Publications

2023

• Error estimates for finite differences approximations of the total variation
  
  • Caillaud Corentin
  • Chambolle Antonin
  IMA Journal of Numerical Analysis, Oxford University Press (OUP), 2023, 43 (2), pp.692--736. We present a convergence rate analysis of the Rudin-Osher-Fatemi (ROF) denoising problem for two different discretizations of the total variation. The first discretization is the well-known isotropic total variation that suffers from a blurring effect in a special diagonal direction. We prove that in the setting corresponding to this direction, the discrete ROF energy converges to the continuous one in O(h^2/3). The second total variation is based on Raviart-Thomas fields and achieves a O(h) convergence rate for the same quantity under some standard hypotheses. (10.1093/imanum/drac001)
  DOI : 10.1093/imanum/drac001
• Design patterns of hierarchies for order structures
  
  • Allamigeon Xavier
  • Canu Quentin
  • Cohen Cyril
  • Sakaguchi Kazuhiko
  • Strub Pierre-Yves
  , 2023. Using order structures in a proof assistant naturally raises the problem of working with multiple instances of a same structure over a common type of elements. This goes against the main design pattern of hierarchies used for instance in Coq's MathComp or Lean's mathlib libraries, where types are canonically associated to at most one instance and instances share a common overloaded syntax. We present new design patterns to leverage these issues, and apply them to the formalization of order structures in the MathComp library. A common idea in these patterns is underloading, i.e., a disambiguation of operators on a common type. In addition, our design patterns include a way to deal with duality in order structures in a convenient way. We hence formalize a large hierarchy which includes partial orders, semilattices, lattices as well as many variants. We finally pay a special attention to order substructures. We introduce a new kind of structure called prelattice. They are abstractions of semilattices, and allow us to deal with finite lattices and their sublattices within a common signature. As an application, we report on significant simplifications of the formalization of the face lattices of polyhedra in the Coq-Polyhedra library.
• Time reversal of spinal processes for linear and non-linear branching processes near stationarity
  
  • Henry Benoît
  • Méléard Sylvie
  • Chi Tran Viet
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2023, 28 (none). We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time $T$, which is the path constituted of the traits of the ancestors of this individual in past times $t\leq T$. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of $T$). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations. (10.1214/23-EJP911)
  DOI : 10.1214/23-EJP911
• Stochastic Approximation Beyond Gradient for Signal Processing and Machine Learning
  
  • Dieuleveut Aymeric
  • Fort Gersende
  • Moulines Eric
  • Wai Hoi-To
  IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2023, 71, pp.3117-3148. Stochastic approximation (SA) is a classical algorithm that has had since the early days a huge impact on signal processing, and nowadays on machine learning, due to the necessity to deal with a large amount of data observed with uncertainties. An exemplar special case of SA pertains to the popular stochastic (sub)gradient algorithm which is the working horse behind many important applications. A lesser-known fact is that the SA scheme also extends to non-stochastic-gradient algorithms such as compressed stochastic gradient, stochastic expectation-maximization, and a number of reinforcement learning algorithms. The aim of this article is to overview and introduce the non-stochastic-gradient perspectives of SA to the signal processing and machine learning audiences through presenting a design guideline of SA algorithms backed by theories. Our central theme is to propose a general framework that unifies existing theories of SA, including its non-asymptotic and asymptotic convergence results, and demonstrate their applications on popular non-stochastic-gradient algorithms. We build our analysis framework based on classes of Lyapunov functions that satisfy a variety of mild conditions. We draw connections between non-stochastic-gradient algorithms and scenarios when the Lyapunov function is smooth, convex, or strongly convex. Using the said framework, we illustrate the convergence properties of the non-stochastic-gradient algorithms using concrete examples. Extensions to the emerging variance reduction techniques for improved sample complexity will also be discussed. (10.1109/TSP.2023.3301121)
  DOI : 10.1109/TSP.2023.3301121
• Tree-Based Diffusion Schrödinger Bridge with Applications to Wasserstein Barycenters
  
  • Noble Maxence
  • de Bortoli Valentin
  • Doucet Arnaud
  • Durmus Alain
  , 2023. Multi-marginal Optimal Transport (mOT), a generalization of OT, aims at minimizing the integral of a cost function with respect to a distribution with some prescribed marginals. In this paper, we consider an entropic version of mOT with a tree-structured quadratic cost, i.e., a function that can be written as a sum of pairwise cost functions between the nodes of a tree. To address this problem, we develop Tree-based Diffusion Schrödinger Bridge (TreeDSB), an extension of the Diffusion Schrödinger Bridge (DSB) algorithm. TreeDSB corresponds to a dynamic and continuous state-space counterpart of the multi-marginal Sinkhorn algorithm. A notable use case of our methodology is to compute Wasserstein barycenters which can be recast as the solution of a mOT problem on a star-shaped tree. We demonstrate that our methodology can be applied in high-dimensional settings such as image interpolation and Bayesian fusion.
• Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach
  
  • Greco Giacomo
  • Noble Maxence
  • Conforti Giovanni
  • Durmus Alain
  , 2023. Computational optimal transport (OT) has recently emerged as a powerful framework with applications in various fields. In this paper we focus on a relaxation of the original OT problem, the entropic OT problem, which allows to implement efficient and practical algorithmic solutions, even in high dimensional settings. This formulation, also known as the Schrödinger Bridge problem, notably connects with Stochastic Optimal Control (SOC) and can be solved with the popular Sinkhorn algorithm. In the case of discrete-state spaces, this algorithm is known to have exponential convergence; however, achieving a similar rate of convergence in a more general setting is still an active area of research. In this work, we analyze the convergence of the Sinkhorn algorithm for probability measures defined on the d-dimensional torus , that admit densities with respect to the Haar measure. In particular, we prove pointwise exponential convergence of Sinkhorn iterates and their gradient. Our proof relies on the connection between these iterates and the evolution along the Hamilton-Jacobi-Bellman equations of value functions obtained from SOC-problems. Our approach is novel in that it is purely probabilistic and relies on coupling by reflection techniques for controlled diffusions on the torus.
• An optimal control-based numerical method for scalar transmission problems with sign-changing coefficients
  
  • Ciarlet Patrick
  • Lassounon David
  • Rihani Mahran
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2023, 61 (3), pp.1316-1339. In this work, we present a new numerical method for solving the scalar transmission problem with sign-changing coefficients. In electromagnetism, such a transmission problem can occur if the domain of interest is made of a classical dielectric material and a metal or a metamaterial, with for instance an electric permittivity that is strictly negative in the metal or metamaterial. The method is based on an optimal control reformulation of the problem. Contrary to other existing approaches, the convergence of this method is proved without any restrictive condition. In particular, no condition is imposed on the a priori regularity of the solution to the problem, and no condition is imposed on the meshes, other than that they fit with the interface between the two media. Our results are illustrated by some (2D) numerical experiments. (10.1137/22M1495998)
  DOI : 10.1137/22M1495998
• The nonlocal isoperimetric problem for polygons: Hardy-Littlewood and Riesz inequalities
  
  • Bogosel Beniamin
  • Bucur Dorin
  • Fragalà Ilaria
  Mathematische Annalen, Springer Verlag, 2023. Given a non-increasing and radially symmetric kernel in $L ^ 1 _{\rm loc} (\Bbb{R} ^ 2 ; \Bbb{R} _+)$, we investigate counterparts of the classical Hardy-Littlewood and Riesz inequalities when the class of admissible domains is the family of polygons with given area and $N$ sides. The latter corresponds to study the polygonal isoperimetric problem in nonlocal version. We prove that, for every $N \geq 3$, the regular $N$-gon is optimal for Hardy-Littlewood inequality. Things go differently for Riesz inequality: while for $N = 3$ and $N = 4$ it is known that the regular triangle and the square are optimal, for $N\geq 5$ we prove that symmetry or symmetry breaking may occur (i.e.\ the regular $N$-gon may be optimal or not), depending on the value of $N$ and on the choice of the kernel. (10.1007/s00208-023-02683-x)
  DOI : 10.1007/s00208-023-02683-x
• Development and clinical validation of real‐time artificial intelligence diagnostic companion for fetal ultrasound examination
  
  • Stirnemann Julien J.
  • Besson Rémy
  • Spaggiari Emmanuel
  • Rojo Sandra
  • Logé Frédéric
  • Saint Paul Hélène Peyro
  • Allassonnière Stéphanie
  • Le Pennec Erwan
  • Hutchinson C.
  • Sebire Neil J
  • Ville Yves
  Ultrasound in Obstetrics and Gynecology = Ultrasound in Obstetrics & Gynecology, Wiley-Blackwell, 2023, 62 (3), pp.353-360. Objective: Prenatal diagnosis of a rare disease on ultrasound relies on a physician's ability to remember an intractable amount of knowledge. We developed a real‐time decision support system (DSS) that suggests, at each step of the examination, the next phenotypic feature to assess, optimizing the diagnostic pathway to the smallest number of possible diagnoses. The objective of this study was to evaluate the performance of this real‐time DSS using clinical data. Methods This validation study was conducted on a database of 549 perinatal phenotypes collected from two referral centers (one in France and one in the UK). Inclusion criteria were: at least one anomaly was visible on fetal ultrasound after 11 weeks' gestation; the anomaly was confirmed postnatally; an associated rare disease was confirmed or ruled out based on postnatal/postmortem investigation, including physical examination, genetic testing and imaging; and, when confirmed, the syndrome was known by the DSS software. The cases were assessed retrospectively by the software, using either the full phenotype as a single input, or a stepwise input of phenotypic features, as prompted by the software, mimicking its use in a real‐life clinical setting. Adjudication of discordant cases, in which there was disagreement between the DSS output and the postnatally confirmed (‘ascertained’) diagnosis, was performed by a panel of external experts. The proportion of ascertained diagnoses within the software's top‐10 differential diagnoses output was evaluated, as well as the sensitivity and specificity of the software to select correctly as its best guess a syndromic or isolated condition. Results: The dataset covered 110/408 (27%) diagnoses within the software's database, yielding a cumulative prevalence of 83%. For syndromic cases, the ascertained diagnosis was within the top‐10 list in 93% and 83% of cases using the full‐phenotype and stepwise input, respectively, after adjudication. The full‐phenotype and stepwise approaches were associated, respectively, with a specificity of 94% and 96% and a sensitivity of 99% and 84%. The stepwise approach required an average of 13 queries to reach the final set of diagnoses. Conclusions: The DSS showed high performance when applied to real‐world data. This validation study suggests that such software can improve perinatal care, efficiently providing complex and otherwise overlooked knowledge to care‐providers involved in ultrasound‐based prenatal diagnosis. (10.1002/uog.26242)
  DOI : 10.1002/uog.26242
• A growth-fragmentation-isolation process on random recursive trees and contact tracing
  
  • Bansaye Vincent
  • Gu Chenlin
  • Yuan Linglong
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2023, 33 (6B), pp.5233-5278. We consider a random process on recursive trees, with three types of events. Vertices give birth at a constant rate (growth), each edge may be removed independently (fragmentation of the tree) and clusters (or trees) are frozen with a rate proportional to their sizes (isolation of connected component). A phase transition occurs when the isolation is able to stop the growthfragmentation process and cause extinction. When the process survives, the number of clusters increases exponentially and we prove that the normalised empirical measure of clusters a.s. converges to a limit law on recursive trees. We exploit the branching structure associated to the size of clusters, which is inherited from the splitting property of random recursive trees. This work is motivated by the control of epidemics and contact tracing where clusters correspond to trees of infected individuals that can be identified and isolated. We complement this work by providing results on the Malthusian exponent to describe the effect of control policies on epidemics. (10.1214/23-AAP1947)
  DOI : 10.1214/23-AAP1947
• Well posedness and stochastic derivation of a diffusion-growth-fragmentation equation in a chemostat
  
  • Tchouanti Josué
  Stochastics and Partial Differential Equations: Analysis and Computations, Springer US, 2023. We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative real valued trait described by a diffusion. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show that the semi-group of the stochastic trait dynamics admits a density by probabilistic arguments, that allows the measure solution of the diffusiongrowth-fragmentation equation to be a function with a certain Besov regularity. (10.1007/s40072-023-00288-8)
  DOI : 10.1007/s40072-023-00288-8
• Cross impact in derivative markets
  
  • Tomas Mehdi
  • Mastromatteo Iacopo
  • Benzaquen Michael
  Wilmott Journal, Wiley, 2023, 123, pp.16–28. We introduce a linear cross-impact framework in a setting in which the price of some given financial instruments (derivatives) is a deterministic function of one or more, possibly tradeable, stochastic factors (underlying). We show that a particular cross-impact model, the multivariate Kyle model, prevents arbitrage and aggregates (potentially non-stationary) traded order flows on derivatives into (roughly stationary) liquidity pools aggregating order flows traded on both derivatives and underlying. Using E-Mini futures and options along with VIX futures, we provide empirical evidence that the price formation process from order flows on derivatives is driven by cross-impact and confirm that the simple Kyle cross-impact model is successful at capturing parsimoniously such empirical phenomenology. Our framework may be used in practice for estimating execution costs, in particular hedging costs.
• Filling the gap between individual-based evolutionary models and Hamilton-Jacobi equations
  
  • Champagnat Nicolas
  • Méléard Sylvie
  • Mirrahimi Sepideh
  • Chi Tran Viet
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2023, 10, pp.1247-1275. We consider a stochastic model for the evolution of a discrete population structured by a trait with values on a finite grid of the torus, and with mutation and selection. Traits are vertically inherited unless a mutation occurs, and influence the birth and death rates. We focus on a parameter scaling where population is large, individual mutations are small but not rare, and the grid mesh for the trait values is much smaller than the size of mutation steps. When considering the evolution of the population in a long time scale, the contribution of small sub-populations may strongly influence the dynamics. Our main result quantifies the asymptotic dynamics of sub-population sizes on a logarithmic scale. We establish that under the parameter scaling the logarithm of the stochastic population size process, conveniently normalized, converges to the unique viscosity solution of a Hamilton-Jacobi equation. Such Hamilton-Jacobi equations have already been derived from parabolic integro-differential equations and have been widely developed in the study of adaptation of quantitative traits. Our work provides a justification of this framework directly from a stochastic individual based model, leading to a better understanding of the results obtained within this approach. The proof makes use of almost sure maximum principles and careful controls of the martingale parts. (10.5802/jep.244)
  DOI : 10.5802/jep.244
• A Posteriori Validation of Generalized Polynomial Chaos Expansions
  
  • Breden Maxime
  SIAM Journal on Applied Dynamical Systems, Society for Industrial and Applied Mathematics, 2023, 22 (2), pp.765-801. Generalized polynomial chaos (gPC) expansions are a powerful tool for studying differential equations with random coefficients, allowing, in particular, one to efficiently approximate random invariant sets associated with such equations. In this work, we use ideas from validated numerics in order to obtain rigorous a posteriori error estimates together with existence results about gPC expansions of random invariant sets. This approach also provides a new framework for conducting validated continuation, i.e., for rigorously computing isolated branches of solutions in parameter-dependent systems, which generalizes in a straightforward way to multiparameter continuation. We illustrate the proposed methodology by rigorously computing random invariant periodic orbits in the Lorenz system, as well as branches and 2 dimensional manifolds of steady states of the Swift–Hohenberg equation. (10.1137/22M1493197)
  DOI : 10.1137/22M1493197
• Foundations of Modern Statistics
  
  • Belomestny Denis
  • Butucea Cristina
  • Mammen Enno
  • Moulines Eric
  • Reiß Markus
  • Ulyanov Vladimir
  , 2023, 425. (10.1007/978-3-031-30114-8)
  DOI : 10.1007/978-3-031-30114-8
• Counter-Examples in First-Order Optimization: A Constructive Approach
  
  • Goujaud Baptiste
  • Dieuleveut Aymeric
  • Taylor Adrien
  IEEE Control Systems Letters, IEEE, 2023, 7, pp.2485-2490. While many approaches were developed for obtaining worst-case complexity bounds for first-order optimization methods in the last years, there remain theoretical gaps in cases where no such bound can be found. In such cases, it is often unclear whether no such bound exists (e.g., because the algorithm might fail to systematically converge) or simply if the current techniques do not allow finding them. In this work, we propose an approach to automate the search for cyclic trajectories generated by first-order methods. This provides a constructive approach to show that no appropriate complexity bound exists, thereby complementing approaches providing sufficient conditions for convergence. Using this tool, we provide ranges of parameters for which the famous Polyak heavy-ball, Nesterov accelerated gradient, inexact gradient descent, and three-operator splitting algorithms fail to systematically converge, and show that it nicely complements existing tools searching for Lyapunov functions. (10.1109/LCSYS.2023.3286277)
  DOI : 10.1109/LCSYS.2023.3286277
• Graph classes with few $P_4$'s: Universality and Brownian graphon limits
  
  • Lenoir Théo
  , 2023. We consider large uniform labeled random graphs in different classes with few induced $P_4$ ($P_4$ is the graph consisting of a single line of $4$ vertices) which generalize the case of cographs. Our main result is the convergence to a Brownian limit object in the space of graphons. As a by-product we obtain new asymptotic enumerative results for all these graph classes. We also obtain typical density results for a wide variety of induced subgraphs. These asymptotics hold at a smaller scale than what is observable through the graphon convergence. Our proofs rely on tree encoding of graphs. We then use mainly combinatorial arguments, including the symbolic method and singularity analysis.
• Continuous and discrete data assimilation with noisy observations for the Rayleigh-Bénard convection: a computational study
  
  • Hammoud Mohamad Abed El Rahman
  • Le Maître Olivier
  • Titi Edriss S
  • Hoteit Ibrahim
  • Knio Omar
  Computational Geosciences, Springer Verlag, 2023, 27, pp.63-79. Obtaining accurate high-resolution representations of model outputs is essential to describe the system dynamics. In general, however, only spatially-and temporally-coarse observations of the system states are available. These observations can also be corrupted by noise. Downscaling is a process/scheme in which one uses coarse scale observations to reconstruct the highresolution solution of the system states. Continuous Data Assimilation (CDA) is a recently introduced downscaling algorithm that constructs an increasingly accurate representation of the system states by continuously nudging the large scales using the coarse observations. We introduce a Discrete Data Assimilation (DDA) algorithm as a downscaling algorithm based on CDA with discrete-in-time nudging. We then investigate the performance of the CDA and DDA algorithms for downscaling noisy observations of the Rayleigh-Bénard convection system in the chaotic regime. In this computational study, a set of noisy observations was generated by perturbing a reference solution with Gaussian noise before downscaling them. The downscaled fields are then assessed using various error-and ensemble-based skill scores. The CDA solution was shown to converge towards the reference solution faster than that of DDA but at the cost of a higher asymptotic error. The numerical results also suggest a quadratic relationship between the 2 error and the noise level for both CDA and DDA. Cubic and quadratic dependences of the DDA and CDA expected errors on the spatial resolution of the observations were obtained, respectively. (10.1007/s10596-022-10180-4)
  DOI : 10.1007/s10596-022-10180-4
• Inversion of Eddy-Current Signals Using a Level-Set Method and Block Krylov Solvers
  
  • Audibert Lorenzo
  • Girardon Hugo
  • Haddar Houssem
  • Jolivet Pierre
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2023, 45 (3), pp.B366-B389. The application motivating this work is related to the identification of deposits inside nuclear power plant steam generators using eddy-current probes. We consider a realistic experimental process that relies on the scan of a domain by sweeping along a tube axis a probe made out of coils, playing the role of the sources/receivers. Solving the inverse shape problem associated with these measurements using a least squares method requires solutions to the eddy-current and the adjoint problems for a large number of right-hand sides at each gradient-descent iteration. Additional cost in the forward solver comes from the use of a potential formulation of the problem that has the advantage of being independent from the topology of the conductive media (that may vary during iterations). We use a level-set approach to avoid remeshing and handle unknown topologies. The crucial ingredient in our algorithm is an optimized way of handling high numbers of right-hand sides for iterative solvers of large-scale problems. We first benchmark various block Krylov methods, block GMRES and block BGCRODR, to test their effectiveness compared to their standard counterpart, i.e., GMRES and GCRODR. Then, we propose for BGCRODR a new implementation for recycling information from previously generated Krylov bases that scales better than traditional approaches. This part is independent from the practical inverse problem at hand. The efficiency of the overall inversion procedure is finally demonstrated on realistic synthetic 3D examples. (10.1137/20M1382064)
  DOI : 10.1137/20M1382064
• The linear sampling method for random sources
  
  • Garnier Josselin
  • Haddar Houssem
  • Montanelli Hadrien
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2023, 16 (3), pp.1572-1593. (10.1137/22M1531336)
  DOI : 10.1137/22M1531336
• Time reversal of diffusion processes under a finite entropy condition
  
  • Cattiaux Patrick
  • Conforti Giovanni
  • Gentil Ivan
  • Léonard Christian
  Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2023, 59 (4), pp.1844-1881. Motivated by entropic optimal transport, time reversal of diffusion processes is revisited. An integration by parts formula is derived for the carré du champ of a Markov process in an abstract space. It leads to a time reversal formula for a wide class of diffusion processes in Rn possibly with singular drifts, extending the already known results in this domain. The proof of the integration by parts formula relies on stochastic derivatives. Then, this formula is applied to compute the semimartingale characteristics of the time-reversed P* of a diffusion measure P provided that the relative entropy of P with respect to another diffusion measure R is finite, and the semimartingale characteristics of the time-reversed R* are known (for instance when the reference path measure R is reversible). As an illustration of the robustness of this method, the integration by parts formula is also employed to derive a time-reversal formula for a random walk on a graph. (10.1214/22-AIHP1320)
  DOI : 10.1214/22-AIHP1320
• TVD analysis of a (pseudo-)staggered scheme for the isentropic Euler equations
  
  • Ait-Ameur Katia
  • Ndjinga Michael
  Springer Proceedings in Mathematics & Statistics, Springer, 2023, 433, pp.249-257. In this paper, we build and analyze the stability of a collocated scheme, involving a specific numerical diffusion operator, for the isentropic Euler equations. This scheme is based on the numerical diffusion operator of a family of staggered finite volume schemes introduced in [1]. The properties of this operator allowed to understand the L 2-stability of staggered finite volume methods. Staggered schemes are popular in the thermal hydraulics community for their reported robustness and lack of spurious oscillations. The contributions of this paper are twofold: we firstly build a colocated scheme with a staggered-based numerical diffusion operator, hence the proposed terminology of "pseudo-staggered" scheme, and we secondly present a rigorous TVD analysis, in order to explain the non-oscillatory behaviour of staggered discretisations. (10.1007/978-3-031-40860-1_26)
  DOI : 10.1007/978-3-031-40860-1_26
• A discontinuous Galerkin spectral element method for a nonconservative compressible multicomponent flow model
  
  • Abgrall Rémi
  • Rai Pratik
  • Renac Florent
  Journal of Computational Physics, Elsevier, 2023, 472, pp.111693. In this work, we propose an accurate, robust (the solution remains in the set of states), and stable discretization of a nonconservative model for the simulation of compressible multicomponent flows with shocks and material interfaces. We consider the gamma-based model by Shyue [J. Comput. Phys., 142 (1998), 208-242] where each component follows a stiffened gas equation of state (EOS). We here extend the framework proposed in Renac [J. Comput. Phys., 382 (2019), 1-26] and Coquel et al. [J. Comput. Phys. 431 (2021), 110135] for the discretization of hyperbolic systems, with both fluxes and nonconservative products, to unstructured meshes with curved elements in multiple space dimensions. The framework relies on a high-order discontinuous Galerkin spectral element method (DGSEM) using collocation of quadrature and interpolation points as proposed by Gassner [SIAM J. Sci. Comput., 35 (2013)] in the case of hyperbolic conservation laws. We modify the integrals over discretization elements where we replace the physical fluxes and nonconservative products by two-point numerical fluctuations. The contributions of this work are threefold. First, we analyze the semi-discrete DGSEM discretization of general hyperbolic systems with conservative and nonconservative terms and derive the conditions to obtain a scheme that is high-order accurate, freestream preserving, and entropy stable when excluding material interfaces. Second, we design a three-point scheme with a HLLC solver for the gamma-based model that does not require a root-finding algorithm for the approximation of the nonconservative products. The scheme is proved to be robust and entropy stable for convex entropies, to preserve uniform profiles of pressure and velocity across material interfaces (material interface preservation), and to satisfy a discrete minimum principle on the specific entropy and maximum principles on the parameters of the EOS. Third, the HLLC solver is applied at interfaces in the DGSEM scheme, while we consider two kinds of fluctuations in the integrals over discretization elements: the former is entropy conservative (EC), while the latter preserves material interfaces (CP). Time integration is performed using high-order strong-stability preserving Runge-Kutta schemes. The fully discrete scheme is shown to preserve material interfaces with CP fluctuations. Under a given condition on the time step, both EC and CP fluctuations ensure that the cell-averaged solution remains in the set of states; satisfy a minimum principle on any convex entropy and maximum principles on the EOS parameters. These results allow to use existing limiters in order to restore positivity, and discrete maximum principles of degrees-of-freedom within elements. Numerical experiments in one and two space dimensions on flows with discontinuous solutions support the conclusions of our analysis and highlight stability, robustness and accuracy of the DGSEM scheme with either CP, or EC fluctuations, while the scheme with CP fluctuations is shown to offer better resolution capabilities. (10.1016/j.jcp.2022.111693)
  DOI : 10.1016/j.jcp.2022.111693
• Variance-based sensitivity analysis of oil spill predictions in the Red Sea region
  
  • Hammoud Mohamad Abed El Rahman
  • Mittal H V R
  • Le Maitre Olivier
  • Hoteit Ibrahim
  • Knio Omar
  Frontiers in Marine Science, Frontiers Media, 2023, 10, pp.1185106. To support accidental spill rapid response efforts, oil spill simulations may generally need to account for uncertainties concerning the nature and properties of the spill, which compound those inherent in model parameterizations. A full detailed account of these sources of uncertainty would however require prohibitive resources needed to sample a large dimensional space. In this work, a variance based sensitivity analysis is conducted to explore the possibility of restricting a priori the set of uncertain parameters, at least in the context of realistic simulations of oil spills in the Red Sea region spanning a two-week period following the oil release. The evolution of the spill is described using the simulation capabilities of Modelo Hidrodinamico, driven by high-resolution metocean fields of the Red Sea (RS) was adopted to simulate accidental oil spills in the RS. Eight spill scenarios are considered in the analysis, which are carefully selected to account for the diversity of meto-cean conditions in the region. Polynomial chaos expansions are employed to propagate parametric uncertainties and efficiently estimate variance-based sensitivities. Attention is focused on integral quantities characterizing the transport, deformation, evaporation and dispersion of the spill. The analysis indicates that variability in these quantities may be suitably captured by restricting the set of uncertain inputs parameters, namely the wind coefficient, interfacial tension, API gravity, and viscosity. Thus, forecast variability and confidence intervals may be reasonably estimated in the corresponding four dimensional input space. (10.3389/fmars.2023.1185106)
  DOI : 10.3389/fmars.2023.1185106