A Robust Gauss‐Newton Algorithm for the Optimization of Hydrological Models: From Standard Gauss‐Newton to Robust Gauss‐Newton

Model calibration using optimization algorithms is a perennial challenge in hydrological modeling. This study explores opportunities to improve the efficiency of a Newton‐type method by making it more robust against problematic features in models' objective functions, including local optima and...

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Vydané v:	Water resources research Ročník 54; číslo 11; s. 9655 - 9683
Hlavní autori:	Qin, Youwei, Kavetski, Dmitri, Kuczera, George
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Washington John Wiley & Sons, Inc 01.11.2018
Predmet:	algorithm efficiency Algorithms Approximation Case studies Catchments coarse gradient approximation Cost analysis Costs Efficiency Empirical analysis empirical research Exploitation Exploration Hydrologic models hydrological model calibration Hydrology industry Industry standards Mathematical models Modelling Objective function Optimization Optimization algorithms parameter optimization Problem solving reliability‐cost tradeoffs robust Gauss‐Newton algorithm Robustness Sampling Tradeoffs water Watersheds
ISSN:	0043-1397, 1944-7973
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Abstract	Model calibration using optimization algorithms is a perennial challenge in hydrological modeling. This study explores opportunities to improve the efficiency of a Newton‐type method by making it more robust against problematic features in models' objective functions, including local optima and other noise. We introduce the robust Gauss‐Newton (RGN) algorithm for least squares optimization, which employs three heuristic schemes to enhance its exploratory abilities while keeping costs low. The large sampling scale (LSS) scheme is a central difference approximation with perturbation (sampling scale) made as large as possible to capture the overall objective function shape; the best‐sampling point (BSP) scheme exploits known function values to detect better parameter locations; and the null‐space jump (NSJ) scheme attempts to escape near‐flat regions. The RGN heuristics are evaluated using a case study comprising four hydrological models and three catchments. The heuristics make synergistic contributions to overall efficiency: the LSS scheme substantially improves reliability albeit at the expense of increased costs, and scenarios where LSS on its own is ineffective are bolstered by the BSP and NSJ schemes. In 11 of 12 modeling scenarios, RGN is 1.4–18 times more efficient in finding the global optimum than the standard Gauss‐Newton algorithm; similar gains are made in finding tolerable optima. Importantly, RGN offers its largest gains when working with difficult objective functions. The empirical analysis provides insights into tradeoffs between robustness versus cost, exploration versus exploitation, and how to manage these tradeoffs to maximize optimization efficiency. In the companion paper, the RGN algorithm is benchmarked against industry standard optimization algorithms. Key Points Robust Gauss‐Newton (RGN) optimization algorithm is introduced for least squares model calibration RGN employs several heuristics including coarse gradient approximations to favor exploration over exploitation RGN heuristics provide synergistic benefits, increasing overall efficiency by factors of 1.4–18 in 11 of 12 hydrological modeling scenarios
AbstractList	Model calibration using optimization algorithms is a perennial challenge in hydrological modeling. This study explores opportunities to improve the efficiency of a Newton‐type method by making it more robust against problematic features in models' objective functions, including local optima and other noise. We introduce the robust Gauss‐Newton (RGN) algorithm for least squares optimization, which employs three heuristic schemes to enhance its exploratory abilities while keeping costs low. The large sampling scale (LSS) scheme is a central difference approximation with perturbation (sampling scale) made as large as possible to capture the overall objective function shape; the best‐sampling point (BSP) scheme exploits known function values to detect better parameter locations; and the null‐space jump (NSJ) scheme attempts to escape near‐flat regions. The RGN heuristics are evaluated using a case study comprising four hydrological models and three catchments. The heuristics make synergistic contributions to overall efficiency: the LSS scheme substantially improves reliability albeit at the expense of increased costs, and scenarios where LSS on its own is ineffective are bolstered by the BSP and NSJ schemes. In 11 of 12 modeling scenarios, RGN is 1.4–18 times more efficient in finding the global optimum than the standard Gauss‐Newton algorithm; similar gains are made in finding tolerable optima. Importantly, RGN offers its largest gains when working with difficult objective functions. The empirical analysis provides insights into tradeoffs between robustness versus cost, exploration versus exploitation, and how to manage these tradeoffs to maximize optimization efficiency. In the companion paper, the RGN algorithm is benchmarked against industry standard optimization algorithms. Model calibration using optimization algorithms is a perennial challenge in hydrological modeling. This study explores opportunities to improve the efficiency of a Newton‐type method by making it more robust against problematic features in models' objective functions, including local optima and other noise. We introduce the robust Gauss‐Newton (RGN) algorithm for least squares optimization, which employs three heuristic schemes to enhance its exploratory abilities while keeping costs low. The large sampling scale (LSS) scheme is a central difference approximation with perturbation ( sampling scale ) made as large as possible to capture the overall objective function shape; the best‐sampling point (BSP) scheme exploits known function values to detect better parameter locations; and the null‐space jump (NSJ) scheme attempts to escape near‐flat regions. The RGN heuristics are evaluated using a case study comprising four hydrological models and three catchments. The heuristics make synergistic contributions to overall efficiency: the LSS scheme substantially improves reliability albeit at the expense of increased costs, and scenarios where LSS on its own is ineffective are bolstered by the BSP and NSJ schemes. In 11 of 12 modeling scenarios, RGN is 1.4–18 times more efficient in finding the global optimum than the standard Gauss‐Newton algorithm; similar gains are made in finding tolerable optima. Importantly, RGN offers its largest gains when working with difficult objective functions. The empirical analysis provides insights into tradeoffs between robustness versus cost, exploration versus exploitation, and how to manage these tradeoffs to maximize optimization efficiency. In the companion paper, the RGN algorithm is benchmarked against industry standard optimization algorithms. Robust Gauss‐Newton (RGN) optimization algorithm is introduced for least squares model calibration RGN employs several heuristics including coarse gradient approximations to favor exploration over exploitation RGN heuristics provide synergistic benefits, increasing overall efficiency by factors of 1.4–18 in 11 of 12 hydrological modeling scenarios Model calibration using optimization algorithms is a perennial challenge in hydrological modeling. This study explores opportunities to improve the efficiency of a Newton‐type method by making it more robust against problematic features in models' objective functions, including local optima and other noise. We introduce the robust Gauss‐Newton (RGN) algorithm for least squares optimization, which employs three heuristic schemes to enhance its exploratory abilities while keeping costs low. The large sampling scale (LSS) scheme is a central difference approximation with perturbation (sampling scale) made as large as possible to capture the overall objective function shape; the best‐sampling point (BSP) scheme exploits known function values to detect better parameter locations; and the null‐space jump (NSJ) scheme attempts to escape near‐flat regions. The RGN heuristics are evaluated using a case study comprising four hydrological models and three catchments. The heuristics make synergistic contributions to overall efficiency: the LSS scheme substantially improves reliability albeit at the expense of increased costs, and scenarios where LSS on its own is ineffective are bolstered by the BSP and NSJ schemes. In 11 of 12 modeling scenarios, RGN is 1.4–18 times more efficient in finding the global optimum than the standard Gauss‐Newton algorithm; similar gains are made in finding tolerable optima. Importantly, RGN offers its largest gains when working with difficult objective functions. The empirical analysis provides insights into tradeoffs between robustness versus cost, exploration versus exploitation, and how to manage these tradeoffs to maximize optimization efficiency. In the companion paper, the RGN algorithm is benchmarked against industry standard optimization algorithms. Key Points Robust Gauss‐Newton (RGN) optimization algorithm is introduced for least squares model calibration RGN employs several heuristics including coarse gradient approximations to favor exploration over exploitation RGN heuristics provide synergistic benefits, increasing overall efficiency by factors of 1.4–18 in 11 of 12 hydrological modeling scenarios
Author	Kavetski, Dmitri Qin, Youwei Kuczera, George
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Snippet	Model calibration using optimization algorithms is a perennial challenge in hydrological modeling. This study explores opportunities to improve the efficiency...
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SubjectTerms	algorithm efficiency Algorithms Approximation Case studies Catchments coarse gradient approximation Cost analysis Costs Efficiency Empirical analysis empirical research Exploitation Exploration Hydrologic models hydrological model calibration Hydrology industry Industry standards Mathematical models Modelling Objective function Optimization Optimization algorithms parameter optimization Problem solving reliability‐cost tradeoffs robust Gauss‐Newton algorithm Robustness Sampling Tradeoffs water Watersheds
Title	A Robust Gauss‐Newton Algorithm for the Optimization of Hydrological Models: From Standard Gauss‐Newton to Robust Gauss‐Newton
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