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In this work, we propose the joint use of a mixed penalty-logarithmic barrier approach and generating set search, for addressing nonlinearly constrained derivative-free optimization problems. A merit function is considered, wherein the set of inequality constraints is divided into two groups: one treated with a logarithmic barrier approach, and another, along with the equality constraints, addressed using a penalization term. This strategy, initially proposed in the framework of LOG-DFL [12], is adapted and incorporated into SID-PSM [14,15] algorithm, a generalized pattern search method, allowing to effectively handle general nonlinear constraints. Under reasonable assumptions regarding the smoothness of the functions, convergence is established, without any convexity assumptions. Using CUTEst test problems, numerical experiments demonstrate the robustness, efficiency, and overall effectiveness of the proposed method, when compared with state-of-art solvers and with the original SID-PSM and LOG-DFL implementations.

In this paper we consider bound-constrained mixed-integer optimization problems where the objective function is differentiable w.r.t.\ the continuous variables for every configuration of the integer variables. We mainly suggest to exploit derivative information when possible in these scenarios: concretely, we propose an algorithmic framework that carries out local optimization steps, alternating searches along gradient-based and primitive directions. The algorithm is shown to match the convergence properties of a derivative-free counterpart. Most importantly, the results of thorough computational experiments show that the proposed method clearly outperforms not only the derivative-free approach but also a state-of-the-art solver both in terms of efficiency and effectiveness.

In this work, we consider smooth unconstrained optimization problems and we deal with the class of gradient methods with momentum, i.e., descent algorithms where the search direction is defined as a linear combination of the current gradient and the preceding search direction. This family of algorithms includes nonlinear conjugate gradient methods and Polyak's heavy-ball approach, and is thus of high practical and theoretical interest in large-scale nonlinear optimization. We propose a general framework where the scalars of the linear combination defining the search direction are computed simultaneously by minimizing the approximate quadratic model in the 2 dimensional subspace. This strategy allows us to define a class of gradient methods with momentum enjoying global convergence guarantees and an optimal worst-case complexity bound in the nonconvex setting. Differently than all related works in the literature, the convergence conditions are stated in terms of the Hessian matrix of the bi-dimensional quadratic model. To the best of our knowledge, these results are novel to the literature. Moreover, extensive computational experiments show that the gradient methods with momentum here presented outperform classical conjugate gradient methods and are (at least) competitive with the state-of-art method for unconstrained optimization, i.e, L-BFGS method.

In this paper, we analyze a derivative-free line search method designed for bound-constrained problems. Our analysis demonstrates that this method exhibits a worst-case complexity comparable to other derivative-free methods for unconstrained and linearly constrained problems. In particular, when minimizing a function with $n$ variables, we prove that at most ${\cal O}(n\epsilon^{-2})$ iterations are needed to drive a criticality measure below a predefined threshold $\epsilon$, requiring at most ${\cal O}(n^2\epsilon^{-2})$ function evaluations. We also show that the total number of iterations where the criticality measure is not below $\epsilon$ is upper bounded by ${\cal O}(n^2\epsilon^{-2})$. Moreover, we investigate the method capability to identify active constraints at the final solutions. We show that, after a finite number of iterations, all the active constraints satisfying the strict complementarity condition are correctly identified.

Techniques to retrieve the atmospheric properties of exoplanets via direct observation of their reflected light have often been limited in scope due to computational constraints imposed by the forward-model calculations. We have developed a new set of techniques which significantly decreases the time required to perform a retrieval while maintaining accurate results. We constructed a grid of 1.4 million pre-computed geometric albedo spectra valued at discrete sets of parameter points. Spectra from this grid are used to produce models for a fast and efficient nested sampling routine called PSGnest. Beyond the upfront time to construct a spectral grid, the amount of time to complete a full retrieval using PSGnest is on the order of seconds to minutes using a personal computer. An extensive evaluation of the error induced from interpolating intermediate spectra from the grid indicates that this bias is insignificant compared to other retrieval error sources, with an average coefficient of determination between interpolated and true spectra of 0.998. We apply these new retrieval techniques to help constrain the optimal bandpass centers for retrieving various atmospheric and bulk parameters from a LuvEx-type mission observing several planetary archetypes. We show that spectral observations made using a 20\% bandpass centered at 0.73 microns can be used alongside our new techniques to make detections of $H_2O$ and $O_2$ without the need to increase observing time beyond what is necessary for a signal-to-noise ratio of 10. The methods introduced here will enable robust studies of the capabilities of future observatories to characterize exoplanets., Comment: 32 pages, 17 figures. Accepted for publication in The Astronomical Journal

The supervised training of a deep neural network on a given dataset consists in the unconstrained minimization of the finite sum of continuously differentiable functions, commonly referred to as loss with respect to the samples. These functions depend on the network parameters and most of the times are non-convex. We develop CMA Light, a globally convergent mini-batch gradient method to tackle this problem. We consider the recently introduced Controlled Minibatch Algorithm (CMA) framework and we overcome its main bottleneck, removing the need for at least one evaluation of the whole objective function per iteration. We prove globally convergence of CMA Light under mild assumptions and we discuss extensive computational results on the same experimental test-bed used for CMA, showing that CMA Light requires less computational effort than most of the state-of-the-art optimizers. Eventually, we present early results on a large-scale Image Classification task.

Enceladus is a prime target in the search for life in our solar system, having an active plume likely connected to a large liquid water subsurface ocean. Using the sensitive NIRSpec instrument onboard JWST, we searched for organic compounds and characterized the plume's composition and structure. The observations directly sample the fluorescence emissions of H2O and reveal an extraordinarily extensive plume (up to 10,000 km or 40 Enceladus radii) at cryogenic temperatures (25 K) embedded in a large bath of emission originating from Enceladus' torus. Intriguingly, the observed outgassing rate (300 kg/s) is similar to that derived from close-up observations with Cassini 15 years ago, and the torus density is consistent with previous spatially unresolved measurements with Herschel 13 years ago, suggesting that the vigor of gas eruption from Enceladus has been relatively stable over decadal timescales. This level of activity is sufficient to maintain a derived column density of 4.5x1017 m-2 for the embedding equatorial torus, and establishes Enceladus as the prime source of water across the Saturnian system. We performed searches for several non-water gases (CO2, CO, CH4, C2H6, CH3OH), but none were identified in the spectra. On the surface of the trailing hemisphere, we observe strong H2O ice features, including its crystalline form, yet we do not recover CO2, CO nor NH3 ice signatures from these observations. As we prepare to send new spacecraft into the outer solar system, these observations demonstrate the unique ability of JWST in providing critical support to the exploration of distant icy bodies and cryovolcanic plumes., Comment: Accepted for publication in Nature Astronomy on May 17th 2023

In this paper we propose an heuristic to improve the performances of the recently proposed derivative-free method for nonsmooth optimization CS-DFN. The heuristic is based on a clustering-type technique to compute a direction { which relies on an estimate of Clarke's generalized gradient} of the objective function. As such, this direction (as it is shown by the numerical experiments) is a good descent direction for the objective function. We report some numerical results and comparison with the original CS-DFN method to show the utility of the proposed improvement on a set of well-known test problems.

This paper is devoted to the analysis of worst case complexity bounds for linesearch-type derivative-free algorithms for the minimization of general non-convex smooth functions. We prove that two linesearch-type algorithms enjoy the same complexity properties which have been proved for pattern and direct search algorithms. In particular, we consider two derivative-free algorithms based on two different linesearch techniques and manage to prove that the number of iterations and of function evaluations required to drive the norm of the gradient of the objective function below a given threshold $\epsilon$ is ${\cal O}(\epsilon^{-2})$ in the worst case.

We present observations of the 1.35+/-0.07 Earth-radius planet L 98-59 c, collected using Wide Field Camera 3 on the Hubble Space Telescope. L 98-59 is a nearby (10.6 pc), bright (H=7.4 mag), M3V star that harbors three small, transiting planets. As one of the closest known transiting multi-planet systems, L 98-59 offers one of the best opportunities to probe and compare the atmospheres of rocky planets that formed in the same stellar environment. We measured the transmission spectrum of L 98-59 c and the extracted spectrum showed marginal evidence (2.1{\sigma}) for wavelength-dependent transit depth variations that could indicate the presence of an atmosphere. We forward-modeled possible atmospheric compositions of the planet based on the transmission spectrum. Although L 98-59 was previously thought to be a fairly quiet star, we have seen evidence for stellar activity, and therefore we assessed a scenario where the source of the signal originates with inhomogeneities on stellar surface. We also see a correlation between transits of L 98-59 c and L 98-59 b collected 12.5 hours apart, which is suggestive (but at <2{\sigma} confidence) of a contaminating component from the star impacting the exoplanet spectrum. While intriguing, our results are inconclusive and additional data is needed to verify any atmospheric signal. Fortunately, additional data has been collected from both HST and JWST. Should this result be confirmed with additional data, L 98-59 c would be the first planet smaller than two Earth-radii with a detected atmosphere., Comment: Accepted for publication in The Astronomical Journal

In this work, we consider minimizing the average of a very large number of smooth and possibly non-convex functions, and we focus on two widely used minibatch frameworks to tackle this optimization problem: Incremental Gradient (IG) and Random Reshuffling (RR). We define ease-controlled modifications of the IG/RR schemes, which require a light additional computational effort {but} can be proved to converge under {weak} and standard assumptions. In particular, we define two algorithmic schemes in which the IG/RR iteration is controlled by using a watchdog rule and a derivative-free linesearch that activates only sporadically to guarantee convergence. The two schemes differ in the watchdog and the linesearch, which are performed using either a monotonic or a non-monotonic rule. The two schemes also allow controlling the updating of the stepsize used in the main IG/RR iteration, avoiding the use of pre-set rules that may drive the stepsize to zero too fast, reducing the effort in designing effective updating rules of the stepsize. We prove convergence under the mild assumption of Lipschitz continuity of the gradients of the component functions and perform extensive computational analysis using different deep neural architectures and a benchmark of varying-size datasets. We compare our implementation with both a full batch gradient method (i.e. L-BFGS) and an implementation of IG/RR methods, proving that our algorithms require a similar computational effort compared to the other online algorithms and that the control on the learning rate may allow a faster decrease of the objective function., Comment: Add author, add references, correct typos, improve imoplementation

The presence of phosphine (PH$_3$) in the atmosphere of Venus was reported by Greaves et al. (2021a), based on observations of the J=1-0 transition at 267 GHz using ground-based, millimeter-wave spectroscopy. This unexpected discovery presents a challenge for our understanding of Venus's atmosphere, and has led to a reappraisal of the possible sources and sinks of atmospheric phosphorous-bearing gases. Here we present results from a search for PH$_3$ on Venus using the GREAT instrument aboard the SOFIA aircraft, over three flights conducted in November 2021. Multiple PH$_3$ transitions were targeted at frequencies centered on 533 GHz and 1067 GHz, but no evidence for atmospheric PH$_3$ was detected. Through radiative transfer modeling, we derived a disk-averaged upper limit on the PH$_3$ abundance of 0.8 ppb in the altitude range 75-110 km, which is more stringent than previous ground-based studies., Comment: Accepted for publication in Geophysical Research Letters

In this work, we consider multiobjective optimization problems with both bound constraints on the variables and general nonlinear constraints, where objective and constraint function values can only be obtained by querying a black box. Furthermore, we consider the case where a subset of the variables can only take integer values. We propose a new linesearch-based solution method and show that it converges to a set of stationary points for the problem. For what concerns the continuous variables, we employ a strategy for the estimation of the Pareto frontier recently proposed in the literature and which takes advantage of dense sequences of search directions. The subset of variables that must assume discrete values are dealt with using primitive directions appropriately modified to take into account the presence of more than one objective functions. Numerical results obtained with the proposed method on a set of test problems and comparison with other solution methods show the viability and efficiency of the proposed approach.

The training phase is the most important stage during the machine learning process. In the case of labeled data and supervised learning, machine training consists in minimizing the loss function subject to different constraints. In an abstract setting, it can be formulated as a multiple criteria optimization model in which each criterion measures the distance between the output associated with a specific input and its label. Therefore, the fitting term is a vector function and its minimization is intended in the Pareto sense. We provide stability results of the efficient solutions with respect to perturbations of input and output data. We then extend the same approach to the case of learning with multiple datasets. The multiple dataset environment is relevant when reducing the bias due to the choice of a specific training set. We propose a scalarization approach to implement this model and numerical experiments in digit classification using MNIST data.

In this paper we consider constrained optimization problems where both the objective and constraint functions are of the black-box type. Furthermore, we assume that the nonlinear inequality constraints are non-relaxable, i.e. their values and that of the objective function cannot be computed outside of the feasible region. This situation happens frequently in practice especially in the black-box setting where function values are typically computed by means of complex simulation programs which may fail to execute if the considered point is outside of the feasible region. For such problems, we propose a new derivative-free optimization method which is based on the use of a merit function that handles inequality constraints by means of a log-barrier approach and equality constraints by means of a quadratic penalty approach. We prove convergence of the proposed method to KKT stationary points of the problem under quite mild assumptions. Furthermore, we also carry out a preliminary numerical experience on standard test problems and comparison with a state-of-the-art solver which shows efficiency of the proposed method., Comment: We dropped the convexity assumption to take into account that convexity is no longer required, we changed the theoretical analysis, exposition of the main algorithm has changed. We first present a simpler method and then the main algorithm. Numerical results have been a lot extended by adding some comparison

In this paper, we consider mixed-integer nonsmooth constrained optimization problems whose objective/constraint functions are available only as the output of a black-box zeroth-order oracle (i.e., an oracle that does not provide derivative information) and we propose a new derivative-free linesearch-based algorithmic framework to suitably handle those problems. We first describe a scheme for bound constrained problems that combines a dense sequence of directions (to handle the nonsmoothness of the objective function) with primitive directions (to handle discrete variables). Then, we embed an exact penalty approach in the scheme to suitably manage nonlinear (possibly nonsmooth) constraints. We analyze the global convergence properties of the proposed algorithms toward stationary points and we report the results of an extensive numerical experience on a set of mixed-integer test problems.

Training a large multilayer neural network can present many difficulties due to the large number of useless stationary points. These points usually attract the minimization algorithm used during the training phase, which therefore results inefficient. Extending some results proposed in literature for shallow networks, we propose the mathematical characterization of a class of such stationary points that arise in deep neural networks training. Availing such a description, we are able to define an incremental training algorithm that avoids getting stuck in the region of attraction of these undesirable stationary points.

Quantum computers promise tremendous impact across applications -- and have shown great strides in hardware engineering -- but remain notoriously error prone. Careful design of low-level controls has been shown to compensate for the processes which induce hardware errors, leveraging techniques from optimal and robust control. However, these techniques rely heavily on the availability of highly accurate and detailed physical models which generally only achieve sufficient representative fidelity for the most simple operations and generic noise modes. In this work, we use deep reinforcement learning to design a universal set of error-robust quantum logic gates on a superconducting quantum computer, without requiring knowledge of a specific Hamiltonian model of the system, its controls, or its underlying error processes. We experimentally demonstrate that a fully autonomous deep reinforcement learning agent can design single qubit gates up to $3\times$ faster than default DRAG operations without additional leakage error, and exhibiting robustness against calibration drifts over weeks. We then show that $ZX(-\pi/2)$ operations implemented using the cross-resonance interaction can outperform hardware default gates by over $2\times$ and equivalently exhibit superior calibration-free performance up to 25 days post optimization using various metrics. We benchmark the performance of deep reinforcement learning derived gates against other black box optimization techniques, showing that deep reinforcement learning can achieve comparable or marginally superior performance, even with limited hardware access., Comment: 12 pages, 5 figures

In this paper, we consider nonlinear optimization problems with nonlinear equality constraints and bound constraints on the variables. For the solution of such problems, many augmented Lagrangian methods have been defined in the literature. Here, we propose to modify one of these algorithms, namely ALGENCAN by Andreani et al., in such a way to incorporate second-order information into the augmented Lagrangian framework, using an active-set strategy. We show that the overall algorithm has the same convergence properties as ALGENCAN and an asymptotic quadratic convergence rate under suitable assumptions. The numerical results confirm that the proposed algorithm is a viable alternative to ALGENCAN with greater robustness.

The detection of phosphine (PH3) in the atmosphere of Venus has been recently reported based on millimeter-wave radio observations (Greaves et al. 2020), and its re-analyses (Greaves et al. 2021a/b). In this Matters Arising we perform an independent reanalysis, identifying several issues in the interpretation of the spectroscopic data. As a result, we determine sensitive upper-limits for PH3 in Venus' atmosphere (>75 km, above the cloud decks) that are discrepant with the findings in G2020 and G2021a/b. The measurements target the fundamental first rotational transition of PH3 (J=1-0) at 266.944513 GHz, which was observed with the James Clerk Maxwell Telescope (JCMT) in June 2017 and with the Atacama Large Millimeter/submillimeter Array (ALMA) in March 2019. This line's center is near the SO2 (J=309,21-318,24) transition at 266.943329 GHz (only 1.3 km/s away from the PH3 line) which represents a potential source of contamination. The JCMT and ALMA data, as presented in G2020, are at spectral resolutions comparable to the frequency separation of the two lines. Moreover, the spectral features identified are several km/s in width, and therefore do not permit distinct spectroscopic separation of the candidate spectral lines of PH3 and SO2. We present the radiative transfer modelling we have performed and then discuss the ALMA and JCMT analyses in turn.

Telescope and detector developments continuously enable deeper and more detailed studies of astronomical objects. Larger collecting areas, improvement in dispersion and detector techniques, and higher sensitivities allow detection of more molecules in a single observation, at lower abundances, resulting in better constraints of the targets physical and chemical conditions. Improvements on current telescopes, and not to mention future observatories, both in space and on the ground, will continue this trend, ever improving our understanding of the Universe. Planetary exploration missions carry instrumentation to unexplored areas, and reveal details impossible to observe from the Earth by performing in-situ measurements. Space based observatories allow observations of object at wavelength ranges absorbed by the Earths atmosphere. The depth of understanding from all of these studies can be greatly enhanced by combining observations: ground-based and space-based, low-resolution and high-resolution, local and global-scale, similar observations over a broader or different spectra range, or by providing temporal information through follow-ups. Combined observations provide context and a broader scope of the studied object, and in this white paper, we outline a number of studies where observations are synergistically applied to increase the scientific value of both datasets. Examples include atmospheric studies of Venus, Mars, Titan, comets, Jupiter, as well as more specific cases describing synergistic studies in the Juno mission, and ground-based radar studies for near Earth objects. The examples aim to serve as inspiration for future synergistic observations, and recommendations are made based on the lessons learned from these examples., Comment: This is a white paper submitted to the Planetary Decadal Survey

Brain-Computer Interfaces (BCIs) are systems allowing people to interact with the environment bypassing the natural neuromuscular and hormonal outputs of the peripheral nervous system (PNS). These interfaces record a user's brain activity and translate it into control commands for external devices, thus providing the PNS with additional artificial outputs. In this framework, the BCIs based on the P300 Event-Related Potentials (ERP), which represent the electrical responses recorded from the brain after specific events or stimuli, have proven to be particularly successful and robust. The presence or the absence of a P300 evoked potential within the EEG features is determined through a classification algorithm. Linear classifiers such as SWLDA and SVM are the most used for ERPs' classification. Due to the low signal-to-noise ratio of the EEG signals, multiple stimulation sequences (a.k.a. iterations) are carried out and then averaged before the signals being classified. However, while augmenting the number of iterations improves the Signal-to-Noise Ratio (SNR), it also slows down the process. In the early studies, the number of iterations was fixed (no stopping), but recently, several early stopping strategies have been proposed in the literature to dynamically interrupt the stimulation sequence when a certain criterion is met to enhance the communication rate. In this work, we explore how to improve the classification performances in P300 based BCIs by combining optimization and machine learning. First, we propose a new decision function that aims at improving classification performances in terms of accuracy and Information Transfer Rate both in a no stopping and early stopping environment. Then, we propose a new SVM training problem that aims to facilitate the target-detection process. Our approach proves to be effective on several publicly available datasets., Comment: 32 pages, research article

In this paper we address game theory problems arising in the context of network security. In traditional game theory problems, given a defender and an attacker, one searches for mixed strategies which minimize a linear payoff functional. In the problems addressed in this paper an additional quadratic term is added to the minimization problem. Such term represents switching costs, i.e., the costs for the defender of switching from a given strategy to another one at successive rounds of a Nash game. The resulting problems are nonconvex QP ones with linear constraints and turn out to be very challenging. We will show that the most recent approaches for the minimization of nonconvex QP functions over polytopes, including commercial solvers such as CPLEX and GUROBI, are unable to solve to optimality even test instances with n = 50 variables. For this reason, we propose to extend with them the current benchmark set of test instances for QP problems. We also present a spatial branch-and-bound approach for the solution of these problems, where a predominant role is played by an optimality-based domain reduction, with multiple solutions of LP problems at each node of the branch-and-bound tree. Of course, domain reductions are standard tools in spatial branch-and-bound approaches. However, our contribution lies in the observation that, from the computational point of view, a rather aggressive application of these tools appears to be the best way to tackle the proposed instances. Indeed, according to our experiments, while they make the computational cost per node high, this is largely compensated by the rather slow growth of the number of nodes in the branch-and-bound tree, so that the proposed approach strongly outperforms the existing solvers for QP problems.

Observations of water ice clouds and aerosols on Mars can provide important insights into the complexity of the water cycle. Recent observations have indicated an important link between dust activity and the water cycle, as intense dust activity can significantly raise the hygropause, and subsequently increase the escape of water after dissociation in the upper atmosphere. Here present observations from NOMAD/TGO that investigate the variation of water ice clouds in the perihelion season of Mars Year 34 (April 2018-19), their diurnal and seasonal behavior, and the vertical structure and microphysical properties of water ice and dust. These observations reveal the recurrent presence of a layer of mesospheric water ice clouds subsequent to the 2018 Global Dust Storm. We show that this layer rose from 45 to 80 km in altitude on a timescale of days from heating in the lower atmosphere due to the storm. In addition, we demonstrate that there is a strong dawn dusk asymmetry in water ice abundance, related to nighttime nucleation and subsequent daytime sublimation. Water ice particle sizes are retrieved consistently and exhibit sharp vertical gradients (from 0.1 to 4.0 um), as well as mesospheric differences between the Global Dust Storm (<0.5 um) and the 2019 regional dust storm (1.0 um), which suggests differing water ice nucleation efficiencies. These results form the basis to advance our understanding of mesospheric water ice clouds on Mars, and further constrain the interactions between water ice and dust in the middle atmosphere., Comment: Submitted to Journal of Geophysical Research - Planets. 36 pages, 9 figures, 1 table

We analyze the implications of transboundary pollution externalities on environmental policymaking in a spatial and finite time horizon setting. We focus on a simple regional optimal pollution control problem in order to compare the global and local solutions in which, respectively, the transboundary externality is and is not taken into account in the determination of the optimal policy by individual local policymakers. We show that the local solution is suboptimal and as such a global approach to environmental problems is effectively needed. Our conclusions hold true in different frameworks, including situations in which the spatial domain is either bounded or unbounded, and situations in which macroeconomic-environmental feedback effects are taken into account. We also show that if every local economy implements an environmental policy stringent enough, then the global average level of pollution will fall. If this is the case, over the long run the entire global economy will be able to achieve a completely pollution-free status.

The Nadir and Occultation for MArs Discovery instrument (NOMAD), onboard the ExoMars Trace Gas Orbiter (TGO) spacecraft was conceived to observe Mars in solar occultation, nadir, and limb geometries, and will be able to produce an outstanding amount of diverse data, mostly focused on properties of the atmosphere. The infrared channels of the instrument operate by combining an echelle grating spectrometer with an Acousto-Optical Tunable Filter (AOTF). Using in-flight data, we characterized the instrument performance and parameterized its calibration. In particular: an accurate frequency calibration was achieved, together with its variability due to thermal effects on the grating. The AOTF properties and transfer function were also quantified, and we developed and tested a realistic method to compute the spectral continuum transmitted through the coupled grating + AOTF system. The calibration results enabled unprecedented insights into the important problem of the sensitivity of NOMAD to methane abundances in the atmosphere. We also deeply characterized its performance under realistic conditions of varying aerosol abundances, diverse albedos and changing illumination conditions as foreseen over the nominal mission. The results show that, in low aerosol conditions, NOMAD single spectrum, 1-sigma sensitivity to CH4 is around 0.33 ppbv at 20 km of altitude when performing solar occultations, and better than 1 ppbv below 30 km. In dusty conditions, we show that the sensitivity drops to 0 below 10 km. In Nadir geometry, results demonstrate that NOMAD will be able to produce seasonal maps of CH4 with a sensitivity around 5 ppbv over most of planet's surface with spatial integration over 5x5 degrees bins. Results show also that such numbers can be improved by a factor of 10 to 30 by data binning. Overall, our results quantify NOMAD's capability to address the variable aspects of Martian climate.