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The skill of global ocean biogeochemical models, and the earth system models in which they are embedded, can be improved by systematic calibration of the parameter values against observations. However, such tuning is seldom undertaken as these models are computationally very expensive. Here we investigate the performance of DFO-LS, a local, derivative-free optimisation algorithm which has been designed for computationally expensive models with irregular model–data misfit landscapes typical of biogeochemical models. We use DFO-LS to calibrate six parameters of a relatively complex global ocean biogeochemical model (MOPS) against synthetic dissolved oxygen, phosphate and nitrate "observations" from a reference run of the same model with a known parameter configuration. The performance of DFO-LS is compared with that of CMA-ES, another derivative-free algorithm that was applied in a previous study to the same model in one of the first successful attempts at calibrating a global model of this complexity. We find that DFO-LS successfully recovers five of the six parameters in approximately 40 evaluations of the misfit function (each one requiring a 3000-year run of MOPS to equilibrium), while CMA-ES needs over 1200 evaluations. Moreover, DFO-LS reached a "baseline" misfit, defined by observational noise, in just 11–14 evaluations, whereas CMA-ES required approximately 340 evaluations. We also find that the performance of DFO-LS is not significantly affected by observational sparsity, however fewer parameters were successfully optimised in the presence of observational uncertainty. The results presented here suggest that DFO-LS is sufficiently inexpensive and robust to apply to the calibration of complex, global ocean biogeochemical models. [ABSTRACT FROM AUTHOR]

The performance of global ocean biogeochemical models, and the Earth System Models in which they are embedded, can be improved by systematic calibration of the parameter values against observations. However, such tuning is seldom undertaken as these models are computationally very expensive. Here we investigate the performance of DFO-LS, a local, derivative-free optimisation algorithm which has been designed for computationally expensive models with irregular model data misfit landscapes typical of biogeochemical models. We use DFO-LS to calibrate six parameters of a relatively complex global ocean biogeochemical model (MOPS) against synthetic dissolved oxygen, inorganic phosphate and inorganic nitrate "observations" from a reference run of the same model with a known parameter configuration. The performance of DFO-LS is compared with that of CMA-ES, another derivative-free algorithm that was applied in a previous study to the same model in one of the first successful attempts at calibrating a global model of this complexity. We find that DFO-LS successfully recovers 5 of the 6 parameters in approximately 40 evaluations of the misfit function (each one requiring a 3000 year run of MOPS to equilibrium), while CMA-ES needs over 1200 evaluations. Moreover, DFO-LS reached a "baseline" misfit, defined by observational noise, in just 11–14 evaluations, whereas CMA-ES required approximately 340 evaluations.We also find that the performance of DFO-LS is not significantly affected by observational sparsity, however fewer parameters were successfully optimised in the presence of observational uncertainty. The results presented here suggest that DFO-LS is sufficiently inexpensive and robust to apply to the calibration of complex, global ocean biogeochemical models. [ABSTRACT FROM AUTHOR]

High-order optimality conditions for convexly constrained nonlinear optimization problems are analysed. A corresponding (expensive) measure of criticality for arbitrary order is proposed and extended to define high-order ϵ-approximate critical points. This new measure is then used within a conceptual trust-region algorithm to show that if derivatives of the objective function up to order q≥1 can be evaluated and are Lipschitz continuous, then this algorithm applied to the convexly constrained problem needs at most O(ϵ-(q+1)) evaluations of f and its derivatives to compute an ϵ-approximate qth-order critical point. This provides the first evaluation complexity result for critical points of arbitrary order in nonlinear optimization. An example is discussed, showing that the obtained evaluation complexity bounds are essentially sharp. [ABSTRACT FROM AUTHOR]

As shown by Blumensath and Davies (2009) and Baraniuk et al. (2010), signals whose wavelet coefficients exhibit a rooted tree structure can be recovered using specially adapted compressed sensing algorithms from just n=\mathcal O(k) measurements, where $k$ is the sparsity of the signal. Motivated by these results, we introduce a simplified proportional-dimensional asymptotic framework, which enables the quantitative evaluation of recovery guarantees for tree-based compressed sensing. In the context of Gaussian matrices, we apply this framework to existing worst-case analysis of the iterative tree projection (ITP) algorithm, which makes use of the tree-based restricted isometry property (RIP). Within the same framework, we then obtain quantitative results based on a new method of analysis, which considers the fixed points of the algorithm. By exploiting the realistic average-case assumption that the measurements are statistically independent of the signal, we obtain significant quantitative improvements when compared with the tree-based RIP analysis. Our results have a refreshingly simple interpretation, explicitly determining a bound on the number of measurements that are required as a multiple of the sparsity. For example, we prove that exact recovery of binary tree-based signals from noiseless Gaussian measurements is asymptotically guaranteed for ITP with constant stepsize provided $n\geq 50k$ . All our results extend to the more realistic case in which measurements are corrupted by noise. [ABSTRACT FROM PUBLISHER]

We present a new recovery analysis for a standard compressed sensing algorithm, Iterative Hard Thresholding (IHT) (Blumensath and Davies, 2008), which considers the fixed points of the algorithm. In the context of arbitrary measurement matrices, we derive a sufficient condition for the convergence of IHT to a fixed point and a necessary condition for the existence of fixed points. These conditions allow us to perform a sparse signal recovery analysis in the deterministic noiseless case by implying that the original sparse signal is the unique fixed point and limit point of IHT, and in the case of Gaussian measurement matrices and noise by generating a bound on the approximation error of the IHT limit as a multiple of the noise level. By generalizing the notion of fixed points, we extend our analysis to the variable stepsize Normalised IHT (Blumensath and Davies, 2010). For both stepsize schemes, we obtain lower bounds on asymptotic phase transitions in a proportional-dimensional framework, quantifying the sparsity/undersampling tradeoff for which recovery is guaranteed. Exploiting the reasonable average-case assumption that the underlying signal and measurement matrix are independent, comparison with previous results within this framework shows a substantial quantitative improvement. [ABSTRACT FROM AUTHOR]

Compressed sensing (CS) seeks to recover an unknown vector with N entries by making far fewer than N measurements; it posits that the number of CS measurements should be comparable to the information content of tile vector, not simply N. CS combines directly the important task of compression with the measurement task. Since its introduction in 2004 there have been hundreds of papers on CS, a large fraction of which develop algorithms to recover a signal from its compressed measurements. Because of the paradoxical nature of CS--exact reconstruction from seemingly undersampled measurements--it is crucial for acceptance of an algorithm that rigorous analyses verify the degree of undersampling the algorithm permits. The restricted isometry property (RIP) has become the dominant tool used for the analysis in such cases. We present here an asymmetric form of RIP that gives tighter bounds than the usual symmetric one. We give the best known bounds on the RIP constants for matrices from the Gaussian ensemble. Our derivations illustrate the way in which the combinatorial nature of CS is controlled. Our quantitative bounds on the RIP allow precise statements as to how aggressively a signal can be undersampled, the essential question for practitioners. We also document the extent to which RIP gives precise information about the true performance limits of CS, by comparison with approaches from high-dimensional geometry. [ABSTRACT FROM AUTHOR]

Abstract: A major enterprise in compressed sensing and sparse approximation is the design and analysis of computationally tractable algorithms for recovering sparse, exact or approximate, solutions of underdetermined linear systems of equations. Many such algorithms have now been proven to have optimal-order uniform recovery guarantees using the ubiquitous Restricted Isometry Property (RIP) (Candès and Tao (2005) ). However, without specifying a matrix, or class of matrices, it is unclear when the RIP-based sufficient conditions on the algorithm are satisfied. Bounds on RIP constants can be inserted into the algorithms RIP-based conditions, translating the conditions into requirements on the signal''s sparsity level, length, and number of measurements. We illustrate this approach for Gaussian matrices on three of the state-of-the-art greedy algorithms: CoSaMP (Needell and Tropp (2009) ), Subspace Pursuit (SP) (Dai and Milenkovic (2009) ) and Iterative Hard Thresholding (IHT) (Blumensath and Davies (2009) ). Designed to allow a direct comparison of existing theory, our framework implies that, according to the best available analysis on these three algorithms, IHT requires the fewest number of compressed sensing measurements, has the best proven stability bounds, and has the lowest per iteration computational cost. [Copyright &y& Elsevier]

Perturbed physics configurations of version 3 of the Hadley Centre Atmosphere Model (HadAM3) driven with observed sea surface temperatures (SST) and sea ice were tuned to outgoing radiation observations using a Gauss-Newton line search optimization algorithm to adjust the model parameters. Four key parameters that previous research found affected climate sensitivity were adjusted to several different target values including two sets of observations. The observations used were the global average reflected shortwave radiation (RSR) and outgoing longwave radiation (OLR) from the Clouds and the Earth's Radiant Energy System instruments combined with observations of ocean heat content. Using the same method, configurations were also generated that were consistent with the earlier Earth Radiation Budget Experiment results. Many, though not all, tuning experiments were successful, with about 2500 configurations being generated and the changes in simulated outgoing radiation largely due to changes in clouds. Clear-sky radiation changes were small, largely due to a cancellation between changes in upper-tropospheric relative humidity and temperature. Changes in other climate variables are strongly related to changes in OLR and RSR particularly on large scales. There appears to be some equifinality with different parameter configurations producing OLR and RSR values close to observed values. These models have small differences in their climatology with the one group being similar to the standard configuration and the other group drier in the tropics and warmer everywhere. [ABSTRACT FROM AUTHOR]

Optimisation methods were successfully used to calibrate parameters in an atmospheric component of a climate model using two variants of the Gauss-Newton line-search algorithm. 1) A standard Gauss-Newton algorithm in which, in each iteration, all parameters were perturbed. 2) A randomized block-coordinate variant in which, in each iteration, a random sub-set of parameters was perturbed. The cost function to be minimized used multiple large-scale observations and was constrained to produce net radiative fluxes close to those observed. These algorithms were used to calibrate the HadAM3 (3rd Hadley Centre Atmospheric Model) model at N48 resolution and the HadAM3P model at N96 resolution. For the HadAM3 model, cases with seven and fourteen parameters were tried. All ten 7-parameter cases using HadAM3 converged to cost function values similar to that of the standard configuration. For the 14-parameter cases several failed to converge, with the random variant in which 6 parameters were perturbed being most successful. Multiple sets of parameter values were found that produced multiple models very similar to the standard configuration. HadAM3 cases that converged were coupled to an ocean model and ran for 20 years starting from a pre-industrial HadCM3 (3rd Hadley Centre Coupled model) state resulting in several models whose global-average temperatures were consistent with pre-industrial estimates. For the 7-parameter cases the Gauss-Newton algorithm converged in about 70 evaluations. For the 14-parameter algorithm with 6 parameters being randomly perturbed about 80 evaluations were needed for convergence. However, when 8 parameters were randomly perturbed algorithm performance was poor. Our results suggest the computational cost for the Gauss-Newton algorithm scales between P and P2 where P is the number of parameters being calibrated. For the HadAM3P model three algorithms were tested. Algorithms in which seven parameters were perturbed and three out of seven parameters randomly perturbed produced final configurations comparable to the standard hand tuned configuration. An algorithm in which six out of thirteen parameters were randomly perturbed failed to converge. These results suggest that automatic parameter calibration using atmospheric models is feasible and that the resulting coupled models are stable. Thus, automatic calibration could replace human driven trial and error. However, convergence and costs are, likely, sensitive to details of the algorithm. [ABSTRACT FROM AUTHOR]

A large number of perturbed-physics simulations of version 3 of the Hadley Centre Atmosphere Model (HadAM3) were compared with the Clouds and the Earth's Radiant Energy System (CERES) estimates of outgoing longwave radiation (OLR) and reflected shortwave radiation (RSR) as well as OLR and RSR from the earlier Earth Radiation Budget Experiment (ERBE) estimates. The model configurations were produced from several independent optimization experiments in which four parameters were adjusted. Model-observation uncertainty was estimated by combining uncertainty arising from satellite measurements, observational radiation imbalance, total solar irradiance, radiative forcing, natural aerosol, internal climate variability, and sea surface temperature and that arising from parameters that were not varied. Using an emulator built from 14 001 'slab' model evaluations carried out using the ensemble, the climate sensitivity for each configuration was estimated. Combining different prior probabilities for model configurations with the likelihood for each configuration and taking account of uncertainty in the emulated climate sensitivity gives, for the HadAM3 model, a 2.5%-97.5% range for climate sensitivity of 2.7-4.2 K if the CERES observations are correct. If the ERBE observations are correct, then they suggest a larger range, for HadAM3, of 2.8-5.6 K. Amplifying the CERES observational covariance estimate by a factor of 20 brings CERES and ERBE estimates into agreement. In this case the climate sensitivity range is 2.7-5.4 K. The results rule out, at the 2.5% level for HadAM3 and several different prior assumptions, climate sensitivities greater than 5.6 K. [ABSTRACT FROM AUTHOR]