Papers by Jeffrey Baggett
Low-dimensional models of subcritical transition to turbulence
Page 1. ( )124 2!# 74! 7)41)# Jeffrey S. Baggett and Lloyd N. Trefethen Center for Applied Mathem... more Page 1. ( )124 2!# 74! 7)41)# Jeffrey S. Baggett and Lloyd N. Trefethen Center for Applied Mathematics, Cornell University Abstract. In the past five years, working largely inde-pendently, five groups of researchers have proposed ...
Physics of Fluids, 1995
A simple model in three real dimensions is proposed, illustrating a possible mechanism of transit... more A simple model in three real dimensions is proposed, illustrating a possible mechanism of transition to turbulence. The linear part of the model is stable but highly non-normal, so that certain inputs experience a great deal of growth before they eventually decay. The nonlinear terms of the model contribute no energy growth, but recycle some of the linear outputs into inputs, closing a feedback loop and allowing initially small solutions to "bootstrap" to a much larger amplitude. Although different choices of parameters in the nonlinearity lead to a variety of long-term behaviors, the bootstrapping process is essentially independent of the details-of the nonlinearity and varies predictably with the Reynolds number. The bootstrapping scenario demonstrated by this model is the basis of some recent explanations for the failure of classical hydrodynamic stability analysis to predict the onset of turbulence in certain tlow configurations. Q 1995 American Institute of Physics,
Low-dimensional models of subcritical transition to turbulence
Physics of Fluids, 1997
Page 1. ( )124 2!# 74! 7)41)# Jeffrey S. Baggett and Lloyd N. Trefethen Center for Applied Mathem... more Page 1. ( )124 2!# 74! 7)41)# Jeffrey S. Baggett and Lloyd N. Trefethen Center for Applied Mathematics, Cornell University Abstract. In the past five years, working largely inde-pendently, five groups of researchers have proposed ...
A simple model in three real dimensions is proposed, illustrating a possible mechanism of transit... more A simple model in three real dimensions is proposed, illustrating a possible mechanism of transition to turbulence. The linear part of the model is stable but highly non-normal, so that certain inputs experience a great deal of growth before they eventually decay. The nonlinear terms of the model contribute no energy growth, but recycle some of the linear outputs into inputs, closing a feedback loop and allowing initially small solutions to "bootstrap" to a much larger amplitude. Although different choices of parameters in the nonlinearity lead to a variety of long-term behaviors, the bootstrapping process is essentially independent of the details-of the nonlinearity and varies predictably with the Reynolds number. The bootstrapping scenario demonstrated by this model is the basis of some recent explanations for the failure of classical hydrodynamic stability analysis to predict the onset of turbulence in certain tlow configurations. Q 1995 American Institute of Physics,
Environmental Modelling and Software, 2009
This article describes some of the capabilities encapsulated within the Model Independent Calibra... more This article describes some of the capabilities encapsulated within the Model Independent Calibration and Uncertainty Analysis Toolbox (MICUT), which was written to support the popular PEST model independent interface. We have implemented a secant version of the Levenberg-Marquardt (LM) method that requires far fewer model calls for local search than the PEST LM methodology. Efficiency studies on three distinct environmental model structures (HSPF, FASST, and GSSHA) show that we can find comparable local minima with 36-84% fewer model calls than a conventional model independent LM application. Using the secant LM method for local search, MICUT also supports global optimization through the use of a slightly modified version of a stochastic global search technique called Multi-Level
An efficient, advanced regularized inversion method for highly parameterized environmental models
The Levenberg-Marquardt method of computer based parameter estimation can be readily modified in ... more The Levenberg-Marquardt method of computer based parameter estimation can be readily modified in cases of high parameter insensitivity and correlation by the inclusion of various regularization devices to maintain numerical stability and robustness, including; for example, Tikhonov regularization and truncated singular value decomposition. With Tikhonov regularization, where parameters or combinations of parameters cannot be uniquely estimated, they are provided with values or assigned relationships with other parameters that are decreed to be realistic by the modeler. Tikhonov schemes provide a mechanism for assimilation of valuable "outside knowledge" into the inversion process, with the result that parameter estimates, thus informed by a modeler's expertise, are more suitable for use in the making of important predictions by that model than would otherwise be the case. However, by maintaining the high dimensionality of the adjustable parameter space, they can potentially be computational burdensome. Moreover, while Tikhonov schemes are very attractive and hence widely used, problems with numerical stability can sometimes arise because the strength with which regularization constraints are applied throughout the regularized inversion process cannot be guaranteed to exactly complement inadequacies in the information content of a given calibration dataset. We will present results associated with development efforts that include an accelerated Levenberg-Marquardt local search algorithm adapted for Tikhonov regularization, and a technique which allows relative regularization weights to be estimated as parameters through the calibration process itself (Doherty and Skahill, 2006). This new method, encapsulated in the MICUT software (Skahill et al., 2008) will be compared, in terms of efficiency and enforcement of regularization relationships, with the SVD Assist method (Tonkin and Doherty, 2005) contained in the popular PEST package by considering various watershed model calibration problems. Doherty, J., Skahill, B.E., 2006. An Advanced Regularization Methodology for Use in Watershed Model Calibration. Journal of Hydrology, 327, 564- 577. Skahill, B.E., Baggett, J.S., Frankenstein, S., and Downer C.W., 2008. Efficient Levenberg-Marquardt Method Based PEST Compatible Model Independent Calibration. Environmental Modelling and Software (conditionally accepted pending revisions). Tonkin, M. J., Doherty, J., 2005. A hybrid regularized inversion methodology for highly parameterized environmental models. Water Resour. Res., 41, W10412, doi:10.1029/2005WR003995.
PEST Compatible Software for More Efficient Levenburg-Marquardt Method Based Model Calibration: HEC-HMS / Watershed Model Applications for Demonstration
Our independent Levenberg-Marquardt (LM) implementation accommodates the PEST model independent a... more Our independent Levenberg-Marquardt (LM) implementation accommodates the PEST model independent and input control file protocol. First, to reduce the number of model calls needed to find a local minimum we use a combination of Broyden rank one updates (secant method) and central and forward finite differences to update the model sensitivity matrix at each optimization iteration. The exact combination of Broyden rank one updates and finite differences is easily specified by the user. While PEST Version 11 does include the ability to utilize Broyden updates, that implementation does not realize the complete efficiency gains that are possible. Second, we have added Multi Level Single Linkage (MLSL), a stochastic global optimization algorithm, to our PEST compatible model independent calibration software. MLSL uses the LM algorithm for local search and a minimum distance threshold to avoid repeated visits to the same local minima. The use of our MLSL implementation requires only a minor addition to a PEST input control file. Efficiencies that can be achieved for LM method based model independent calibration from a properly implemented secant version of the LM method will be demonstrated by examining the reduction in the total number of model calls for single HEC-HMS / watershed model inversion runs associated with the use of our independent LM implementation that accommodates the PEST model independent and input control file protocol. Using HEC-HMS / watershed models, we will also compare the efficiencies, in terms of the number of model calls required to achieve a given objective function value, of our implementations of Multistart, Trajectory Repulsion, and MLSL with that of Shuffled Complex Evolution (SCE) and Covariance Matrix Adaption Evolutionary Strategy (CMAES), as interfaced to PEST.
More efficient evolutionary strategies for model calibration with watershed model for demonstration
Evolutionary strategies allow automatic calibration of more complex models than traditional gradi... more Evolutionary strategies allow automatic calibration of more complex models than traditional gradient based approaches, but they are more computationally intensive. We present several efficiency enhancements for evolution strategies, many of which are not new, but when combined have been shown to dramatically decrease the number of model runs required for calibration of synthetic problems. To reduce the number of expensive model runs we employ a surrogate objective function for an adaptively determined fraction of the population at each generation (Kern et al., 2006). We demonstrate improvements to the adaptive ranking strategy that increase its efficiency while sacrificing little reliability and further reduce the number of model runs required in densely sampled parts of parameter space. Furthermore, we include a gradient individual in each generation that is usually not selected when the search is in a global phase or when the derivatives are poorly approximated, but when selected near a smooth local minimum can dramatically increase convergence speed (Tahk et al., 2007). Finally, the selection of the gradient individual is used to adapt the size of the population near local minima. We show, by incorporating these enhancements into the Covariance Matrix Adaption Evolution Strategy (CMAES; Hansen, 2006), that their synergetic effect is greater than their individual parts. This hybrid evolutionary strategy exploits smooth structure when it is present but degrades to an ordinary evolutionary strategy, at worst, if smoothness is not present. Calibration of 2D-3D synthetic models with the modified CMAES requires approximately 10%-25% of the model runs of ordinary CMAES. Preliminary demonstration of this hybrid strategy will be shown for watershed model calibration problems. Hansen, N. (2006). The CMA Evolution Strategy: A Comparing Review. In J.A. Lozano, P. Larrañga, I. Inza and E. Bengoetxea (Eds.). Towards a new evolutionary computation. Advances in estimation of distribution algorithms. pp. 75-102, Springer Kern, S., N. Hansen and P. Koumoutsakos (2006). Local Meta-Models for Optimization Using Evolution Strategies. In Ninth International Conference on Parallel Problem Solving from Nature PPSN IX, Proceedings, pp.939-948, Berlin: Springer. Tahk, M., Woo, H., and Park. M, (2007). A hybrid optimization of evolutionary and gradient search. Engineering Optimization, (39), 87-104.
More Efficient Derivative-based Watershed Model Calibration
Recent publications have demonstrated the efficacy of model independent, derivative-based calibra... more Recent publications have demonstrated the efficacy of model independent, derivative-based calibration of watershed models (Skahill and Doherty 2006; Doherty and Skahill 2006; Guti¨¦rrez-Magness and McCuen 2005). In the works referenced above, the Jacobian at each optimization iteration was approximated using software that requires between m and 2m forward model calls (see below). The objective of this work is to describe and demonstrate a potentially more efficient approach to approximating the Jacobian. Let the matrix X represent the action of a linear model. Let the vector p represent its m parameters, the vector h represent the n observations comprising the calibration dataset, and the n-dimensional vector ¦Å represent the observation noise. These quantities are related by: Xp = h + ¦Å. Minimization of a weighted least squares objective function is achieved for p calculated as p = ((XtQX)-1XtQh (1), where Q is proportional to the inverse of the covariance matrix of measurement noise. For a nonlinear model, implementation of equation 1 becomes an iterative process starting from a user- supplied set of initial parameter estimates. Furthermore, the Jacobian matrix replaces X. The nonlinear parameter estimation process then becomes one of successive linearization, requiring construction of the Jacobian matrix at each optimization iteration for computation of the upgrade vector. Although the Levenburg- Marquardt method of computer-based parameter estimation can generally complete a single inversion run with a high level of efficiency, even if the column space of the Jacobian must be populated based on model runs with incrementally varied parameter values, the process of constructing the Jacobian is usually the most computationally demanding aspect of the inversion process. Where model run times are high, model run efficiency of the calibration process becomes of paramount concern. One way to reduce the cost of each optimization iteration is to iteratively update the Jacobian matrix by using a secant approximation to the derivative along each search direction. Over the course of many iterations, the accuracy of the approximated Jacobian may degrade, so the Jacobian is occasionally recalculated, as necessary, using finite differences. As we demonstrate by calibrating two different surface hydrology models, this reduced the total number of model calls by approximately 45 percent relative to conventional updating of the Jacobian, and in each case with no loss in objective function improvement. Skahill, B., and Doherty, J. 2006. Efficient accommodation of local minima in watershed model calibration. Journal of Hydrology (in press). Doherty, J., and Skahill, B.E. 2006. "An Advanced Regularization Methodology for Use in Watershed Model Calibration." Journal of Hydrology, (327), 564¨C 577. Guti¨¦rrez-Magness, A.L., and McCuen, R.H. 2005. "Effect of Flow Proportions on HSPF Model Calibration Accuracy." Journal of Hydrologic Engineering, Vol. 10, No. 5.
Journal of Fluid Mechanics, 1998
Streak breakdown caused by a spanwise inflectional instability is one phase of the following tran... more Streak breakdown caused by a spanwise inflectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette flow. The streamwise vortex scenario is described by
Journal of Fluid Mechanics, 1998
Streak breakdown caused by a spanwise in ectional instability is one phase of the following trans... more Streak breakdown caused by a spanwise in ectional instability is one phase of the following transition scenarios, which occur in plane Poiseuille and Couette ow. The streamwise vortex scenario is described by (SV) streamwise vortices =) streamwise streaks =) streak breakdown =) transition:
Streak Breakdown and Transition to Turbulence in Plane Channel Flows^1
Streak breakdown caused by a spanwise inflectional instability is one phase of the following tran... more Streak breakdown caused by a spanwise inflectional instability is one phase of the following transition scenario: streamwise vortices ==> streamwise streaks ==> streak breakdown ==> transition. This work investigates streak breakdown in plane Poiseuille (PPF) and Couette flow (PCF) by a linear stability analysis and direct numerical simulations. We find that breakdown is confined to disturbances with low streamwise wavenumber. The cutoff is on the order of the spanwise wavenumber of the streaks. Growth rates for instability increase with streak amplitude. Using the linear stability analysis, we find that a lower bound on the threshold vortex amplitude for transition is O(R-1) for PCF and O(R-1.6) for PPF, where R is the Reynolds number footnotetext[1]PJS was supported in part by NSF Grant DMS-9406636 and JSB was supported in part by NSF grant DMS-950090-75CS and DOE grant DE-FG02-94ER25199.
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Papers by Jeffrey Baggett