The Airsea team is involved in the modeling and algorithmic aspects of ocean-atmosphere (OA) coupling. We have been actively working on the analysis of such coupling both in terms of continuous and numerical formulations. Particular attention is paid to the inclusion of physical parameterizations in our theoretical framework. Our activities have recently led to practical implementations in state-of-the-art oceanic and Earth system models. Our focus during the last few years has been on the following topics :

These topics are addressed through strong collaborations between the applied mathematicians and the climate and operational community (Meteo-France, Ifremer, SHOM, Mercator-Ocean, LMD, and LOCEAN). Airsea team members play a major role in the structuration of a multi-disciplinary scientific community working on ocean-atmosphere coupling spanning a broad range from mathematical theory to practical implementations in climate and operational models.

The AIRSEA team works on topics around physics-dynamics coupling 52. Schematically, numerical models consist of two blocks generally identified as “physics” and “dynamics” which are often developed separately. The “Physics” represents unresolved or under-resolved processes with typical scales below model resolution while the “dynamics” corresponds to a discrete representation in space and time of resolved processes. Unresolved processes cannot be ignored because they directly influence the resolved part of the flow since energy is continuously transferred between scales. The interplay between resolved and unresolved scales is a large, incomplete and complex topic for which there is still much to do within the Earth system modeling community 63. During the last year we worked on the following topics :

Those topics are addressed through collaborations with the climate and operational community (Meteo-France, SHOM, Mercator-Ocean, and IGE). Two projects are currently funded, one on the energetically consistent discretization aspect (SHOM 19CP07, 2020-2024, PI: F. Lemarié) and one on the convection parameterization (Institut des Mathématiques pour la Planète Terre, 2021-2024, PIs: F. Lemarié and G. Madec). Furthermore, an ANR project, in which the Airsea team is involved, entitled PLUME was selected for funding in the latest ANR call. One of the objectives of this project is to use LES numerical simulations and laboratory experiments of deep convection to calibrate and evaluate physical parameterizations.

Artificial intelligence and machine learning may be considered as a potential way to address unresolved model scales and to approximate poorly known processes such as dissipation that occurs essentially at small scales. In order to understand the possibility to combine numerical model and neural network learned with the aid of external data, we develop a network generation and learning algorithm and use it to approximate nonlinear model operators.

A potential way to reconstruct subgrid scales consists in application of the Image Super-Resolution methods that refer to the process of recovering high-resolution images from low-resolution image in computer vision and image processing. Recent years have shown remarkable progress of image super-resolution using machine learning techniques 79. We try to use this methodology in order to identify fine structure of the chaotic turbulent solution of a simple barotropic ocean model. After the learning the flow patterns obtained by the high resolution model, the neuron net can identify fine structure in the low-resolution model solution with better precision than bicubic interpolation.

Different techniques of neuron net construction have been analyzed. Full-connected networks, basic convolutional and encoder-decoder processes 61 as well as mixed architectures were compared with each other and with the classical interpolation of model solution on a low-resolution grid.

In 32 we propose a gradient-enhanced algorithm for high-dimensional function approximation. The algorithm proceeds in two steps: firstly, we reduce the input dimension by learning the relevant input features from gradient evaluations, and secondly, we regress the function output against the pre-learned features. To ensure theoretical guarantees, we construct the feature map as the first components of a diffeomorphism, which we learn by minimizing an error bound obtained using Poincaré Inequality applied either in the input space or in the feature space. This leads to two different strategies, which we compare both theoretically and numerically and relate to existing methods in the literature. In addition, we propose a dimension augmentation trick to increase the approximation power of feature detection. A generalization to vector-valued functions demonstrate that our methodology directly applies to learning autoencoders. Here, we approximate the identity function over a given dataset by a composition of feature map (encoder) with the regression function (decoder). In practice, we construct the diffeomorphism using coupling flows, a particular class of invertible neural networks. Numerical experiments on various high-dimensional functions show that the proposed algorithm outperforms state-of-the-art competitors, especially with small datasets.

In a joint work 69 with Didier Georges (GIPSA Lab, Grenoble) and Mathieu Oliver (internship student), we proposed a spatialized extension of a SIR model that accounts for undetected infections and recoveries as well as the load on hospital services. The spatialized compartmental model we introduced is governed by a set of partial differential equations (PDEs) defined on a spatial domain with complex boundary. We proposed to solve the set of PDEs defining our model by using a meshless numerical method based on a finite difference scheme in which the spatial operators were approximated by using radial basis functions. Then we calibrated our model on the French department of Isère during the first period of lockdown, using daily reports of hospital occupancy in France. Our methodology allowed to simulate the spread of Covid-19 pandemic at a departmental level, and for each compartment. However, the simulation cost prevented from online short-term forecast. Therefore, we proposed to rely on reduced order modeling tools to compute short-term forecasts of infection number. The strategy consisted in learning a time-dependent reduced order model with few compartments from a collection of evaluations of our spatialized detailed model, varying initial conditions and parameter values. A set of reduced bases was learnt in an offline phase while the projection on each reduced basis and the selection of the best projection was performed online, allowing short-term forecast of the global number of infected individuals in the department. This work is going on in the framework of Robin Vaudry's PhD (co-supervised with Didier Georges). We are investigating the more complex setting of spatialized models taking into account the vaccination and the loss of immunity.

In the framework of Arthur Macherey's PhD (defended in June 2021), in collaboration with Anthony Nouy and Marie Billaud-Friess (Ecole Centrale Nantes), we have proposed algorithms for solving high-dimensional Partial Differential Equations (PDEs) that combine a probabilistic interpretation of PDEs, through Feynman-Kac representation, with sparse interpolation 40. Monte-Carlo methods and time-integration schemes are used to estimate pointwise evaluations of the solution of a PDE. We use a sequential control variates algorithm, where control variates are constructed based on successive approximations of the solution of the PDE. Then we turned to parametrized PDE and proposed stochastic algorithms in the framework of potentially high dimensional parameter space 23. A preliminary step was the development of a PAC algorithm in relative precision for bandit problem with costly sampling 41.

External collaborators: Arthur Macherey, Didier Georges and Matthieu Oliver

In the current context of rapid climate change, numerical models are important tools for predicting climate change and assisting decision making by policy makers (e.g. in terms of protection of marine areas, land use or definition of fishing quotas). The huge complexity of the models and the generally very high cost of numerical simulations make an exhaustive exploration of the parameter space, corresponding to all possible scenarios and all model internal options, completely illusory. The idea is therefore to exploit statistical tools for the design of experiments. These tools make it possible to identify specific combinations that provide maximum information on a given quantity of interest (for example an indicator of ecosystem health) calculated from the simulation performed. The design of experiments also has the advantage that it can be built adaptively, with the aim of taking into account the results of pre-existing simulations, performed with various models, under various scenarios. In Angelique Saillet's PhD, we aim at developing methodologies to address this open research domain, exploiting theoretical tools such as sequential design of experiments, enrichment strategies, and multi-fidelity Gaussian process regression.

Incremental Variational Data Assimilation addresses the non-linear least-square optimization challenge inherent in variational data assimilation by iteratively minimizing a sequence of linear least-squares cost functions. In the context of Hélène Hénon's PhD research, we explore the potential application of a multifidelity approach to tackle these linear least-square problems. This involves considering varying levels of fidelity or accuracy in the computational models, offering a nuanced strategy to enhance the efficiency and effectiveness of the iterative optimization process.

Transport map methods offer a powerful statistical learning tool that can couple a target high-dimensional random variable with some reference random variable using invertible transformations. The paper 24 presents new computational techniques for building the Knothe–Rosenblatt (KR) rearrangement based on general separable functions. We first introduce a new construction of the KR rearrangement—with guaranteed invertibility in its numerical implementation—based on approximating the density of the target random variable using tensor-product spectral polynomials and downward closed sparse index sets. Compared to other constructions of KR arrangements based on either multi-linear approximations or nonlinear optimizations, our new construction only relies on a weighted least square approximation procedure. Then, inspired by the recently developed deep tensor trains (Cui and Dolgov, Found. Comput. Math. 22:1863–1922, 2022), we enhance the approximation power of sparse polynomials by preconditioning the density approximation problem using compositions of maps. This is particularly suitable for high-dimensional and concentrated probability densities commonly seen in many applications. We approximate the complicated target density by a composition of self-reinforced KR rearrangements, in which previously constructed KR rearrangements—based on the same approximation ansatz—are used to precondition the density approximation problem for building each new KR rearrangement. We demonstrate the efficiency of our proposed methods and the importance of using the composite map on several inverse problems governed by ordinary differential equations (ODEs) and partial differential equations (PDEs).

In the paper 3 we present a novel offline-online method to mitigate the computational burden of the characterization of posterior random variables in statistical learning. In the offline phase, the proposed method learns the joint law of the parameter random variables and the observable random variables in the tensor-train (TT) format. In the online phase, the resulting order-preserving conditional transport can characterize the posterior random variables given newly observed data in real time. Compared with the state-of-the-art normalizing flow techniques, the proposed method relies on function approximation and is equipped with a thorough performance analysis. The function approximation perspective also allows us to further extend the capability of transport maps in challenging problems with high-dimensional observations and high-dimensional parameters. On the one hand, we present novel heuristics to reorder and/or reparametrize the variables to enhance the approximation power of TT. On the other hand, we integrate the TT-based transport maps and the parameter reordering/reparametrization into layered compositions to further improve the performance of the resulting transport maps. We demonstrate the efficiency of the proposed method on various statistical learning tasks in ordinary differential equations (ODEs) and partial differential equations (PDEs).

In the paper 1, we address the problem of the parametrization and the learning of monotone triangular transport maps. Transportation of measure provides a versatile approach for modeling complex probability distributions, with applications in density estimation, Bayesian inference, generative modeling, and beyond. Monotone triangular transport maps—approximations of the Knothe–Rosenblatt (KR) rearrangement are a canonical choice for these tasks. Yet the representation and parameterization of such maps have a significant impact on their generality and expressiveness, and on properties of the optimization problem that arises in learning a map from data (e.g., via maximum likelihood estimation). We present a general framework for representing monotone triangular maps via invertible transformations of smooth functions. We establish conditions on the transformation such that the associated infinite-dimensional minimization problem has no spurious local minima, i.e., all local minima are global minima; and we show for target distributions satisfying certain tail conditions that the unique global minimizer corresponds to the KR map. Given a sample from the target, we then propose an adaptive algorithm that estimates a sparse semi-parametric approximation of the underlying KR map. We demonstrate how this framework can be applied to joint and conditional density estimation, likelihood-free inference, and structure learning of directed graphical models, with stable generalization performance across a range of sample sizes.

External collaborators: Sergey Dolgov, Tiangang Cui, Youssef Marzouk, Ricardo Baptista

In the paper 28 we investigate the approximation of high-dimensional target measures as low-dimensional updates of a dominating reference measure. This approximation class replaces the associated density with the composition of: (i) a feature map that identifies the leading principal components or features of the target measure, relative to the reference, and (ii) a low-dimensional profile function. When the reference measure satisfies a subspace $\phi$-Sobolev inequality, we construct a computationally tractable approximation that yields certifiable error guarantees with respect to the Amari $\alpha$-divergences. Our construction proceeds in two stages. First, for any feature map and any $\alpha$-divergence, we obtain an analytical expression for the optimal profile function. Second, for linear feature maps, the principal features are obtained from eigenvectors of a matrix involving gradients of the log-density. Neither step requires explicit access to normalizing constants. Notably, by leveraging the $\phi$-Sobolev inequalities, we demonstrate that these features universally certify approximation errors across the range of $\alpha$-divergences $\alpha \in (0,1]$. We then propose an application to Bayesian inverse problems and provide an analogous construction with approximation guarantees that hold in expectation over the data. We conclude with an extension of the proposed dimension reduction strategy to nonlinear feature maps.

In the paper 26 we consider high-dimensional Bayesian inverse problems with arbitrary likelihood and product-form Laplace prior for which we provide a certified approximation of the posterior density in the Hellinger distance. The approximate posterior density differs from the prior density only in a small number of relevant coordinates that contribute the most to the update from the prior to the posterior. We propose and analyze a gradient-based diagnostic to identify these relevant coordinates. Although this diagnostic requires computing an expectation with respect to the posterior, we propose tractable methods for the classical case of a linear forward model with Gaussian likelihood. Our methods can be employed to estimate the diagnostic before solving the Bayesian inverse problem via, e.g., Markov chain Monte Carlo (MCMC) methods. After selecting the coordinates, the approximate posterior density can be efficiently inferred since most of its coordinates are only informed by the prior. Moreover, specialized MCMC methods, such as the pseudo-marginal MCMC algorithm, can be used to obtain less correlated samples when sampling the exact posterior density. We show the applicability of our method using a 1D signal deblurring problem and a high-dimensional 2D super-resolution problem. In Qiao Chen's PhD project, we propose novel approach to jointly reducing parameter and data dimensions in Bayesian inverse problems, addressing a gap in existing research that typically treats these reductions separately. The proposed method leverages gradient information to ensure controlled posterior approximation errors and enables goal-oriented dimension reduction by specifying a parameter subspace of interest. An efficient "Alternating Eigendecomposition" algorithm is presented for solving both coupled and goal-oriented problems. Demonstrating applicability in Bayesian optimal experimental design, the method connects goal-oriented data dimension reduction to maximizing the expected information gain. This overcomes the limitations of existing approaches by accommodating nonlinear models and non Gaussian posteriors. Additionally, the method offers a relaxed perspective on the NP-hard combinatorial optimization problem in optimal experimental design, paving the way for the identification of optimal subspaces and the use of advanced sampling techniques for sensor placement.

Reduced models are also developed in the framework of robust inversion. In 46, we have combined a new greedy algorithm for functional quantization with a Stepwise Uncertainty Reduction strategy to solve a robust inversion problem under functional uncertainties. In a more recent work, we further reduced the number of simulations required to solve the same robust inversion problem, based on Gaussian process meta-modeling on the joint input space of deterministic control parameters and functional uncertain variable 47. These results are applied to automotive depollution. This research axis was conducted in the framework of the Chair OQUAIDO. This research axis is till active in the team through Clément Duhamel's PhD, in collaboration with Céline Helbert (Ecole Centrale Lyon) and Miguel Munoz Zuniga, Delphine Sinoquet (IFPEN, Rueil Malmaison). In particular in 4 we proposed a SUR version of the Bichon criterion for excursion set estimation.

External collaborators: Youssef Marzouk, Matthew Li, Yiqiu Dong, Felipe Uribe

In Variational Data Assimilation, the analysis step boils down to solving a high-dimensional nonlinear least-squares problem. In practical terms, this minimization process involves iterative inversions of large matrices, which may be ill-conditioned, formed through linearizations of the forward model. To enhance the convergence rate of these methods and thereby reduce computational demands, preconditioning techniques are commonly employed to obtain better-conditioned matrices. However, these techniques typically rely on either the sparsity pattern of the matrix to be inverted or some spectral information.

In our approach, we propose the utilization of Deep Neural Networks to construct a preconditioner. This preconditioner is trained using properties derived from the singular value decomposition, and the training dataset can be dynamically constructed online as needed. This innovative approach aims to improve the efficiency of the inversion process and address the challenges posed by the high dimensionality and potential ill-conditioning of the matrices involved. This work was presented in two conferences 35, 34.

Forecasting geophysical  systems require complex models, which sometimes need to be coupled, and which make use of data assimilation. The objective of this project is, for a given output of such a system, to identify the most influential parameters, and to evaluate the effect of uncertainty in input parameters on model output. Existing stochastic tools are not well suited for high dimension problems (in particular time-dependent problems), while deterministic tools are fully applicable but only provide limited information. So the challenge is to gather expertise on one hand on numerical approximation and control of Partial Differential Equations, and on the other hand on stochastic methods for sensitivity analysis, in order to develop and design innovative stochastic solutions to study high dimension models and to propose new hybrid approaches combining the stochastic and deterministic methods. We took part to the writing of a position paper on the future of sensitivity analysis 75.

An important challenge for stochastic sensitivity analysis is to develop methodologies which work for dependent inputs. Recently, the Shapley value, from econometrics, was proposed as an alternative to quantify the importance of random input variables to a function. Owen 71 derived Shapley value importance for independent inputs and showed that it is bracketed between two different Sobol' indices. Song et al. 77 recently advocated the use of Shapley value for the case of dependent inputs. In a recent work 70, in collaboration with Art Owen (Standford's University), we showed that Shapley value removes the conceptual problems of functional ANOVA for dependent inputs. We also investigated further the properties of Shapley effects in 60. By the end of 2021, Clémentine Prieur started a collaboration with Elmar Plischke (TU Clausthal, Germany) and Emanuele Borgonovo (Bocconi University, Milan, Italy) to estimate total Sobol' indices as a measure for variable selection even in the framework of dependent inputs. In particular, it allows to estimate total Sobol' indices for inputs defined on a non rectangular domain 2. This setting is of particular interest for applications where the input space is reduced due to physical constraints on the quantity of interest. This last setting was encountered, e.g., in Maria Belén Heredia's PhD thesis (defended in december, 2020), and analyzed by estimating Shapley effects with a nonparametric procedure based on nearest neighbors 38 (see Section 8.4.3 for more details). In October 2021, Ri Wang has started a PhD, cosupervised by Clémentine Prieur and Véronique Maume-Deschamps (ICJ, Lyon 1) on the estimation of quantile oriented sensitivity indices in the framework of dependent inputs, by means of random forests or other machine learning tools. Ri Wang has received a funding from the Chinese Scientific Council.

External collaborators: Maria Belén Heredia, Adrien Hirvoas, Alexandre Janon

Pesticide transfer models play a crucial role in predicting and preventing water body pollution. However, deploying these models in operational settings demands a thorough understanding of their structure, particularly the influential parameters. The objective of E. Rouziès PhD (defended in march 2023 76)) was to conduct a global sensitivity analysis (GSA) on the PESHMELBA model (pesticide and hydrology: modeling at the catchment scale). The challenge lies in the modular and intricate structure of the model, which couples various physical processes, resulting in a high-dimensional input space and significant computational costs that limit the number of feasible runs. An extensive use of different metamodelling techniques made it feasible for such an application 9.

The complexity is further compounded when considering the temporal dimension. We explore this aspect in K. Radisic's PhD, using polynomial chaos extension to conduct a global sensitivity analysis on the small agricultural catchment of Morcille during a winter rainstorm, as detailed in 8. This study expanded to include a comparison of sensitivity analyses under different rainfall forcings and proposed a global index that considers rainfall stochasticity 17.

These initial steps paved the way for developing a method for robust calibration through stochastic metamodelling of PESHMELBA, utilizing stochastic polynomial chaos expansion. Subsequent enhancements were observed by refining the inference of marginal distributions through a Gaussian mixture.

Another research direction for global SA algorithm starts with the report that most of the algorithms to compute sensitivity measures require special sampling schemes or additional model evaluations so that available data from previous model runs (e.g., from an uncertainty analysis based on Latin Hypercube Sampling) cannot be reused. One challenging task for estimating global sensitivity measures consists in recycling an available finite set of input/output data. Green sensitivity, by recycling, avoids wasting. These given data have been discussed, e.g., in 72, 73. Most of the given data procedures depend on parameters (number of bins, truncation argument…) not easy to calibrate with a bias-variance compromise perspective. Adaptive selection of these parameters remains a challenging issue for most of these given-data algorithms. In the context of María Belén Heredia's PhD thesis, we have proposed 39 a non-parametric given data estimator for aggregated Sobol' indices, introduced in 62 and further developed in 50 for multivariate or functional outputs. We also introduced aggregated Shapley effects and we have extended a nearest neighbor estimation procedure to estimate these indices 38. We also started a collaboration with Sébastien Da Veiga (Safran Tech), Agnès Lagnoux, Thierry Klein and Fabrice Gamboa (Institut de Mathématiques de Toulouse) on a new nonparametric estimation procedure for closed Sobol' indices of any order based on U-statistics 25.

Many models are stochastic in nature, and some of them may be driven by parametrized stochastic differential equations (SDE). It is important for applications to propose a strategy to perform global sensitivity analysis (GSA) for such models, in presence of uncertainties on the parameters. In collaboration with Pierre Etoré (DATA department in Grenoble), Clémentine Prieur proposed an approach based on Feynman-Kac formulas 48. The research on GSA for stochastic simulators is still ongoing, first in the context of the MATH-AmSud project FANTASTIC (Statistical inFerence and sensitivity ANalysis for models described by sTochASTIC differential equations) with Chile and Uruguay, secondly through the PhD thesis of Henri Mermoz Kouye, co-supervised by Clémentine Prieur, in collaboration with Gildas Mazo and Eliza Vergu (INRAE, département MIA,  Jouy). Note that our recent developments with P. Etoré on GSA for parametrized SDEs and are strongly related to reduced order modeling (see Section 8.2), as GSA requires jose leon intensive computations of the quantity of interest. In collaboration with Pierre Etoré and Joël Andrepont (master internship started in spring 2021), Clémentine Prieur is working on GSA for parametrized SDEs based on Fokker-Planck equation and kernel based sensitivity indices. Note that a joint work between Pierre Etoré, Clémentine Prieur and Jose R. Leon has been published 6, related to exact or approximated computation of Kolmogorov hypoelliptic equations (KHE). Even if not dealing with GSA, it could be a starting point for analyzing sensitivity for models described by a parametrized version of KHE. Concerning Henri Mermoz Kouye's PhD thesis, the approach was different. We are interested in GSA for compartmental stochastic models. Our methodology relies on a deterministic representation of continuous time Markov chains stochastic compartmental models 27.

Physically-based avalanche propagation models must still be locally calibrated to provide robust predictions, e.g. in long-term forecasting and subsequent risk assessment. Friction parameters cannot be measured directly and need to be estimated from observations. Rich and diverse data is now increasingly available from test-sites, but for measurements made along flow propagation, potential autocorrelation should be explicitly accounted for. In the context of María Belén Heredia's PhD, in collaboration with IRSTEA Grenoble, we have proposed in 55 a comprehensive Bayesian calibration and statistical model selection framework with application to an avalanche sliding block model with the standard Voellmy friction law and high rate photogrammetric images. An avalanche released at the Lautaret test-site and a synthetic data set based on the avalanche were used to test the approach. Results have demonstrated i) the efficiency of the proposed calibration scheme, and ii) that including autocorrelation in the statistical modelling definitely improves the accuracy of both parameter estimation and velocity predictions. In the context of the energy transition, wind power generation is developing rapidly in France and worldwide. Research and innovation on wind resource characterisation, turbin control, coupled mechanical modelling of wind systems or technological development of offshore wind turbines floaters are current research topics. In particular, the monitoring and the maintenance of wind turbine is becoming a major issue. Current solutions do not take full advantage of the large amount of data provided by sensors placed on modern wind turbines in production. These data could be advantageously used in order to refine the predictions of production, the life of the structure, the control strategies and the planning of maintenance. In this context, it is interesting to optimally combine production data and numerical models  in order to obtain highly reliable models of wind turbines. This process is of interest to many industrial and academic groups and is known in many fields of the industry, including the wind industry, as "digital twin”. The objective of Adrien Hirvoas's PhD work was to develop of data assimilation methodology to build the "digital twin" of an onshore wind turbine. Based on measurements, the data assimilation should allow to reduce the uncertainties of the physical parameters of the numerical model developed during the design phase to obtain a highly reliable model. Various ensemble data assimilation approches are currently under consideration to address the problem.  In the context of this work, it is necessary to develop algorithms of identification quantifying and ranking all the uncertainty sources. This work was done in collaboration with IFPEN 58, 57. The work on model calibration is still ongoing in our team, through the PhD thesis of Adama Barry. We are finishing a work on designs for physical and numerical calibration in the framework of calibration of costly to evaluate numerical models.

Due to the sanitary context, Clémentine Prieur decided to join a working group, SEEPIA Simulation & Estimation of EPIdemics with Algorithms, animated by Didier Georges (Gipsa-lab). A first work has been published 53. An extension of the classical pandemic SIRD model was considered for the regional spread of COVID-19 in France under lockdown strategies. This compartment model divides the infected and the recovered individuals into undetected and detected compartments respectively. By fitting the extended model to the real detected data during the lockdown, an optimization algorithm was used to derive the optimal parameters, the initial condition and the epidemics start date of regions in France. Considering all the age classes together, a network model of the pandemic transport between regions in France was presented on the basis of the regional extended model and was simulated to reveal the transport effect of COVID-19 pandemic after lockdown. Using the the measured values of displacement of people mobilizing between each city, the pandemic network of all cities in France was simulated by using the same model and method as the pandemic network of regions. Finally, a discussion on an integro-differential equation was given and a new model for the network pandemic model of each age was provided. As already mentioned in Section 8.2, Clémentine Prieur went on working on the pandemic, in collaboration with Didier Georges (GIPSA Lab, Grenoble). Both of them supervised the internship of Matthieu Oliver, submitting a work proposing time-dependent reduced order modeling for short-term forecast from a spatialized SIR model 69. Robin Vaudry started in October 2021 a PhD, funded by the CNR research platform MODCOV19, and cosupervised by Clémentine Prieur and Didier Georges. The objective of this PhD is to solve inverse problems related to spatialized and ages structured commpartmental models of COVID19 pandemic.

Estimating key parameters in numerical models is a crucial aspect of numerical simulation, particularly when some parameters are not directly observable. Traditional estimation methods infer these parameters indirectly from their effects on observable variables, thus introducing uncertainties. In addition to the parameters to be estimated, there are often uncertain and uncontrollable nuisance parameters in the numerical model. Although variational inference is effective in ideal scenarios, its application in the context of nuisance parameters is less explored. Not accounting for the stochastic nature of these nuisance parameters leads to suboptimal estimation of said parameters due to compensation of errors. These nuisance parameters can be modeled as a random variable, and therefore the numerical model is redefined as a random variable. We formulate the problem in a Bayesian framework where we aim at estimating the posterior distribution minimizing the Kullback Leibler distance over a family of parametrized distributions. To do so, we integrate generative neural networks such as normalizing flows to enhance the expressiveness of this family. We apply these methods to the shallow water model, aiming at estimating the friction coefficient, a key parameter in coastal regions that defines the roughness of the seabed. The nuisance parameters represent the boundary forcing due to tidal frequencies and amplitudes.

The way how applications are executed on supercomputers still follows a traditional static resource allocation pattern: Computing resources are allocated at the start of a job which executes the application and are only released at the end of the job's runtime. This still follows the way of running jobs since decades where a dynamic resource allocation over the application's runtime would lead to several benefits: higher utilization of the computing resources, ad-hoc allocation of AI accelerator cards, less energy consumption, faster response for interactive jobs, improved data locality and I/O over the full runtime, support of urgent computing without necessarily killing running jobs, etc. Various attempts have been conducted under different terminologies used such as “evolving” jobs (application-driven dynamic resource changes) and “malleability” (system-driven dynamic resource changes) where we see a hybridization of them required for reaching optimal results.

We currently investigate possibilities to extend the MPI parallel programming model (which is the de-facto standard in HPC) and also models beyond MPI with such interfaces. As part of that, we participate in regular MPI Session working group meetings where certain effort has been done to investigate such interfaces. Based on previous work 49, 59 we worked on “A Case Study on PMIx-Usage for Dynamic Resource Management interface to a resource manager” 13.

External collaborators: Daniel Holmes, Dominik Huber, Howard Pritchard, Martin Schulz,

We made further progress toward a Domain-Specific Language (DSL) for the CROCO ocean simulation model to close the increasing gap between the numerics and HPC with a separation of concerns. Such a DSL would allow applied mathematicians to express their model equations in a high-level language and HPC experts to develop tools to automatically transform this into highly performing code on the target programming model and corresponding architecture. For our DSL, we work with PSyclone, a code generation and transformation system for Fortran-based developments, and we would like to gratefully acknowledge various developments (partly unpublished) of PSyclone developers that accelerated our success. Progress has been made on three pillars:

The first pillar is a proof-of-concept on directly using PSyclone in the support of GPU-based architectures: Using the PSyclone NEMO API and our extensions, we can parse, analyze, and automatically insert OpenACC statements in the dynamical core of the CROCO ocean model. With this, we could reproduce a formerly hand-ported and highly optimized OpenACC version with a similar performance, showing the potential to develop a DSL around this.

As a second pillar, we also started with a proof-of-concept to automatically insert code statements implementing MPI communication. This prototype analyzes deep data dependencies to determine communication dependencies and implements the required halo exchanges using asynchronous MPI operations, including in-depth studies of this (see Master's thesis of Anna Mittermair 2023_MA_anna_mittermair.pdf).

The third pillar is on the data assimilation requiring the computation of adjoints of time steps. Realizing this by hand is a non-trivial process and requires maintenance by an expert every time the numerics are updated. We successfully investigated using PSyclone for automatically generating code for adjoints as a proof-of-concept for simple ODE examples psyclone-autodiff.readthedocs.io.

These three pillars will form the fundament for the next steps in developing a DSL for the Croco ocean model which will be embedded into Fortran.

External collaborators: Rupert Ford, Anna Mittermair, Andrew Porter, Sergi Siso

We investigate new time-integration methods by considering HPC requirements, targeting a better wallclock time vs.  error ratio. These numerical methods include exponential, semi-Lagrangian, and parallel-in-time integration methods. Our team and collaborators have made significant progress over the last years. However, publishable results have not yet been generated or are currently in production.

As part of a collaboration with the University of São Paulo (USP), we investigated exponential integration methods based on Faber polynomials with a recently accepted publication 74. Although this paper mainly focuses on seismic wave propagation, these results can also be applied to the linearized fast modes in atmospheric models, e.g., as part of time-splitting methods.

In previous work, we have been able to show significant speedups using the Parallel Full Approximation Scheme in Space and Time (PFASST) parallel-in-time method for the shallow-water equations on the rotating sphere with an IMEX-SDC scheme using benchmarks which are used for the development of dynamical cores 54. Motivated by this work, we investigated this in collaboration with USP using the MGRIT parallel-in-time method. We developed a novel filtering method specific to each multi-level; see 31. It allows running the shallow-water equations on the rotating sphere without artificial viscosity on the target level and with more than two levels, which was considered a significant challenge.

On a related topic, Rishabh Bhatt successfully defended his PhD in December 2023 20, his research focused on the application of time parallelization algorithms to enhance variational data assimilation techniques. In this framework, each iteration of the optimization algorithm involves the integration of the direct model through a time-parallel method, specifically employing the Parareal algorithm 65. The optimization process utilizes a modified version of the inexact conjugate gradient method 51, where matrix-vector multiplications are facilitated through Parareal, introducing a level of inexactness. The convergence conditions of the inexact conjugate gradient method enable adaptive use of Parareal by monitoring errors in matrix-vector products, maintaining accuracy levels comparable to those achieved with the conventional conjugate gradient method. To ensure practical implementation, computationally challenging norms are replaced with easily calculable approximations. The efficacy of this approach was validated by its application to both one and two-dimensional shallow water models. For the more intricate two-dimensional model, a Krylov-enhanced subspace Parareal version was employed, demonstrating accelerated convergence and reduced iteration counts.

External collaborators: Pedro S. Peixoto, João C. Steinstraesser