The last decade has witnessed a remarkable convergence between several sub-domains of the calculus of variations, namely optimal transport (and its many generalizations), infinite dimensional geometry of diffeomorphisms groups and inverse problems in imaging (in particular sparsity-based regularization). This convergence is due to (i) the mathematical objects manipulated in these problems, namely sparse measures (e.g. coupling in transport, edge location in imaging, displacement fields for diffeomorphisms) and (ii) the use of similar numerical tools from non-smooth optimization and geometric discretization schemes. Optimal Transportation, diffeomorphisms and sparsity-based methods are powerful modeling tools, that impact a rapidly expanding list of scientific applications and call for efficient numerical strategies. Our research program shows the important part played by the team members in the development of these numerical methods and their application to challenging problems.

Optimal Mass Transportation is a mathematical research topic which started two centuries ago with Monge's work on the “Théorie des déblais et des remblais" (see 101).
This engineering problem consists in minimizing the transport cost between two given mass densities. In the 40's, Kantorovich 110 introduced a powerful linear relaxation and introduced its dual formulation. The Monge-Kantorovich problem became a specialized research topic in optimization and Kantorovich obtained the 1975 Nobel prize in economics for his contributions to resource allocations problems. Since the seminal discoveries of Brenier in the 90's 64, Optimal Transportation has received renewed attention from mathematical analysts and the Fields Medal awarded in 2010 to C. Villani, who gave important contributions to Optimal Transportation and wrote the modern reference monographs 134, 135, arrived at a culminating moment for this theory. Optimal Mass Transportation is today a mature area of mathematical analysis with a constantly growing range of applications. Optimal Transportation has also received a lot of attention from probabilists (see for instance the recent survey 114 for an overview of the Schrödinger problem which is a stochastic variant of the Benamou-Brenier dynamical formulation of optimal transport). The development of numerical methods for Optimal Transportation and Optimal Transportation related problems is a difficult topic and comparatively underdeveloped. This research field has experienced a surge of activity in the last five years, with important contributions of the Mokaplan group (see the list of important publications of the team). We describe below a few of recent and less recent Optimal Transportation concepts and methods which are connected to the future activities of Mokaplan :

Brenier's theorem 67 characterizes the unique optimal map as the gradient of a convex
potential. As such Optimal Transportation may be interpreted as an infinite dimensional optimisation problem under “convexity constraint": i.e. the solution of this infinite dimensional optimisation problem is a convex potential. This connects Optimal Transportation to “convexity constrained" non-linear variational problems such as, for instance, Newton's problem of the body of minimal resistance.
The value function of the optimal transport problem is also known to define a distance between source and target densities called the Wasserstein distance which plays a key role in many applications such as image processing.

A formal substitution of the optimal transport map as the gradient of a convex potential in the mass conservation constraint (a Jacobian equation) gives a non-linear Monge-Ampère equation. Caffarelli 71 used this result to extend the regularity theory for the Monge-Ampère equation. In the last ten years, it also motivated new research on numerical solvers for non-linear degenerate Elliptic equations 9311857 56 and the references therein. Geometric approaches based on Laguerre diagrams and discrete data 121 have also been developed. Monge-Ampère based Optimal Transportation solvers have recently given the first linear cost computations of Optimal Transportation (smooth) maps.

In recent years, the classical Optimal Transportation problem has been extended in several directions. First, different ground costs measuring the “physical" displacement have been considered. In particular, well posedness for a large class of convex and concave costs has been established by McCann and Gangbo 100. Optimal Transportation techniques have been applied for example to a Coulomb ground cost in Quantum chemistry in relation with Density Functional theory 87. Given the densities of electrons Optimal Transportation models the potential energy and their relative positions. For more than more than 2 electrons (and therefore more than 2 densities) the natural extension of Optimal Transportation is the so called Multi-marginal Optimal Transport (see 126 and the references therein). Another instance of multi-marginal Optimal Transportation arises in the so-called Wasserstein barycenter problem between an arbitrary number of densities 42. An interesting overview of this emerging new field of optimal transport and its applications can be found in the recent survey of Ghoussoub and Pass 125.

Optimal transport has found many applications, starting from its relation with several physical models such as the semi-geostrophic equations in meteorology 105, 90, 89, 53, 117, mesh adaptation 116, the reconstruction of the early mass distribution of the Universe 98, 65 in Astrophysics, and the numerical optimisation of reflectors following the Optimal Transportation interpretation of Oliker 70 and Wang 136. Extensions of OT such as multi-marginal transport has potential applications in Density Functional Theory , Generalized solution of Euler equations 66 (DFT) and in statistics and finance 51, 99 .... Recently, there has been a spread of interest in applications of OT methods in imaging sciences 60, statistics 58 and machine learning 91. This is largely due to the emergence of fast numerical schemes to approximate the transportation distance and its generalizations, see for instance 55. Figure 1 shows an example of application of OT to color transfer. Figure 2 shows an example of application in computer graphics to interpolate between input shapes.

While the optimal transport problem, in its original formulation, is a static problem (no time evolution is considered), it makes sense in many applications to rather consider time evolution. This is relevant for instance in applications to fluid dynamics or in medical images to perform registration of organs and model tumor growth.

In this perspective, the optimal transport in Euclidean space corresponds to an evolution where each particule of mass evolves in straight line. This interpretation corresponds to the Computational Fluid Dynamic (CFD) formulation proposed by Brenier and Benamou in 52. These solutions are time curves in the space of densities and geodesics for the Wasserstein distance. The CFD formulation relaxes the non-linear mass conservation constraint into a time dependent continuity equation, the cost function remains convex but is highly non smooth. A remarkable feature of this dynamical formulation is that it can be re-cast as a convex but non smooth optimization problem. This convex dynamical formulation finds many non-trivial extensions and applications, see for instance 54. The CFD formulation also appears to be a limit case of Mean Fields games (MFGs), a large class of economic models introduced by Lasry and Lions 112 leading to a system coupling an Hamilton-Jacobi with a Fokker-Planck equation. In contrast, the Monge case where the ground cost is the euclidan distance leads to a static system of PDEs 61.

Another extension is, instead of considering geodesic for transportation metric (i.e. minimizing the Wasserstein distance to a target measure), to make the density evolve in order to minimize some functional. Computing the steepest descent direction with respect to the Wasserstein distance defines a so-called Wasserstein gradient flow, also known as JKO gradient flows after its authors 109. This is a popular tool to study a large class of non-linear diffusion equations. Two interesting examples are the Keller-Segel system for chemotaxis 108, 79 and a model of congested crowd motion proposed by Maury, Santambrogio and Roudneff-Chupin 120. From the numerical point of view, these schemes are understood to be the natural analogue of implicit scheme for linear parabolic equations. The resolution is however costly as it involves taking the derivative in the Wasserstein sense of the relevant energy, which in turn requires the resolution of a large scale convex but non-smooth minimization.

To tackle more complicated warping problems, such as those encountered in medical image analysis, one unfortunately has to drop the convexity of the functional involved in defining the gradient flow. This gradient flow can either be understood as defining a geodesic on the (infinite dimensional) group of diffeomorphisms 50, or on a (infinite dimensional) space of curves or surfaces 137. The de-facto standard to define, analyze and compute these geodesics is the “Large Deformation Diffeomorphic Metric Mapping” (LDDMM) framework of Trouvé, Younes, Holm and co-authors 50, 104. While in the CFD formulation of optimal transport, the metric on infinitesimal deformations is just the

Beside image warping and registration in medical image analysis, a key problem in nearly all imaging applications is the reconstruction of high quality data from low resolution observations. This field, commonly referred to as “inverse problems”, is very often concerned with the precise location of features such as point sources (modeled as Dirac masses) or sharp contours of objects (modeled as gradients being Dirac masses along curves). The underlying intuition behind these ideas is the so-called sparsity model (either of the data itself, its gradient, or other more complicated representations such as wavelets, curvelets, bandlets 119 and learned representation 138).

The huge interest in these ideas started mostly from the introduction of convex methods to serve as proxy for these sparse regularizations. The most well known is the

However, the theoretical analysis of sparse reconstructions involving real-life acquisition operators (such as those found in seismic imaging, neuro-imaging, astro-physical imaging, etc.) is still mostly an open problem. A recent research direction, triggered by a paper of Candès and Fernandez-Granda 74, is to study directly the infinite dimensional problem of reconstruction of sparse measures (i.e. sum of Dirac masses) using the total variation of measures (not to be mistaken for the total variation of 2-D functions). Several works 73, 95, 94 have used this framework to provide theoretical performance guarantees by basically studying how the distance between neighboring spikes impacts noise stability.

In image processing, one of the most popular methods is the total variation regularization 129, 68. It favors low-complexity images that are piecewise constant, see Figure 4 for some examples on how to solve some image processing problems. Beside applications in image processing, sparsity-related ideas also had a deep impact in statistics 131 and machine learning 46. As a typical example, for applications to recommendation systems, it makes sense to consider sparsity of the singular values of matrices, which can be relaxed using the so-called nuclear norm (a.k.a. trace norm) 45. The underlying methodology is to make use of low-complexity regularization models, which turns out to be equivalent to the use of partly-smooth regularization functionals 115, 133 enforcing the solution to belong to a low-dimensional manifold.

The dynamical formulation of optimal transport creates a link between optimal transport and geodesics on diffeomorphisms groups. This formal link has at least two strong implications that Mokaplan will elaborate on: (i) the development of novel models that bridge the gap between these two fields ; (ii) the introduction of novel fast numerical solvers based on ideas from both non-smooth optimization techniques and Bregman metrics.

In a similar line of ideas, we believe a unified approach is needed to tackle both sparse regularization in imaging and various generalized OT problems. Both require to solve related non-smooth and large scale optimization problems. Ideas from proximal optimization has proved crucial to address problems in both fields (see for instance 52, 128). Transportation metrics are also the correct way to compare and regularize variational problems that arise in image processing (see for instance the Radon inversion method proposed in 55) and machine learning (see 91).

Since its creation, the Mokaplan team has made important contributions in Optimal Transport both on the theoretical and the numerical side, together with applications such as fluid mechanics, the simulation biological systems, machine learning. We have also contributed to to the field of inverse problems in signal and image processing (super-resolution, nonconvex low rank matrix recovery). In 2022, the team was renewed with the following research program which broadens our spectrum and addresses exciting new problems.

FreeForm Optics, Fluid Mechanics (Incompressible Euler, Semi-Geostrophic equations), Quantum Chemistry (Density Functional Theory), Statistical Physics (Schroedinger problem), Porous Media.

Full Waveform Inversion (Geophysics), Super-resolution microscopy (Biology), Satellite imaging (Meteorology)

Mean-field games, spatial economics, principal-agent models, taxation, nonlinear pricing.

We prove in 17 the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [Colombo, De Rosa, Marchese ; 2021], extending it to the Lagrangian framework.

When expressed in Lagrangian variables, the equations of motion for compressible (barotropic) fluids have the structure of a classical Hamiltonian system in which the potential energy is given by the internal energy of the fluid. The dissipative counterpart of such a system coincides with the porous medium equation, which can be cast in the form of a gradient flow for the same internal energy. Motivated by these related variational structures, we propose in 19 a particle method for both problems in which the internal energy is replaced by its Moreau-Yosida regularization in the L2 sense, which can be efficiently computed as a semi-discrete optimal transport problem. Using a modulated energy argument which exploits the convexity of the problem in Eulerian variables, we prove quantitative convergence estimates towards smooth solutions. We verify such estimates by means of several numerical tests.

We address in 11 the “freeform optics” inverse problem of designing a reﬂector surface mapping a prescribed source distribution of light to a prescribed far ﬁeld distribution, for a ﬁnite light source. When the ﬁnite source reduces to a point source, the light source distribution has support only on the optics ray directions. In this setting the inverse problem is well posed for arbitrary source and target probability distributions. It can be recast as an Optimal Transportation problem and has been studied both mathematically and nu-merically. We are not aware of any similar mathematical formulation in the ﬁnite source case: i.e. the source has an “´etendue” with support both in space and directions. We propose to leverage the well-posed variational formulation of the point source problem to build a smooth parameterization of the reﬂec-tor and the reﬂection map. Under this parameterization we can construct a smooth loss/misﬁt function to optimize for the best solution in this class of reﬂectors. Both steps, the parameterization and the loss, are related to Optimal Transportation distances. We also take advantage of recent progress in the numerical approximation and resolution of these mathematical objects to perform a numerical study.

The aim of 13 is to give an elementary proof of linear convergence of the Sinkhorn algorithm for the entropic regularization of multi-marginal optimal transport. The proof simply relies on: i) the fact that Sinkhorn iterates are bounded, ii) strong convexity of the exponential on bounded intervals and iii) the convergence analysis of the coordinate descent (Gauss-Seidel) method of Beck and Tetruashvili.

FISTA is a popular convex optimisation algorithm which is known to converge at an optimal rate whenever the optimisation domain is contained in a suitable Hilbert space. We propose in 16 a modified algorithm where each iteration is performed in a subspace, and that subspace is allowed to change at every iteration. Analytically, this allows us to guarantee convergence in a Banach space setting, although at a reduced rate depending on the conditioning of the specific problem. Numerically we show that a greedy adaptive choice of discretisation can greatly increase the time and memory efficiency in infinite dimensional Lasso optimisation problems.

We discuss in 15 the efficiency of Hybrid Primal-Dual (HPD) type algorithms to approximate solve discrete Optimal Transport (OT) and Wasserstein Barycenter (WB) problems, with and without entropic regularization. Our first contribution is an analysis showing that these methods yield state-of-the-art convergence rates, both theoretically and practically. Next, we extend the HPD algorithm with linesearch proposed by Malitsky and Pock in 2018 to the setting where the dual space has a Bregman divergence, and the dual function is relatively strongly convex to the Bregman's kernel. This extension yields a new method for OT and WB problems based on smoothing of the objective that also achieves state-of-the-art convergence rates. Finally, we introduce a new Bregman divergence based on a scaled entropy function that makes the algorithm numerically stable and reduces the smoothing, leading to sparse solutions of OT and WB problems. We complement our findings with numerical experiments and comparisons.

We introduce in 14 an algorithm to solve linear inverse problems regularized with the total (gradient) variation in a gridless manner. Contrary to most existing methods, that produce an approximate solution which is piecewise constant on a fixed mesh, our approach exploits the structure of the solutions and consists in iteratively constructing a linear combination of indicator functions of simple polygons.

We consider in 39 first order local minimization problems

In 40, we consider algorithms for reconstructing time-varying data into a finite sum of discrete trajectories, alternatively, an off-the-grid sparse-spikes decomposition which is continuous in time. Recent work showed that this decomposition was possible by minimising a convex variational model which combined a quadratic data fidelity with dynamical Optimal Transport. We generalise this framework and propose new numerical methods which leverage efficient classical algorithms for computing shortest paths on directed acyclic graphs. Our theoretical analysis confirms that these methods converge to globally optimal reconstructions which represent a finite number of discrete trajectories. Numerically, we show new examples for unbalanced Optimal Transport penalties, and for balanced examples we are 100 times faster in comparison to the previously known method.

A classical tool for approximating integrals is the Laplace method. The first-order, as well as the higher-order Laplace formula is most often written in coordinates without any geometrical interpretation. In

8, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation of the first-order term of the Laplace method. The central tool is the Kim–McCann Riemannian metric which was introduced in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two different metrics arising naturally in the Kim–McCann framework. Passing by, we give an explicitly quantified version of the Laplace formula, as well as examples of applications.

We investigate in

32the convergence rate of the optimal entropic cost vε to the optimal transport cost as the noise parameter

. We show that for a large class of cost functions

on

(for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and

marginals, one has

. Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on c, i.e. invertibility of

, we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty's trick.

ROMSOC project on cordis.europa.eu

The development of high quality products and processes is essential for the future competitiveness of the European economy. In most key technology areas product development is increasingly based on simulation and optimization via mathematical models that allow to optimize design and functionality using free design parameters. Best performance of modelling, simulation and optimization (MSO) techniques is obtained by using a model hierarchy ranging from very fine to very coarse models obtained by model order reduction (MOR) techniques and to adapt the model and the methods to the user-defined requirements in accuracy and computational speed.

ROMSOC will work towards this goal for high dimensional and coupled systems that describe different physical phenomena on different scales; it will derive a common framework for different industrial applications and train the next generation of researchers in this highly interdisciplinary field. It will focus on the three major methodologies: coupling methods, model reduction methods, and optimization methods, for industrial applications in well selected areas, such as optical and electronic systems, economic processes, and materials. ROMSOC will develop novel MSO techniques and associated software with adaptability to user-defined accuracy and efficiency needs in different scientific disciplines. It will transfer synergies between different industrial sectors, in particular for SMEs.

To lift this common framework to a new qualitative level, a joint training programme will be developed which builds on the strengths of the academic and industrial partners and their strong history of academic/industrial cooperation. By delivering early-career training embedded in a cutting-edge research programme, ROMSOC will educate highly skilled interdisciplinary researchers in mathematical MSO that will become facilitators in the transfer of innovative concepts to industry. It will thus enhance the capacity of European research and development.