program instability on acquisition programs; and 3) some strategies recommended by program managers for overcoming and/or mitigating the negative effects of program instability on their programs. Because this report comes as this research is underway, this is not meant to be a definitive document on the subject. Rather, is it anticipated that this research may potentially produce a number of reports on program instability-related topics. The government managers of military acquisition programs rated annual budget or production rate changes, changes in requirements, and technical difficulties as the three top contributors, respectively, to program instability. When asked to partition actual variance in their program’s planned cost and schedule to each of these factors, it was found that the combined effects of unplanned budget and requirement changes accounted for 5.2% annual cost growth and 20% total program schedule slip. At a rate of approximately 5% annual cost growth from these factors, it is easy to see that even conservative estimates of the cost benefits to be gained from acquisition reforms and process improvements can quickly be eclipsed by the added cost associated with program instability. Program management practices involving the integration of stakeholders from throughout the value chain into the decision making process were rated the most effective at avoiding program instability. The use of advanced information technologies was rated the most effective at mitigating the negative impact of program instability.