Optimization of model parameters and experimental designs with the Optimal Experimental Design Toolbox (v1.0) exemplified by sedimentation in salt marshes

Abstract. The geosciences are a highly suitable field of application for optimizing model parameters and experimental designs especially because many data are collected. In this paper, the weighted least squares estimator for optimizing model parameters is presented together with its asymptotic properties. A popular approach to optimize experimental designs called local optimal experimental designs is described together with a lesser known approach which takes into account the potential nonlinearity of the model parameters. These two approaches have been combined with two methods to solve their underlying discrete optimization problem. All presented methods were implemented in an open-source MATLAB toolbox called the Optimal Experimental Design Toolbox whose structure and application is described. In numerical experiments, the model parameters and experimental design were optimized using this toolbox. Two existing models for sediment concentration in seawater and sediment accretion on salt marshes of different complexity served as an application example. The advantages and disadvantages of these approaches were compared based on these models. Thanks to optimized experimental designs, the parameters of these models could be determined very accurately with significantly fewer measurements compared to unoptimized experimental designs. The chosen optimization approach played a minor role for the accuracy; therefore, the approach with the least computational effort is recommended.

[1]  Changbao Wu,et al.  Asymptotic Theory of Nonlinear Least Squares Estimation , 1981 .

[2]  W. V. Leussen,et al.  The variability of settling velocities of suspended fine-grained sediment in the Ems estuary , 1999 .

[3]  S. Stigler.  Gauss and the Invention of Least Squares , 1981 .

[4]  M. Pejrup,et al.  Factors controlling the field settling velocity of cohesive sediment in estuaries , 2010 .

[5]  H. White,et al.  NONLINEAR REGRESSION ON CROSS-SECTION DATA , 1980 .

[6]  D. Shanno.  Conditioning of Quasi-Newton Methods for Function Minimization , 1970 .

[7]  M. M. E. Abd El-Monsef,et al.  Relationships among Several Optimality Criteria , 2009 .

[8]  T. Burt.  Field Settling Velocities of Estuary Muds , 1986 .

[9]  H. L. Miller,et al.  Climate Change 2007: The Physical Science Basis , 2007 .

[10]  Kjell Johnson,et al.  Nonlinear Regression Models , 2013 .

[11]  C. G. Broyden.  The Convergence of a Class of Double-rank Minimization Algorithms 2. The New Algorithm , 1970 .

[12]  Jorge J. Moré,et al.  The Levenberg-Marquardt algo-rithm: Implementation and theory , 1977 .

[13]  T. Coleman,et al.  On the Convergence of Reflective Newton Methods for Large-scale Nonlinear Minimization Subject to Bounds , 1992 .

[14]  T. Amemiya.  Non-linear regression models , 1983 .

[15]  Stephen J. Wright,et al.  Numerical Optimization (Springer Series in Operations Research and Financial Engineering) , 2000 .

[16]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[17]  Eric Walter,et al.  Identification of Parametric Models: from Experimental Data , 1997 .

[18]  E. Malinvaud.  The Consistency of Nonlinear Regressions , 1970 .

[19]  R. Krone.  A Method for Simulating Historic Marsh Elevations , 1987 .

[20]  Andrew Curtis,et al.  Iteratively constructive sequential design of experiments and surveys with nonlinear parameter-data relationships , 2009 .

[21]  C. R. Rao,et al.  Information and the Accuracy Attainable in the Estimation of Statistical Parameters , 1992 .

[22]  T. Wilbanks,et al.  Contribution of Working Group II to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change , 2007 .

[23]  Patrick Meire,et al.  Modelling estuarine variations in tidal marsh sedimentation: response to changing sea level and suspended sediment concentrations , 2004 .

[24]  R. Jennrich.  Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .

[25]  Thomas Slawig,et al.  Modeling the influence of changing storm patterns on the ability of a salt marsh to keep pace with sea level rise , 2013 .

[26]  Thomas F. Coleman,et al.  On the convergence of interior-reflective Newton methods for nonlinear minimization subject to bounds , 1994, Math. Program..

[27]  Kenneth Levenberg.  A METHOD FOR THE SOLUTION OF CERTAIN NON – LINEAR PROBLEMS IN LEAST SQUARES , 1944 .

[28]  R. Fletcher,et al.  A New Approach to Variable Metric Algorithms , 1970, Comput. J..

[29]  P. Meirec,et al.  Modelling estuarine variations in tidal marsh sedimentation : response to changing sea level and suspended sediment concentrations , 2004 .

[30]  Stefan Körkel,et al.  Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen , 2002 .

[31]  D. Goldfarb.  A family of variable-metric methods derived by variational means , 1970 .

[32]  Johannes P. Schlöder,et al.  Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes , 2004, Optim. Methods Softw..

[33]  Patrick Meire,et al.  Modelling long-term tidal marsh growth under changing tidal conditions and suspended sediment concentrations, Scheldt estuary, Belgium , 2003 .

[34]  Thomas F. Coleman,et al.  An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds , 1993, SIAM J. Optim..

[35]  H. White,et al.  Nonlinear Regression with Dependent Observations , 1984 .

[36]  J. Reimer,et al.  Optimization of Model Parameters and Experimental Designs for a Global Marine Biogeochemical Model , 2014 .

[37]  J. Bartholdy,et al.  Storm surge effects on a back-barrier tidal flat of the Danish Wadden Sea , 2001 .