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In this paper, we study the null controllability of a nonlinear age, space and two-sex structured population dynamics model. This model is such that the nonlinearity and the couplage are at birth level. We consider a population with males and females and we are dealing with two cases of null controllability problems. The first problem is related to the total extinction, which means that, we estimate a time \begin{document}$ T $\end{document} to bring the male and female subpopulation density to zero. The second case concerns null controllability of male or female subpopulation. Since the absence of males or females in the population stops births; so, if we have the total extinction of the females at time \begin{document}$ T, $\end{document} and if \begin{document}$ A $\end{document} is the life span of the individuals, at time \begin{document}$ T+A $\end{document} one will get certainly the total extinction of the population. Our method uses first an observability inequality related to the adjoint of an auxiliary system, a null controllability of the linear auxiliary system, and after the Schauder's fixed point theorem.

In this article, we investigate some of the fine properties of the value function associated to an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems., Comment: 55 pages

Combinatorial samplers are algorithmic schemes devised for the approximate- and exact-size generation of large random combinatorial structures, such as context-free words, various tree-like data structures, maps, tilings, RNA molecules. They can be adapted to combinatorial specifications with additional parameters, allowing for a more flexible control over the output profile of parametrised combinatorial patterns. One can control, for instance, the number of leaves, profile of node degrees in trees or the number of certain sub-patterns in generated strings. However, such a flexible control requires an additional and nontrivial tuning procedure. Using techniques of convex optimisation, we present an efficient tuning algorithm for multi-parametric combinatorial specifications. Our algorithm works in polynomial time in the system description length, the number of tuning parameters, the number of combinatorial classes in the specification, and the logarithm of the total target size. We demonstrate the effectiveness of our method on a series of practical examples, including rational, algebraic, and so-called P\'olya specifications. We show how our method can be adapted to a broad range of less typical combinatorial constructions, including symmetric polynomials, labelled sets and cycles with cardinality lower bounds, simple increasing trees or substitutions. Finally, we discuss some practical aspects of our prototype tuner implementation and provide its benchmark results., Comment: 44 pages, an extended version of the paper "Polynomial tuning of multiparametric combinatorial samplers" presented at ANALCO'18. arXiv admin note: text overlap with arXiv:1708.01212

International audience; In this article, we give an in-depth analysis of the problem of optimising the total population size for a standard logistic-diffusive model. This optimisation problem stems from the study of spatial ecology and amounts to the following question: assuming a species evolves in a domain, what is the best way to spread resources in order to ensure a maximal population size at equilibrium? {In recent years, many authors contributed to this topic.} We settle here the proof of two fundamental properties of optimisers: the bang-bang one which had so far only been proved under several strong assumptions, and the other one is the fragmentation of maximisers. Here, we prove the bang-bang property in all generality using a new spectral method. The technique introduced to demonstrate the bang-bang character of optimizers can be adapted and generalized to many optimization problems with other classes of bilinear optimal control problems where the state equation is semilinear and elliptic. We comment on it in a conclusion section.Regarding the geometry of maximisers, we exhibit a blow-up rate for the $BV$-norm of maximisers as the diffusivity gets smaller: if $\Omega$ is an orthotope and if $m_\mu$ is an optimal control, then $\Vert m_\mu\Vert_{BV}\gtrsim \sqrt{\mu}$. The proof of this results relies on a very fine energy argument.

This paper considers the Lurye system of a discrete-time, linear time-invariant plant in negative feedback with a nonlinearity. Both monotone and slope-restricted nonlinearities are considered. The main result is a procedure to construct destabilizing nonlinearities for the Lurye system. If the plant satisfies a certain phase condition then a monotone nonlinearity can be constructed so that the Lurye system has a non-trivial periodic cycle. Several examples are provided to demonstrate the construction. This represents a contribution for absolute stability analysis since the constructed nonlinearity provides a less conservative upper bound than existing bounds in the literature., 6 pages, 8 figures, submitted to IEEE CSL

In this article, we propose in-depth analysis and characterisation of the optimisers of the following optimisation problem: how to choose the initial condition u 0 in order to maximise the spatial integral at a given time of the solution of the semilinear equation u t −Δu = f(u), under L ∞ and L 1 constraints on u 0? Our contribution in the present paper is to give a characterisation of the behaviour of the optimiser u ¯ 0 when it does not saturate the L ∞ constraints, which is a key step in implementing efficient numerical algorithms. We give such a characterisation under mild regularity assumptions by proving that in that case u ¯ 0 can only take values in the ‘zone of concavity’ of f. This is done using two-scale asymptotic expansions. We then show how well-known isoperimetric inequalities yield a full characterisation of maximisers when f is convex. Finally, we provide several numerical simulations in one and two dimensions that illustrate and exemplify the fact that such characterisations significantly improve the computational time. All our theoretical results are in the one-dimensional case and we offer several comments about possible generalisations to other contexts, or obstructions that may prohibit doing so.

International audience; The goal of this paper is to provide a selection principle for potential mean field games on a finite state space and, in this respect, to show that equilibria that do not minimize the corresponding mean field control problem should be ruled out. Our strategy is a tailor-made version of the vanishing viscosity method for partial differential equations. Here, the viscosity has to be understood as the square intensity of a common noise that is inserted in the mean field game or, equivalently, as the diffusivity parameter in the related parabolic version of the master equation. As established in the recent contribution (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98-162), the randomly forced mean field game becomes indeed uniquely solvable for a relevant choice of a Wright-Fisher common noise, the counterpart of which in the master equation is a Kimura operator on the simplex. We here elaborate on (Bayraktar et al., 2021, J. Math. Pures Appl. 147:98-162) to make the mean field game with common noise both uniquely solvable and potential, meaning that its unique solution is in fact equal to the unique minimizer of a suitable stochastic mean field control problem. Taking the limit as the intensity of the common noise vanishes, we obtain a rigorous proof of the aforementioned selection principle. As a byproduct, we get that the classical solution to the viscous master equation associated with the mean field game with common noise converges to the gradient of the value function of the mean field control problem without common noise. We hence select a particular weak solution of the master equation of the original mean field game. Lastly, we establish an intrinsic uniqueness criterion for this solution within a suitable class of weak solutions to the master equation satisfying a weak one-sided Lipschitz inequality.

The paper is devoted to deriving necessary optimality conditions in a general optimal control problem for dynamical systems governed by controlled sweeping processes with hard-constrained control actions entering both polyhedral moving sets and additive perturbations. By using the first-order and mainly second-order tools of variational analysis and generalized differentiation, we develop a well-posed method of discrete approximations, obtain optimality conditions for solutions to discrete-time control systems, and then establish by passing to the limit verifiable necessary optimality conditions for local minimizers of the original controlled sweeping process that are expressed entirely in terms of its given data. The efficiency of the obtained necessary optimality conditions for the sweeping dynamics is illustrated by solving three nontrivial examples of their own interest., Comment: 33 pages, 3 figures

In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the “strengthening” of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs. FJAA and RC were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22. MKT is supported in part by ARC grant DE200100063.

In this paper, we solve a separation of singularities problem in the Bergman space. More precisely, we show that if $P\subset \mathbb{C}$ is a convex polygon which is the intersection of $n$ half planes, then the Bergman space on $P$ decomposes into the sum of the Bergman spaces on these half planes. The result applies to the characterization of the reachable space of the one-dimensional heat equation on a finite interval with boundary controls. We prove that this space is a Bergman space of the square which has the given interval as a diagonal. This gives an affirmative answer to a conjecture raised in [HKT20]., Comment: Journal de Math{\'e}matiques Pures et Appliqu{\'e}es, Elsevier, 2021

We deal with a class of semilinear parabolic PDEs on the space of continuous functions that arise, for example, as Kolmogorov equations associated to the infinite-dimensional lifting of path-dependent SDEs. We investigate existence of smooth solutions through their representation via forward-backward stochastic systems, for which we provide the necessary regularity theory. Because of the lack of smoothing properties of the parabolic operators at hand, solutions in general will only share the same regularity as the coefficients of the equation. To conclude we exhibit an application to Hamilton-Jacobi-Bellman equations associated to suitable optimal control problems., Comment: 40 pages

This paper studies constrained information projections on Banach spaces with respect to a Gaussian reference measure. Specifically our interest lies in characterizing projections of the reference measure, with respect to the KL-divergence, onto sets of measures corresponding to changes in the mean (or {\it shift measures}). As our main result, we give a portmanteau theorem that characterizes the relationship among several different formulations of this problem. In the general setting of Gaussian measures on a Banach space, we show that this information projection problem is equivalent to minimization of a certain Onsager-Machlup (OM) function with respect to an associated stochastic process. We then construct several reformulations in the more specific setting of classical Wiener space. First, we show that KL-weighted optimization over shift measures can also be expressed in terms of an OM function for an associated stochastic process that we are able to characterize. Next, we show how to encode the feasible set of shift measures through an explicit functional constraint by constructing an appropriate penalty function. Finally, we express our information projection problem as a calculus of variations problem, which suggests a solution procedure via the Euler-Lagrange equation. We work out the details of these reformulations for several specific examples., Comment: To Appear in Applied Mathematics and Optimization

We propose two algorithms for solving global optimization problems on a hyperrectangle with an objective function satisfying the Vanderbei condition (this function is also called an $\varepsilon$-Lipschitz continuous function). The algorithms belong to the class of non-uniform cover-ings methods. For the algorithms we prove propositions about convergence to an $\varepsilon$-solution in terms of the objective function. We illustrate the performance of the algorithms using several test numerical examples with non-Lipschitz continuous objective functions., 21 pages, 4 tables, 29 figure, in Russian. (v4) corrected the third numerical example

In this paper we consider an impulsive extension of an optimal control problem with unbounded controls, subject to endpoint and state constraints. We show that the existence of an extended-sense minimizer that is a normal extremal for a constrained Maximum Principle ensures that there is no gap between the infima of the original problem and of its extension. Furthermore, we translate such relation into verifiable sufficient conditions for normality in the form of constraint and endpoint qualifications. Links between existence of an infimum gap and normality in impulsive control have previously been explored for problems without state constraints. This paper establishes such links in the presence of state constraints and of an additional ordinary control, for locally Lipschitz continuous data.

In this article we prove the existence of classical solutions to a system of mean field games arising in the study of exhaustible resource production under market competition. Individual trajectories are modeled by a controlled diffusion process with jumps, which adds a nonlocal term to the PDE system. The assumptions on the Hamiltonian are sufficiently general to cover a large class of examples proposed in the literature on Bertrand and Cournot mean field games. Uniqueness also holds under a sufficient restriction on the structure of the Hamiltonian, which in practice amounts to a small upper bound on the substitutability of goods.

Certifying function nonnegativity is a ubiquitous problem in computational mathematics, with especially notable applications in optimization. We study the question of certifying nonnegativity of signomials based on the recently proposed approach of Sums-of-AM/GM-Exponentials (SAGE) decomposition due to the second author and Shah. The existence of a SAGE decomposition is a sufficient condition for nonnegativity of a signomial, and it can be verified by solving a tractable convex relative entropy program. We present new structural properties of SAGE certificates such as a characterization of the extreme rays of the cones associated to these decompositions as well as an appealing form of sparsity preservation. These lead to a number of important consequences such as conditions under which signomial nonnegativity is equivalent to the existence of a SAGE decomposition; our results represent the broadest-known class of nonconvex signomial optimization problems that can be solved efficiently via convex relaxation. The analysis in this paper proceeds by leveraging the interaction between the convex duality underlying SAGE certificates and the face structure of Newton polytopes. While our primary focus is on signomials, we also discuss how our results provide efficient methods for certifying polynomial nonnegativity, with complexity independent of the degree of a polynomial., Body shortened from 29 to 24 pages. Additional consideration to related work. Some claims made in Section 5 have been formalized. Revised within 2 months of first-round reviews

Choose $N$ unoriented lines through the origin of ${\bf R}^{d+1}$. The sum of the angles between these lines is conjectured to be maximized if the lines are distributed as evenly as possible amongst the coordinate axes of some orthonormal basis for ${\bf R}^{d+1}$. For $d \ge 2$ we embed the conjecture into a one-parameter family of problems, in which we seek to maximize the sum of the $\alpha$-th power of the renormalized angles between the lines. We show the conjecture is equivalent to this same configuration becoming the {\em unique} optimizer (up to rotations) for all $\alpha>1$. We establish both the asserted optimality and uniqueness in the limiting case $\alpha =\infty$ of mildest repulsion. The same conclusions extend to $N=\infty$, provided we assume only finitely many of the lines are distinct., Comment: The published paper in AMO unfortunately has a consistent typos on integer bracket notation...this arXiv print does not have this issue

This paper is concerned with the theory and applications of varifolds to the representation, approximation and diffeomorphic registration of shapes. One of its purpose is to synthesize and extend several prior works which, so far, have made use of this framework mainly in the context of submanifold comparison and matching. In this work, we instead consider deformation models acting on general varifold spaces, which allow to formulate and tackle diffeomorphic registration problems for a much wider class of geometric objects and lead to a more versatile algorithmic pipeline. We study in detail the construction of kernel metrics on varifold spaces and the resulting topological properties of those metrics and then propose a mathematical model for diffeomorphic registration of varifolds under a specific group action which we formulate in the framework of optimal control theory. A second important part of the paper focuses on the discrete aspects. Specifically, we address the problem of optimal finite approximations (quantization) for those metrics and show a $$\varGamma $$ -convergence property for the corresponding registration functionals. Finally, we develop numerical pipelines for quantization and registration before showing a few preliminary results for one- and two-dimensional varifolds.

This paper examines mean field linear-quadratic-Gaussian (LQG) social optimum control with volatility-uncertain common noise. The diffusion terms in the dynamics of agents contain an unknown volatility process driven by a common noise. We apply a robust optimization approach in which all agents view volatility uncertainty as an adversarial player. Based on the principle of person-by-person optimality and a two-step-duality technique for stochastic variational analysis, we construct an auxiliary optimal control problem for a representative agent. Through solving this problem combined with a consistent mean field approximation, we design a set of decentralized strategies, which are further shown to be asymptotically social optimal by perturbation analysis.

We present a backward diffusion flow (i.e. a backward-in-time stochastic differential equation) whose marginal distribution at any (earlier) time is equal to the smoothing distribution when the terminal state (at a latter time) is distributed according to the filtering distribution. This is a novel interpretation of the smoothing solution in terms of a nonlinear diffusion (stochastic) flow. This solution contrasts with, and complements, the (backward) deterministic flow of probability distributions (viz. a type of Kushner smoothing equation) studied in a number of prior works. A number of corollaries of our main result are given including a derivation of the time-reversal of a stochastic differential equation, and an immediate derivation of the classical Rauch-Tung-Striebel smoothing equations in the linear setting.

The paper studies the open-loop saddle point and the open-loop lower and upper values, as well as their relationship for two-person zero-sum stochastic linear-quadratic (LQ, for short) differential games with deterministic coefficients. It derives a necessary condition for the finiteness of the open-loop lower and upper values and a sufficient condition for the existence of an open-loop saddle point. It turns out that under the sufficient condition, a strongly regular solution to the associated Riccati equation uniquely exists, in terms of which a closed-loop representation is further established for the open-loop saddle point. Examples are presented to show that the finiteness of the open-loop lower and upper values does not ensure the existence of an open-loop saddle point in general. But for the classical deterministic LQ game, these two issues are equivalent and both imply the solvability of the Riccati equation, for which an explicit representation of the solution is obtained., Comment: 26 pages

We introduce new parametrized classes of shape admissible domains in R^n , n $\ge$ 2, and prove that they are compact with respect to the convergence in the sense of characteristic functions, the Hausdorff sense, the sense of compacts and the weak convergence of their boundary volumes. The domains in these classes are bounded ($\epsilon$, $\infty$)-domains with possibly fractal boundaries that can have parts of any non-uniform Hausdorff dimension greater or equal to n -- 1 and less than n. We prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acous-tics. A by-product of our proof is the result that the class of bounded ($\epsilon$, $\infty$)-domains with fixed $\epsilon$ is stable under Hausdorff convergence. An additional and related result is the Mosco convergence of Robin-type energy functionals on converging domains.

In statistical decision theory involving a single decision-maker, an information structure is said to be better than another one if for any cost function involving a hidden state variable and an action variable which is restricted to be conditionally independent from the state given some measurement, the solution value under the former is not worse than that under the latter. For finite spaces, a theorem due to Blackwell leads to a complete characterization on when one information structure is better than another. For stochastic games, in general, such an ordering is not possible since additional information can lead to equilibria perturbations with positive or negative values to a player. However, for zero-sum games in a finite probability space, P\k{e}ski introduced a complete characterization of ordering of information structures. In this paper, we obtain an infinite dimensional (standard Borel) generalization of P\k{e}ski's result. A corollary is that more information cannot hurt a decision maker taking part in a zero-sum game. We establish two supporting results which are essential and explicit though modest improvements on prior literature: (i) a partial converse to Blackwell's ordering in the standard Borel setup and (ii) an existence result for equilibria in zero-sum games with incomplete information., Comment: To appear in SIAM Journal on Control and Optimization. A short conference version appeared at the 2020 IEEE International Symposium on Information Theory

An optimal ergodic control problem (EC problem, for short) is investigated for a linear stochastic differential equation with quadratic cost functional. Constant nonhomogeneous terms, not all zero, appear in the state equation, which lead to the asymptotic limit of the state non-zero. Under the stabilizability condition, for any (admissible) closed-loop strategy, an invariant measure is proved to exist, which makes the ergodic cost functional well-defined and the EC problem well-formulated. Sufficient conditions, including those allowing the weighting matrices of cost functional to be indefinite, are introduced for finiteness and solvability for the EC problem. Some comparisons are made between the solvability of EC problem and the closed-loop solvability of stochastic linear quadratic optimal control problem in the infinite horizon. Regularized EC problem is introduced to be used to obtain the optimal value of the EC problem.

We study compressible and incompressible nonlinear elasticity variational problems in a general context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer, in terms of convexity properties of the pressure in the deformed configuration. We also provide a convex relaxation of the problem together with its dual formulation, based on measure-valued mappings, which coincides with the original problem under our condition., Comment: 17 pages

We introduce a mean field game model for pedestrians moving in a given domain and choosing their trajectories so as to minimize a cost including a penalization on the difference between their own velocity and that of the other agents they meet. We prove existence of an equilibrium in a Lagrangian setting by using its variational structure, and then study its properties and regularity.

We consider robust pricing and hedging for options written on multiple assets given market option prices for the individual assets. The resulting problem is called the multi-marginal martingale optimal transport problem. We propose two numerical methods to solve such problems: using discretisation and linear programming applied to the primal side and using penalisation and deep neural networks optimisation applied to the dual side. We prove convergence for our methods and compare their numerical performance. We show how adding further information about call option prices at additional maturities can be incorporated and narrows down the no-arbitrage pricing bounds. Finally, we obtain structural results for the case of the payoff given by a weighted sum of covariances between the assets., Comment: Forthcoming in SIAM Journal on Financial Mathematics

International audience; We consider a continuous time stochastic optimal control problem under both equality and inequality constraints on the expectation of some functionals of the controlled process. Under a qualification condition, we show that the problem is in duality with an optimization problem involving the Lagrange multiplier associated with the constraints. Then by convex analysis techniques, we provide a general existence result and some a priori estimation of the dual optimizers. We further provide a necessary and sufficient optimality condition for the initial constrained control problem. The same results are also obtained for a discrete time constrained control problem. Moreover, under additional regularity conditions, it is proved that the discrete time control problem converges to the continuous time problem, possibly with a convergence rate. This convergence result can be used to obtain numerical algorithms to approximate the continuous time control problem, which we illustrate by two simple numerical examples.

It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional nonlinear systems. In this paper, we restrict ourselves to semilinear infinite-dimensional systems, and show that the exact controllability of the linearized system implies exact controllability of the nonlinear system. The restrictions concerning the nonlinear operator are similar to those that can be found in the literature about the linearized stability analysis of semilinear systems., accepted as full paper for MTNS 2020

Nonconvex optimal-control problems governed by evolution problems in infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromising convex compactification is here devised, combining classical techniques for Young measures with Choquet theory. This is applied to parabolic optimal control problems as far as existence and optimality conditions concerns.

Controlling the shape and position of moving and pinned droplets on a solid surface is an important feature often found in microfluidic applications. In this work, we consider a well investigated phase field model including contact line dynamics as the state system for an (open-loop) optimal control problem. Here the spatially and temporally changeable contact angles between droplet and solid are considered as the control variables. We consider a suitable, energy stable, time discrete version of the state equation in our optimal control problem. We discuss regularity of the solution to the time discrete state equation and its continuity and differentiability properties. Furthermore, we show existence of solutions and state first order optimality conditions to the optimal control problem. We illustrate our results by actively pushing a droplet uphill against gravity in an optimal way.

A linear control system with quadratic cost functional over infinite time horizon is considered without assuming controllability/stabilizability condition and the global integrability condition for the nonhomogeneous term of the state equation and the weight functions in the linear terms in the running cost rate function. Classical approaches do not apply for such kind of problems. Existence and non-existence of overtaking optimal controls in various cases are established. Some concrete examples are presented. These results show that the overtaking optimality approach can be used to solve some of the above-mentioned problems and at the same time, the limitation of this approach is also revealed.

We prove a small-gain theorem for interconnections of $n$ nonlinear heterogeneous input-to-state stable (ISS) control systems of a general nature, covering partial, delay and ordinary differential equations. Furthermore, for the same class of control systems, we derive small-gain theorems for asymptotic gain, uniform global stability and weak input-to-state stability properties. We show that our technique is applicable for different formulations of ISS property (summation, maximum, semimaximum) and discuss tightness of achieved small-gain theorems. Finally, we introduce variations of uniform asymptotic gain and uniform limit properties, which are particularly useful for small-gain arguments and characterize ISS in terms of these notions., Accepted to SICON, 2021

In this paper, we study an optimal control problem for a macroscopic mechanical tumour model based on the phase field approach. The model couples a Cahn--Hilliard type equation to a system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. By taking advantage of previous analytical well-posedness results established by the authors, we seek optimal controls in the form of a boundary nutrient supply, as well as concentrations of cytotoxic and antiangiogenic drugs that minimise a cost functional involving mechanical stresses. Special attention is given to sparsity effects, where with the inclusion of convex non-differentiable regularisation terms to the cost functional, we can infer from the first-order optimality conditions that the optimal drug concentrations can vanish on certain time intervals., 26 pages

In this paper we study the asymptotic behaviour of the quasilinear $curl$-$curl$ equation of 3D magnetostatics with respect to a singular perturbation of the differential operator and prove the existence of the topological derivative using a Lagrangian approach. We follow the strategy proposed in our recent previous work (arXiv:1907.13420) where a systematic and concise way for the derivation of topological derivatives for quasi-linear elliptic problems in $H^1$ is introduced. In order to prove the asymptotics for the state equation we make use of an appropriate Helmholtz decomposition. The evaluation of the topological derivative at any spatial point requires the solution of a nonlinear transmission problem. We discuss an efficient way for the numerical evaluation of the topological derivative in the whole design domain using precomputation in an offline stage. This allows us to use the topological derivative for the design optimization of an electrical machine., Comment: 26 pages, 5 figures

A framework is proposed that addresses both conditional density estimation and latent variable discovery. The objective function maximizes explanation of variability in the data, achieved through the optimal transport barycenter generalized to a collection of conditional distributions indexed by a covariate --either given or latent-- in any suitable space. Theoretical results establish the existence of barycenters, a minimax formulation of optimal transport maps, and a general characterization of variability via the optimal transport cost. This framework leads to a family of non-parametric neural network-based algorithms, the BaryNet, with a supervised version that estimates conditional distributions and an unsupervised version that assigns latent variables. The efficacy of BaryNets is demonstrated by tests on both artificial and real-world data sets. A parallel drawn between autoencoders and the barycenter framework leads to the Barycentric autoencoder algorithm (BAE).

The clustering problem, and more generally, latent factor discovery --or latent space inference-- is formulated in terms of the Wasserstein barycenter problem from optimal transport. The objective proposed is the maximization of the variability attributable to class, further characterized as the minimization of the variance of the Wasserstein barycenter. Existing theory, which constrains the transport maps to rigid translations, is extended to affine transformations. The resulting non-parametric clustering algorithms include k-means as a special case and exhibit more robust performance. A continuous version of these algorithms discovers continuous latent variables and generalizes principal curves. The strength of these algorithms is demonstrated by tests on both artificial and real-world data sets., Improved clarity of presentation

Motivated by gradient methods in optimization theory, we give methods based on $\psi$-fractional derivatives of order $\alpha$ in order to solve unconstrained optimization problems. The convergence of these methods is analyzed in detail. This paper also presents an Adams-Bashforth-Moulton (ABM) method for the estimation of solutions to equations involving $\psi$-fractional derivatives. Numerical examples using the ABM method show that the fractional order $\alpha$ and weight $\psi$ are tunable parameters, which can be helpful for improving the performance of gradient descent methods., Comment: 27 pages

Mean-field games (MFGs) and the best-reply strategy (BRS) are two methods of describing competitive optimisation of systems of interacting agents. The latter can be interpreted as an approximation of the respective MFG system. In this paper, we present an analysis and comparison of the two approaches in the stationary case. We provide novel existence and uniqueness results for the stationary boundary value problems related to the MFG and BRS formulations, and we present an analytical and numerical comparison of the two paradigms in some specific modelling situations.

We introduce and study a variant of the Wasserstein distance on the space of probability measures, specially designed to deal with measures whose support has a dendritic, or treelike structure with a particular direction of orientation. Our motivation is the comparison of and interpolation between plants' root systems. We characterize barycenters with respect to this metric, and establish that the interpolations of root-like measures, using this new metric, are also root like, in a certain sense; this property fails for conventional Wasserstein barycenters. We also establish geodesic convexity with respect to this metric for a variety of functionals, some of which we expect to have biological importance., Comment: 25 pages

One of the difficulties in 3D reconstruction of molecules from images in single particle Cryo-Electron Microscopy (Cryo-EM), in addition to high levels of noise and unknown image orientations, is heterogeneity in samples: in many cases, the samples contain a mixture of molecules, or multiple conformations of one molecule. Many algorithms for the reconstruction of molecules from images in heterogeneous Cryo-EM experiments are based on iterative approximations of the molecules in a non-convex optimization that is prone to reaching suboptimal local minima. Other algorithms require an alignment in order to perform classification, or vice versa. The recently introduced Non-Unique Games framework provides a representation theoretic approach to studying problems of alignment over compact groups, and offers convex relaxations for alignment problems which are formulated as semidefinite programs (SDPs) with certificates of global optimality under certain circumstances. In this manuscript, we propose to extend Non-Unique Games to the problem of simultaneous alignment and classification with the goal of simultaneously classifying Cryo-EM images and aligning them within their respective classes. Our proposed approach can also be extended to the case of continuous heterogeneity.

In the simplest case, we obtain a general solution to a problem of minimizing an integral of a nondecreasing right continuous stochastic process from zero to some nonnegative random variable tau, under the constraints that for some nonnegative random variable T, tau is between zero and T a.s. and the expected value of tau is some alpha. The nondecreasing process and T are allowed to be dependent. In fact a more general setup involving sigma-finite measures, rather than just probability measures is considered and some consequences for families of stochastic processes are given as special cases. Various applications are provided., Comment: 22 pages

We consider 2- or 3-dimensional incompressible Navier-Stokes equations defined on a bounded domain $\Omega$, with no-slip boundary conditions and subject to an external force, assumed to cause instability. We then seek to uniformly stabilize such N-S system, in the vicinity of an unstable equilibrium solution, in critical $L^q$-based Sobolev and Besov spaces, by finite dimensional feedback controls. These spaces are `close' to $L^3(\Omega)$ for $d=3$. This functional setting is significant. In fact, in the case of the uncontrolled N-S dynamics, extensive research efforts have recently lead to the space $L^3(\mathbb{R}^3)$ as being a critical space for the issue of well-posedness in the full space. Thus, our present work manages to solve the stated uniform stabilization problem for the controlled N-S dynamics in a correspondingly related function space setting. In this paper, the feedback controls are localized on an arbitrarily small open interior subdomain $\omega$ of $\Omega$. In addition to providing a solution of the uniform stabilization problem in such critical function space setting, this paper manages also to much improve and simplify, at both the conceptual and computational level, the solution given in the more restrictive Hilbert space setting in the literature. Moreover, such treatment sets the foundation for the authors' final goal in a subsequent paper. Based critically on said low functional level where compatibility conditions are not recognized, the subsequent paper solves in the affirmative a presently open problem: whether uniform stabilization by localized tangential boundary feedback controls, which-in addition-are finite dimensional, is also possible in dim $\Omega = 3$., Comment: arXiv admin note: text overlap with arXiv:2011.03825, arXiv:2202.03435

Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization is very well understood when the penalty involves a Hilbertian norm. Another popular configuration is the use of an $$\ell _1$$ ℓ 1 -norm (or some variant thereof) that favors sparse solutions. In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces. We present a general representer theorem that characterizes the solutions of a remarkably broad class of optimization problems. We then use our theorem to retrieve a number of known results in the literature such as the celebrated representer theorem of machine leaning for RKHS, Tikhonov regularization, representer theorems for sparsity promoting functionals, the recovery of spikes, as well as a few new ones.

For a class of finite horizon first order mean field games and associated N-player games, we give a simple proof of convergence of symmetric N-player Nash equilibria in distributed open-loop strategies to solutions of the mean field game in Lagrangian form. Lagrangian solutions are then connected with those determined by the usual mean field game system of two coupled first order PDEs, and convergence of Nash equilibria in distributed Markov strategies is established., Comment: 18 pages

The ancient concept of circumcenter has recently given birth to the Circumcentered-Reflection method (CRM). CRM was first employed to solve best approximation problems involving affine subspaces. In this setting, it was shown to outperform the most prestigious projection based schemes, namely, the Douglas-Rachford method (DRM) and the method of alternating projections (MAP). We now prove convergence of CRM for finding a point in the intersection of a finite number of closed convex sets. This striking result is derived under a suitable product space reformulation in which a point in the intersection of a closed convex set with an affine subspace is sought. It turns out that CRM skillfully tackles the reformulated problem. We also show that for a point in the affine set the CRM iteration is always closer to the solution set than both the MAP and DRM iterations. Our theoretical results and numerical experiments, showing outstanding performance, establish CRM as a powerful tool for solving general convex feasibility problems.

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This implies that DNNs can break the curse of dimensionality in numerical approximations and optimal control of PDEs and SPDEs. The main ingredient of our proof is to construct a suitable discrete-time system to effectively approximate the evolution of the underlying stochastic dynamics. Similar ideas can also be applied to obtain expression rates of DNNs for value functions induced by stiff systems with regime switching coefficients and driven by general L\'{e}vy noise., Comment: This revised version has been accepted for publication in Analysis and Applications

Let $K$ be a convex polyhedron and $\mathscr F$ its Wulff energy, and let $\mathscr C(K)$ denote the set of convex polyhedra close to $K$ whose faces are parallel to those of $K$. We show that, for sufficiently small $\epsilon$, all $\epsilon$-minimizers belong to $\mathscr C(K)$. As a consequence of this result we obtain the following sharp stability inequality for crystalline norms: There exist $\gamma=\gamma(K,n)>0$ and $\sigma=\sigma(K,n)>0$ such that, whenever $|E|=|K|$ and $|E\Delta K|\le \sigma$ then $$ \mathscr F(E) - \mathscr F(K^a)\ge \gamma |E \Delta K^a| \qquad \text{for some }K^a \in \mathscr C(K). $$ In other words, the Wulff energy $\mathscr F$ grows very fast (with power $1$) away from the set $\mathscr C(K).$ The set $K^a \in \mathscr C(K)$ appearing in the formula above can be informally thought as a sort of "projection" of $E$ on the set $\mathscr C(K).$ Another corollary of our result is a very strong rigidity result for crystals: For crystalline surface tensions, minimizers of $\mathscr F(E)+\int_E g$ with small mass are polyhedra with sides parallel to the ones of $K$. In other words, for small mass, the potential energy cannot destroy the crystalline structure of minimizers. This extends to arbitrary dimensions a two-dimensional result obtained in [9]., Comment: 21 pages, 4 figures

The models introduced in this paper describe a uniform distribution of plant stems competing for sunlight. The shape of each stem, and the density of leaves, are designed in order to maximize the captured sunlight, subject to a cost for transporting water and nutrients from the root to all the leaves. Given the intensity of light, depending on the height above ground, we first solve the optimization problem determining the best possible shape for a single stem. We then study a competitive equilibrium among a large number of similar plants, where the shape of each stem is optimal given the shade produced by all others. Uniqueness of equilibria is proved by analyzing the two-point boundary value problem for a system of ODEs derived from the necessary conditions for optimality., Comment: 36 pages, 6 figures

In an incomplete model, where under an appropriate num\'eraire, the stock price process is driven by a sigma-bounded semimartingale, we investigate the behavior of the expected utility maximization problem under small perturbations of the num\'eraire. We establish a quadratic approximation of the value function and a first-order expansion of the terminal wealth. Relying on a description of the base return process in terms of its semimartingale characteristics, we also construct wealth processes and nearly optimal strategies that allow for matching the primal value function up to the second order. We also link perturbations of the num\'eraire to distortions of the finite-variation part and martingale part of the stock price return and characterize the asymptotic expansions in terms of the risk-tolerance wealth process., Comment: 31 pages. Theorem 4.2 is strengthened by providing corrections to optimal strategies (without a boundedness assumption). The paper is accepted in Stochastic Processes and Their Applications