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We introduce a novel relaxation strategy for denoising hyperbolic-valued data. The main challenge is here the non-convexity of the hyperbolic sheet. Instead of considering the denoising problem directly on the hyperbolic space, we exploit the Euclidean embedding and encode the hyperbolic sheet using a novel matrix representation. For denoising, we employ the Euclidean Tikhonov and total variation (TV) model, where we incorporate our matrix representation. The major contribution is then a convex relaxation of the variational ans\"atze allowing the utilization of well-established convex optimization procedures like the alternating directions method of multipliers (ADMM). The resulting denoisers are applied to a real-world Gaussian image processing task, where we simultaneously restore the pixelwise mean and standard deviation of a retina scan series., Comment: 12 pages, 4 figures, 1 table

This paper considers the problem of computing the operator norm of a linear map between finite dimensional Hilbert spaces when only evaluations of the linear map are available and under restrictive storage assumptions. We propose a stochastic method of random search type for the maximization of the Rayleigh quotient and employ an exact line search in the random search directions. Moreover, we show that the proposed algorithm converges to the global maximum (the operator norm) almost surely and illustrate the performance of the method with numerical experiments.

This article concerns the mathematical justification of an averaged system of partial differential equations governing the evolution of a two-phase mixture of heat non-conductive, compressible fluids, in space dimension 1 with periodic boundary conditions. The first step is to write a system of partial differential equations governing the physics at a mesoscopical level where the two phases are separated by sharp interfaces. In a second step we study the behavior of a sequence of solutions corresponding to highly oscillating initial densities and temperatures. The limit of such a sequence is described byan averaged system of partial differential equations that has a different algebraic structure than the mesoscopic model and that reflects the macroscopical properties of the mixture. The originality of the paper consists in the fact that both the density and temperature are allowed to oscillate, so that the limiting model is a six-equation, two-pressures, two-temperatures model. The key point is to show the strong convergence of the stress tensor in $L^2((0, T) \times (0, 1))$. In order to obtain uniform estimates in spite of the presence of oscillating coefficients in the energy equation, we look at solutions with low regularity for the density and the temperature.For the readers convenience, we provide in a separated section the modelization of the mesoscopic system and the formal derivation of the averaged two-phase system. We also provide, at the end of the paper, numerical illustrations comparing the unknowns solving the mesoscopic system and the one satisfying the averaged two-phase system.

Regularized optimal transport (OT) has received much attention in recent years starting from Cuturi's paper with Kullback-Leibler (KL) divergence regularized OT. In this paper, we propose to regularize the OT problem using the family of $\alpha$-R\'enyi divergences for $\alpha \in (0, 1)$. R\'enyi divergences are neither $f$-divergences nor Bregman distances, but they recover the KL divergence in the limit $\alpha \nearrow 1$. The advantage of introducing the additional parameter $\alpha$ is that for $\alpha \searrow 0$ we obtain convergence to the unregularized OT problem. For the KL regularized OT problem, this was achieved by letting the regularization parameter tend to zero, which causes numerical instabilities. We present two different ways to obtain premetrics on probability measures, namely by R\'enyi divergence constraints and penalization. The latter premetric interpolates between the unregularized and KL regularized OT problem with weak convergence of the minimizer, generalizing the interpolating property of KL regularized OT. We use a nested mirror descent algorithm for solving the primal formulation. Both on real and synthetic data sets R\'enyi regularized OT plans outperform their KL and Tsallis counterparts in terms of being closer to the unregularized transport plans and recovering the ground truth in inference tasks better., Comment: 40 pages, 9 figures, 3 tables, comments welcome

Circle- and sphere-valued data play a significant role in inverse problems like magnetic resonance phase imaging and radar interferometry, in the analysis of directional information, and in color restoration tasks. In this paper, we aim to restore $(d-1)$-sphere-valued signals exploiting the classical anisotropic total variation on the surrounding $d$-dimensional Euclidean space. For this, we propose a novel variational formulation, whose data fidelity is based on inner products instead of the usually employed squared norms. Convexifying the resulting non-convex problem and using ADMM, we derive an efficient and fast numerical denoiser. In the special case of binary (0-sphere-valued) signals, the relaxation is provable tight, i.e. the relaxed solution can be used to construct a solution of the original non-convex problem. Moreover, the tightness can be numerically observed for barcode and QR code denoising as well as in higher dimensional experiments like the color restoration using hue and chromaticity and the recovery of SO(3)-valued signals.

We introduce a new approach to justify mean-field limits for first-and second-order particle systems with singular interactions. It is based on a duality approach combined with the analysis of linearized dual correlations, and it allows to cover for the first time arbitrary square-integrable interaction forces at possibly vanishing temperature. In case of first-order systems, it allows to recover in particular the mean-field limit to the 2d Euler and Navier-Stokes equations. We postpone to a forthcoming work the development of quantitative estimates and the extension to more singular interactions.

Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, a convex relaxation of the otherwise nonconvex Tikhonov-regularization for denoising circle-valued data has been proposed by Condat (2022). The circle constraints are here encoded in a series of low-dimensional, positive semi-definite matrices. Using Schur complement arguments, we show that the resulting variational model can be simplified while leading to the same solution. The simplified model can be generalized to higher dimensional spheres and to SO(3)-valued data, where we rely on the quaternion representation of the latter. Standard algorithms from convex analysis can be applied to solve the resulting convex minimization problem. As proof-of-the-concept, we use the alternating direction method of multipliers to demonstrate the denoising behavior of the proposed method. In a series of experiments, we demonstrate the numerical convergence of the signal- or image values to the underlying manifold.

This paper concerns the existence of global weak solutions {\it \`a la Leray} for compressible Navier--Stokes--Fourier systems with periodic boundary conditions and the truncated virial pressure law which is assumed to be thermodynamically unstable. More precisely, the main novelty is that the pressure law is not assumed to be monotone with respect to the density. This provides the first global weak solutions result for the compressible Navier-Stokes-Fourier system with such kind of pressure law which is strongly used as a generalization of the perfect gas law. The paper is based on a new construction of approximate solutions through an iterative scheme and fixed point procedure which could be very helpful to design efficient numerical schemes. Note that our method involves the recent paper by the authors published in Nonlinearity (2021) for the compactness of the density when the temperature is given.

This work is devoted to investigating a compressible fluid system with low stratification, which is driven by fast acoustic waves and internal waves. The approximation using a soundproof model is justified. More precisely, the soundproof model captures the dynamics of both the non-oscillating mean flows and the oscillating internal waves, while filters out the fast acoustic waves, of the compressible system with or without initial acoustic waves. Moreover, the fast-slow oscillation structure is investigated.

Extreme weather events can have severe impacts on national economies, leading the recovery of low- to middle-income countries to become reliant on foreign financial aid. Foreign aid, however, is slow and uncertain. Therefore, the Sendai Framework and the Paris Agreement advocate for more resilient financial instruments like sovereign catastrophe risk pools. Existing pools, however, might not fully exploit financial resilience potentials because they were not designed with the goal of maximizing risk diversification and they pool risk only regionally. To address this, we introduce a method that forms pools maximizing risk diversification and which selects countries with low bilateral correlations or low shares in the pool risk. We apply the method to explore the benefits of global pooling with respect to regional pooling. We find that global pooling increases risk diversification, it lowers countries shares in the pool risk and it increases the number of countries profiting from risk pooling.

This article introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension~$2$ together with a partial result in dimension~$3$. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted $L^p$ norms of the marginals or observables of the system, uniformly in the number of particles. This allows to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in $L^2$.

Background and Objectives: In a significant portion of surgeries, blood pressure (BP) is often measured non-invasively in an intermittent manner. This practice has a risk of missing clinically relevant BP changes between two adjacent intermittent BP measurements. This study proposes a method to non-invasively estimate systolic blood pressure (SBP) with high accuracy in patients undergoing surgery. Methods: Continuous arterial BP, electrocardiography (ECG), and photoplethysmography (PPG) signals were acquired from 29 patients undergoing surgery. After extracting 9 features from the PPG and ECG signals, we dynamically selected features upon each intermittent measurement (every 10 min) of SBP based on feature robustness and the principle of correlation-based feature selection. Finally, multiple linear regression models were built to combine these features to estimate SBP every 30 s. Results: Compared to the reference SBP, the proposed method achieved a mean of difference at 0.08 mmHg, a standard deviation of difference at 7.97 mmHg, and a correlation coefficient at 0.89 (p < 0.001). Conclusions: This study demonstrates the feasibility of non-invasively estimating SBP every 30 s with high accuracy during surgery by using ECG, PPG, and intermittent SBP measurements every 10 min, which meets the standard of the Association for the Advancement of Medical Instrumentation. The proposed method has the potential to enhance BP monitoring in the operating room, improving patient outcomes and experiences.

This paper deals with the Navier-Stokes system governing the evolution of a compressible barotropic fluid. We extend D. Hoff's intermediate regularity solutions framework by relaxing the integrability needed for the initial density which is usually assumed to be $L^{\infty}$. By achieving this, we are able to take into account general fourth order symmetric viscous-stress tensors with coefficients depending smoothly on the time-space variables. More precisely, in space dimensions $d=2,3$, under periodic boundary conditions, considering a pressure law $p(\rho)=a\rho^{\gamma}$ whith $a>0$ respectively $\gamma\geq d/(4-d)$) and under the assumption that the norms of the initial data $\left( \rho_{0}-M,u_{0}\right) \in L^{2\gamma}\left(\mathbb{T}^{d}\right) \times(H^{1}(\mathbb{T}^{d}))^{d}$ are sufficiently small, we are able to construct global weak solutions. Above, $M$ denotes the total mass of the fluid while $\mathbb{T}$ with $d=2,3$ stands for periodic box. When comparing to the results known for the global weak solutions \`{a} la Leray, i.e. constructed assuming only the basic energy bounds, we obtain a relaxed condition on the range of admissible adiabatic coefficients $\gamma$., Comment: 31 pages

In this note, we propose the first mathematical derivation of a macroscopic Baer-Nunziato type system for compressible two-phase flows allowing two pressure state laws depending on the different phases. By doing so, we extend the results obtained by the first author and M. Hillairet (Annales ENS 2019) to cover this important physical situation. A relaxation term in the mass fraction equation is obtained without closure assumptions contrarily to theoretical-physics literature dedicated to mixture theory, see for instance "Thermo-Fluid dynamics of Two Fluid Flows" by M. Ishii. The relaxation parameter is linked to the viscosities of the different fluids (which may be small for applications) and the relaxed quantity is linked to the laws chosen at interfaces of the two-fluid system at a mesoscale.

This paper proves the mean field limit and quantitative estimates for many-particle systems with singular attractive interactions between particles. As an important example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in optimal subcritical regimes is obtained for the first time. To give an answer to this longstanding problem, we take advantage of a new modulated free energy and we prove some precise large deviation estimates encoding the competition between diffusion and attraction. Combined with the range of repulsive kernels, already treated in the s{\'e}minaire Laurent Schwartz proceeding [https://slsedp.centre-mersenne.org/journals/SLSEDP/ ], we provide the full proof of results announced by the authors in [C. R. Acad. Sciences Section Maths (2019)].

This paper concerns the existence of global weak solutions \`a la Leray for compressible Navier-Stokes equations with a pressure law that depends on the density and on time and space variables $t$ and $x$. The assumptions on the pressure contain only locally Lipschitz assumption with respect to the density variable and some hypothesis with respect to the extra time and space variables. It may be seen as a first step to consider heat-conducting Navier-Stokes equations with physical laws such as the truncated virial assumption. The paper focuses on the construction of approximate solutions through a new regularized and fixed point procedure and on the weak stability process taking advantage of the new method introduced by the two first authors with a careful study of an appropriate regularized quantity linked to the pressure.

In this paper, we consider global weak solutions to com-pressible Navier-Stokes-Korteweg equations with density dependent viscosities , in a periodic domain $\Omega = \mathbb T^3$, with a linear drag term with respect to the velocity. The main result concerns the exponential decay to equilibrium of such solutions using log-sobolev type inequalities. In order to show such a result, the starting point is a global weak-entropy solutions definition introduced in D. Bresch, A. Vasseur and C. Yu [12]. Assuming extra assumptions on the shear viscosity when the density is close to vacuum and when the density tends to infinity, we conclude the exponential decay to equilibrium. Note that our result covers the quantum Navier-Stokes system with a drag term.

This paper is motivated by the study of Lyapunov functionals for four equations describing free surface flows in fluid dynamics: the Hele-Shaw and Mullins-Sekerka equations together with their lubrication approximations, the Boussinesq and thin-film equations. We identify new Lyapunov functionals, including some which decay in a convex manner (these are called strong Lyapunov functionals). For the Hele-Shaw equation and the Mullins-Sekerka equation, we prove that the $L^2$-norm of the free surface elevation and the area of the free surface are Lyapunov functionals, together with parallel results for the thin-film and Boussinesq equations. The proofs combine exact identities for the dissipation rates with functional inequalities. For the thin-film and Boussinesq equations, we introduce a Sobolev inequality of independent interest which revisits some known results and exhibits strong Lyapunov functionals. For the Hele-Shaw and Mullins-Sekerka equations, we introduce a functional which controls the $L^2$-norm of three-half spatial derivative. Under a mild smallness assumption on the initial data, we show that the latter quantity is also a Lyapunov functional for the Hele-Shaw equation, implying that the area functional is a strong Lyapunov functional. Precise lower bounds for the dissipation rates are established, showing that these Lyapunov functionals are in fact entropies. Other quantities are also studied such as Lebesgue norms or the Boltzmann's entropy., Comment: 46 pages, 1 table

In this paper, we prove global existence of weak solutions for the stationary compressible Navier-Stokes equations with an anisotropic and nonlocal viscous term in a periodic domain. This gives an answer to an open problem important for applications in geophysics or in microuidics. The main idea is to adapt in a non-trivial way the new idea developped by the authors in a previous paper, see [2] which allowed them to treat the anisotropic compressible quasi-stationary Stokes system.

This is the document corresponding to the talk the first author gave at IH{\'E}S for the Laurent Schwartz seminar on November 19, 2019. It concerns our recent introduction of a modulated free energy in mean-field theory in BrJaWa [4]. This physical object may be seen as a combination of the modulated potential energy introduced by S. Serfaty [See Proc. Int. Cong. Math. (2018)] and of the relative entropy introduced in mean field limit theory by P.-E. Jabin, Z. Wang [See Inventiones 2018]. It allows to obtain, for the first time, a convergence rate in the mean field limit for Riesz and Coulomb repulsive kernels in the presence of viscosity using the estimates in Du [8] and Se1 [20]. The main objective in this paper is to explain how it is possible to cover more general repulsive kernels through a Fourier transform approach as announced in BrJaWa [4] first in the case $\sigma$N $\rightarrow$ 0 when N $\rightarrow$ +$\infty$ and then if $\sigma$ > 0 is fixed. Then we end the paper with comments on the particle approximation of the Patlak-Keller-Segel system which is associated to an attractive kernel and refer to [C.R.

In this paper, we introduce a new extended version of the shallow water equations with surface tension which is skew-symmetric with respect to the L2 scalar product and allows for large gradients of fluid height. This result is a generalization of the results published by P. Noble and J.-P. Vila in [SIAM J. Num. Anal. (2016)] and by D. Bresch, F. Couderc, P. Noble and J.P. Vila in [C.R. Acad. Sciences Paris (2016)] which are restricted to quadratic forms of the capillary energy respectively in the one dimensional and two dimensional setting.This is also an improvement of the results by J. Lallement, P. Villedieu et al. published in [AIAA Aviation Forum 2018] where the augmented version is not skew-symetric with respect to the L2 scalar product. Based on this new formulation, we propose a new numerical scheme and perform a nonlinear stability analysis.Various numerical simulations of the shallow water equations are presented to show differences between quadratic (w.r.t the gradient of the height) and general surface tension energy when high gradients of the fluid height occur.

In this paper, we study the problem of global existence of weak solutions for the quasi-stationary compressible Stokes equations with an anisotropic viscous tensor. The key element of our proof is the control of a particular defect measure associated to the pressure which avoids the use of the eective ux. Using this new tool, we solve an open problem namely global existence of solutions {\`a} la Leray for such a system without assuming any restriction on the anisotropy amplitude. It provides a exible and natural way to treat compressible quasilinear Stokes systems which are important for instance in biology, porous media, supra-conductivity or other applications in the low Reynolds number regime., Comment: Compared to the version in Ann. IHP. : Revised argument in the proof of Theorem 2.10, main results unchanged : the proof of the main result was incomplete because we cannot prove that the defect measure vanishes at initial time. We prove instead that the mean value of the defect measure over any interval [0,T] is 0. This is sufficient in order to identify the pressure. See Lemmas 2.11 and 2.12

In this note, we propose a new relative entropy combination of the methods developed by P.--E. Jabin and Z.~Wang [Inventiones (2018)] and by S. Serfaty [Proc. Int. Cong. of Math, (2018) and references therein] to treat more general kernels in mean field limit theory. This new relative entropy may be understood as introducing appropriate weights in the relative entropy developed by P.-E. Jabin and Z. Wang (in the spirit of what has been recently developed by D.~Bresch and P.--E. Jabin [Annals of Maths (2018)]) to cancel the more singular terms involving the divergence of the flow. As an example, a full rigorous derivation (with quantitative estimates) of the Patlak-Keller-Segel model in some subcritical regimes is obtained. Our new relative entropy allows to treat singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part.

In this paper, we extend considerably the global existence results of entropy-weak solutions related to compressible Navier-Stokes system with density dependent viscosities obtained, independently (using different strategies), by Vasseur-Yu [Inventiones mathematicae (2016) and arXiv:1501.06803 (2015)] and by Li-Xin [arXiv:1504.06826 (2015)].More precisely we are able to consider a physical symmetric viscous stress tensor $\sigma=2\mu(\rho)\,{\mathbb{D}}(u)+\bigl(\lambda(\rho){\rm div}u -P(\rho)\bigr)\, {\rm Id}$ where ${\mathbb D}(u) = [\nabla u + \nabla^T u]/2$ with a shear and bulk viscosities (respectively $\mu(\rho)$ and $\lambda(\rho)$) satisfying the BD relation $\lambda(\rho)=2(\mu'(\rho)\rho - \mu(\rho))$ and a pressure law $P(\rho)=a\rho^\gamma$ (with $a>0$ a given constant) for any adiabatic constant $\gamma>1$. The nonlinear shear viscosity $\mu(\rho)$ satisfies some lower and upper bounds for low and high densities (our mathematical result includes the case $\mu(\rho)= \mu\rho^\alpha$ with $2/3 < \alpha < 4$ and $\mu>0$ constant). This provides an answer to a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by F. Rousset in the Bourbaki 69\`eme ann\'ee, 2016--2017, no 1135.