Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

Radiative forcing since preindustrial times due to ozone change in the troposphere and the lower stratosphere

Changes in atmospheric ozone have occurred since the preindustrial era as a result of increasing anthropogenic emissions. Within ACCENT, a European Network of Excellence, ozone changes between 1850 and 2000 are assessed for the troposphere and the lower stratosphere ( up to 30 km) by a variety of seven chemistry-climate models and three chemical transport models. The modeled ozone changes are taken as input for detailed calculations of radiative forcing. When only changes in chemistry are considered ( constant climate) the modeled global-mean tropospheric ozone column increase since preindustrial times ranges from 7.9 DU to 13.8 DU among the ten participating models, while the stratospheric column reduction lies between 14.1 DU and 28.6 DU in the models considering stratospheric chemistry. The resulting radiative forcing is strongly dependent on the location and altitude of the modeled ozone change and varies between 0.25 Wm(-2) and 0.45 Wm(-2) due to ozone change in the troposphere and - 0.123 Wm(-2) and + 0.066 Wm(-2) due to the stratospheric ozone change. Changes in ozone and other greenhouse gases since preindustrial times have altered climate. Six out of the ten participating models have performed an additional calculation taking into account both chemical and climate change. In most models the isolated effect of climate change is an enhancement of the tropospheric ozone column increase, while the stratospheric reduction becomes slightly less severe. In the three climate-chemistry models with detailed tropospheric and stratospheric chemistry the inclusion of climate change increases the resulting radiative forcing due to tropospheric ozone change by up to 0.10 Wm(-2), while the radiative forcing due to stratospheric ozone change is reduced by up to 0.034 Wm(-2). Considering tropospheric and stratospheric change combined, the total ozone column change is negative while the resulting net radiative forcing is positive.

Direct radiative effect of aerosols emitted by transport from road, shipping and aviation

Aerosols and their precursors are emitted abundantly by transport activities. Transportation constitutes one of the fastest growing activities and its growth is predicted to increase significantly in the future. Previous studies have estimated the aerosol direct radiative forcing from one transport sub-sector, but only one study to our knowledge estimated the range of radiative forcing from the main aerosol components (sulphate, black carbon (BC) and organic carbon) for the whole transportation sector. In this study, we compare results from two different chemical transport models and three radiation codes under different hypothesis of mixing: internal and external mixing using emission inventories for the year 2000. The main results from this study consist of a positive direct radiative forcing for aerosols emitted by road traffic of +20±11 mW m−2 for an externally mixed aerosol, and of +32±13 mW m−2 when BC is internally mixed. These direct radiative forcings are much higher than the previously published estimate of +3±11 mW m−2. For transport activities from shipping, the net direct aerosol radiative forcing is negative. This forcing is dominated by the contribution of the sulphate. For both an external and an internal mixture, the radiative forcing from shipping is estimated at −26±4 mW m−2. These estimates are in very good agreement with the range of a previously published one (from −46 to −13 mW m−2) but with a much narrower range. By contrast, the direct aerosol forcing from aviation is estimated to be small, and in the range −0.9 to +0.3 mW m−2.

Variational calculations of rovibrational states: a precise high-energy potential surface for HCN [and discussion]

We report the results of variational calculations of the rovibrational energy levels of HCN for J = 0, 1 and 2, where we reproduce all the ca. 100 observed vibrational states for all observed isotopic species, with energies up to 18000 cm$^{-1}$, to about $\pm $1 cm$^{-1}$, and the corresponding rotational constants to about $\pm $0.001 cm$^{-1}$. We use a hamiltonian expressed in internal coordinates r$_{1}$, r$_{2}$ and $\theta $, using the exact expression for the kinetic energy operator T obtained by direct transformation from the cartesian representation. The potential energy V is expressed as a polynomial expansion in the Morse coordinates y$_{i}$ for the bond stretches and the interbond angle $\theta $. The basis functions are built as products of appropriately scaled Morse functions in the bond-stretches and Legendre or associated Legendre polynomials of cos $\theta $ in the angle bend, and we evaluate matrix elements by Gauss quadrature. The hamiltonian matripx is factorized using the full rovibrational symmetry, and the basis is contracted to an optimized form; the dimensions of the final hamiltonian matrix vary from 240 $\times $ 240 to 1000 $\times $ 1000.We believe that our calculation is converged to better than 1 cm$^{-1}$ at 18 000 cm$^{-1}$. Our potential surface is expressed in terms of 31 parameters, about half of which have been refined by least squares to optimize the fit to the experimental data. The advantages and disadvantages and the future potential of calculations of this type are discussed.

Radiative forcing due to changes in ozone and methane caused by the transport sector

The year 2000 radiative forcing (RF) due to changes in O3 and CH4 (and the CH4-induced stratospheric water vapour) as a result of emissions of short-lived gases (oxides of nitrogen (NOx), carbon monoxide and non-methane hydrocarbons) from three transport sectors (ROAD, maritime SHIPping and AIRcraft) are calculated using results from five global atmospheric chemistry models. Using results from these models plus other published data, we quantify the uncertainties. The RF due to short-term O3 changes (i.e. as an immediate response to the emissions without allowing for the long-term CH4 changes) is positive and highest for ROAD transport (31mWm-2) compared to SHIP (24 mWm-2) and AIR (17 mWm-2) sectors in four of the models. All five models calculate negative RF from the CH4 perturbations, with a larger impact from the SHIP sector than for ROAD and AIR. The net RF of O3 and CH4 combined (i.e. including the impact of CH4 on ozone and stratospheric water vapour) is positive for ROAD (+16(±13)(one standard deviation) mWm-2) and AIR (+6(±5) mWm-2) traffic sectors and is negative for SHIP (-18(±10) mWm-2) sector in all five models. Global Warming Potentials (GWP) and Global Temperature change Potentials (GTP) are presented for AIR NOx emissions; there is a wide spread in the results from the 5 chemistry models, and it is shown that differences in the methane response relative to the O3 response drive much of the spread.

A new cyano-bridged one-dimensional (GdFeIII)-Fe-III coordination polymer with o-phenanthroline as the blocking ligand: synthesis, structure, and magnetic properties

Synthesis, structural characterization, and magnetic properties of a new cyano-bridged one-dimensional iron (III)-gadolinium (III) compound, trans-[Gd(o-phen)(2)(H2O)(2)(mu-CN)(2)Fe(CN)(4)], - 2no-phen (o-phen = 1,10-phenanthroline), have been described. The compound crystallizes in the triclinic P (1) over bar space group with the following unit cell parameters: a = 10.538(14) angstrom, b = 12.004(14) angstrom, c = 20.61(2) angstrom, alpha = 92.41(1)degrees, beta = 92.76(1)degrees, gamma = 11 2.72(1)degrees, and Z = 2. In this complex, each gadolinium (III) is coordinated to two nitrile nitrogens of the CN groups coming from two different ferricyanides, the mutually trans cyanides of each of which links another different Gd-III to create -NC-Fe(CN)(4)-CN-Gd-NC- type 1-D chain structure. The one-dimensional chains are self-assembled in two-dimensions via weak C-H center dot center dot center dot N hydrogen bonds. Both the variable-temperature (2-300 K, 0.01 T and 0.8 T) and variable-field (0-50 000 Gauss, 2 K) magnetic measurements reveal the existence of very weak interaction in this molecule. The temperature dependence of the susceptibilities has been analyzed using a model for a chain of alternating classic (7/2) and quantum (1/2) spins. (c) 2005 Elsevier B.V. All rights reserved.

A fast identification algorithm for Box-Cox transformation based radial basis function neural network

In this letter, a Box-Cox transformation-based radial basis function (RBF) neural network is introduced using the RBF neural network to represent the transformed system output. Initially a fixed and moderate sized RBF model base is derived based on a rank revealing orthogonal matrix triangularization (QR decomposition). Then a new fast identification algorithm is introduced using Gauss-Newton algorithm to derive the required Box-Cox transformation, based on a maximum likelihood estimator. The main contribution of this letter is to explore the special structure of the proposed RBF neural network for computational efficiency by utilizing the inverse of matrix block decomposition lemma. Finally, the Box-Cox transformation-based RBF neural network, with good generalization and sparsity, is identified based on the derived optimal Box-Cox transformation and a D-optimality-based orthogonal forward regression algorithm. The proposed algorithm and its efficacy are demonstrated with an illustrative example in comparison with support vector machine regression.

Modified radial basis function neural network using output transformation

A modified radial basis function (RBF) neural network and its identification algorithm based on observational data with heterogeneous noise are introduced. The transformed system output of Box-Cox is represented by the RBF neural network. To identify the model from observational data, the singular value decomposition of the full regression matrix consisting of basis functions formed by system input data is initially carried out and a new fast identification method is then developed using Gauss-Newton algorithm to derive the required Box-Cox transformation, based on a maximum likelihood estimator (MLE) for a model base spanned by the largest eigenvectors. Finally, the Box-Cox transformation-based RBF neural network, with good generalisation and sparsity, is identified based on the derived optimal Box-Cox transformation and an orthogonal forward regression algorithm using a pseudo-PRESS statistic to select a sparse RBF model with good generalisation. The proposed algorithm and its efficacy are demonstrated with numerical examples.

A forward-constrained regression algorithm for sparse kernel density estimation

Using the classical Parzen window (PW) estimate as the target function, the sparse kernel density estimator is constructed in a forward-constrained regression (FCR) manner. The proposed algorithm selects significant kernels one at a time, while the leave-one-out (LOO) test score is minimized subject to a simple positivity constraint in each forward stage. The model parameter estimation in each forward stage is simply the solution of jackknife parameter estimator for a single parameter, subject to the same positivity constraint check. For each selected kernels, the associated kernel width is updated via the Gauss-Newton method with the model parameter estimate fixed. The proposed approach is simple to implement and the associated computational cost is very low. Numerical examples are employed to demonstrate the efficacy of the proposed approach.

Hammerstein model identification algorithm using Bezier-Bernstein approximation

A new identification algorithm is introduced for the Hammerstein model consisting of a nonlinear static function followed by a linear dynamical model. The nonlinear static function is characterised by using the Bezier-Bernstein approximation. The identification method is based on a hybrid scheme including the applications of the inverse of de Casteljau's algorithm, the least squares algorithm and the Gauss-Newton algorithm subject to constraints. The related work and the extension of the proposed algorithm to multi-input multi-output systems are discussed. Numerical examples including systems with some hard nonlinearities are used to illustrate the efficacy of the proposed approach through comparisons with other approaches.

The impact of traffic emissions on atmospheric ozone and OH: results from QUANTIFY

To estimate the impact of emissions by road, aircraft and ship traffic on ozone and OH in the present-day atmosphere six different atmospheric chemistry models have been used. Based on newly developed global emission inventories for road, ship and aircraft emission data sets each model performed sensitivity simulations reducing the emissions of each transport sector by 5%. The model results indicate that on global annual average lower tropospheric ozone responds most sensitive to ship emissions (50.6%±10.9% of the total traffic induced perturbation), followed by road (36.7%±9.3%) and aircraft exhausts (12.7%±2.9%), respectively. In the northern upper troposphere between 200–300 hPa at 30–60° N the maximum impact from road and ship are 93% and 73% of the maximum effect of aircraft, respectively. The latter is 0.185 ppbv for ozone (for the 5% case) or 3.69 ppbv when scaling to 100%. On the global average the impact of road even dominates in the UTLS-region. The sensitivity of ozone formation per NOx molecule emitted is highest for aircraft exhausts. The local maximum effect of the summed traffic emissions on the ozone column predicted by the models is 0.2 DU and occurs over the northern subtropical Atlantic extending to central Europe. Below 800 hPa both ozone and OH respond most sensitively to ship emissions in the marine lower troposphere over the Atlantic. Based on the 5% perturbation the effect on ozone can exceed 0.6% close to the marine surface (global zonal mean) which is 80% of the total traffic induced ozone perturbation. In the southern hemisphere ship emissions contribute relatively strongly to the total ozone perturbation by 60%–80% throughout the year. Methane lifetime changes against OH are affected strongest by ship emissions up to 0.21 (± 0.05)%, followed by road (0.08 (±0.01)%) and air traffic (0.05 (± 0.02)%). Based on the full scale ozone and methane perturbations positive radiative forcings were calculated for road emissions (7.3±6.2 mWm−2) and for aviation (2.9±2.3 mWm−2). Ship induced methane lifetime changes dominate over the ozone forcing and therefore lead to a net negative forcing (−25.5±13.2 mWm−2).