Reconstruction in limited-angle digital breast tomosynthesis (DBT) suffers from slow convergence of low spatial-frequency components when using weighted data-fidelity terms within primal-dual optimization. We introduce a two-channel fidelity strategy that decomposes the sinogram residual into complementary low-pass and high-pass bands using square-root Hanning (Hann^{1/2}) filter families, each driven by an independent \ell_2-ball constraint and dual update in the PDHG (Chambolle-Pock) algorithm with He-Yuan predictor-corrector relaxation. By assigning a larger dual step size and slightly looser tolerance to the low-frequency channel, the method delivers stronger per-iteration correction to the near-DC band without violating global PDHG stability. Experiments on a 2D digital breast phantom across multiple resolutions demonstrate that the two-channel approach yields 19%--61% RMSE improvement over the single-channel baseline, with larger gains at coarser discretizations where problem conditioning is more favorable, supporting more balanced spectral convergence in clinically realistic limited-angle regimes.
We consider transport processes that are modeled by first order hyperbolic partial differential equations. Our goal is to find a full state feedback that makes a given reference profile locally asymptotically stable. To accomplish this we employ Linear Quadratic Regulation (LQR) with finite dimensional patch or point control actuation. We derive the Riccati partial differential equation whose solution is the kernel of the optimal cost. The optimal state feedback is also found. The derivation is accomplished by elementary techniques such as integration by parts and completing the square. We apply this theory to two examples that have appeared in the literature and that were solved by a modification of LQR. The first example deals with a model of a fixed-bed chemical reactor and the second example deals with traffic congestion on a stretch of freeway.
We prove that for impulsive exposure patterns there is no uniform exponential energy law in wall-clock time t, which explains why past t-based unifications of continuous damping with impulses fail. We therefore replace t by a measure-valued clock, sigma, that aggregates absolutely continuous exposure and atomic doses within a single Lyapunov ledger. On this ledger we prove an observability-dissipation principle in the sense of the Hilbert Uniqueness Method (HUM): there exists a structural constant c_sigma > 0 such that the energy decays at least at a product-exponential rate with respect to sigma. When sigma = t, the statement reduces to classical exponential stabilization with the same constant. For the damped wave under the Geometric Control Condition (GCC), the constant is calibrated by the usual observability and geometric factors. The framework yields a monotonicity principle ("more sigma-mass implies faster decay") and unifies intermittent regimes where quiescent intervals are punctuated by impulses. As robustness, secondary to the main contribution, the same decay law persists under structure-compatible discretizations and along compact variational limits; a stochastic extension supplies expectation and pathwise envelopes via the compensator. The contribution is a qualitative dynamics backbone: observability implies sigma-exponential decay with sharp constants.
The non-isothermal analysis of materials with the application of the Arrhenius equation involves temperature integration. If the frequency factor in the Arrhenius equation depends on temperature with a power-law relationship, the integral is known as the general temperature integral. This integral which has no analytical solution is estimated by the approximation functions with different accuracies. In this article, the rational approximations of the integral were obtained based on the minimization of the maximal deviation of bivariate functions. Mathematically, these problems belong to the class of quasiconvex optimization and can be solved using the bisection method. The approximations obtained in this study are more accurate than all approximates available in the literature.
Neural collapse, a newly identified characteristic, describes a property of solutions during model training. In this paper, we explore neural collapse in the context of imbalanced data. We consider the $L$-extended unconstrained feature model with a bias term and provide a theoretical analysis of global minimizer. Our findings include: (1) Features within the same class converge to their class mean, similar to both the balanced case and the imbalanced case without bias. (2) The geometric structure is mainly on the left orthonormal transformation of the product of $L$ linear classifiers and the right transformation of the class-mean matrix. (3) Some rows of the left orthonormal transformation of the product of $L$ linear classifiers collapse to zeros and others are orthogonal, which relies on the singular values of $\hat Y=(I_K-1/N\mathbf{n}1^\top_K)D$, where $K$ is class size, $\mathbf{n}$ is the vector of sample size for each class, $D$ is the diagonal matrix whose diagonal entries are given by $\sqrt{\mathbf{n}}$. Similar results are for the columns of the right orthonormal transformation of the product of class-mean matrix and $D$. (4) The $i$-th row of the left orthonormal transformation of the product of $L$ linear classifiers aligns with the $i$-th column of the right orthonormal transformation of the product of class-mean matrix and $D$. (5) We provide the estimation of singular values about $\hat Y$. Our numerical experiments support these theoretical findings.
Vladimir Anisimov, Guillaume Mijoule, Armando Turchetta
et al.
This paper focuses on statistical modelling and prediction of patient recruitment in clinical trials accounting for patients dropout. The recruitment model is based on a Poisson-gamma model introduced by Anisimov and Fedorov (2007), where the patients arrive at different centres according to Poisson processes with rates viewed as gamma-distributed random variables. Each patient can drop the study during some screening period. Managing the dropout process is of a major importance but data related to dropout are rarely correctly collected. In this paper, a few models of dropout are proposed. The technique for estimating parameters and predicting the number of recruited patients over time and the recruitment time is developed. Simulation results confirm the applicability of the technique and thus, the necessity to account for patients dropout at the stage of forecasting recruitment in clinical trials.
Andrea Pesare, Michele Palladino, Maurizio Falcone
We deal with the convergence of the value function of an approximate control problem with uncertain dynamics to the value function of a nonlinear optimal control problem. The assumptions on the dynamics and the costs are rather general and we assume to represent uncertainty in the dynamics by a probability distribution. The proposed framework aims to describe and motivate some model-based Reinforcement Learning algorithms where the model is probabilistic. We also show some numerical experiments which confirm the theoretical results.
Residual deep neural networks (ResNets) are mathematically described as interacting particle systems. In the case of infinitely many layers the ResNet leads to a system of coupled system of ordinary differential equations known as neural differential equations. For large scale input data we derive a mean--field limit and show well--posedness of the resulting description. Further, we analyze the existence of solutions to the training process by using both a controllability and an optimal control point of view. Numerical investigations based on the solution of a formal optimality system illustrate the theoretical findings.
Andrea Pesare, Michele Palladino, Maurizio Falcone
In this paper, we will deal with a Linear Quadratic Optimal Control problem with unknown dynamics. As a modeling assumption, we will suppose that the knowledge that an agent has on the current system is represented by a probability distribution $Ο$ on the space of matrices. Furthermore, we will assume that such a probability measure is opportunely updated to take into account the increased experience that the agent obtains while exploring the environment, approximating with increasing accuracy the underlying dynamics. Under these assumptions, we will show that the optimal control obtained by solving the "average" Linear Quadratic Optimal Control problem with respect to a certain $Ο$ converges to the optimal control driven related to the Linear Quadratic Optimal Control problem governed by the actual, underlying dynamics. This approach is closely related to model-based Reinforcement Learning algorithms where prior and posterior probability distributions describing the knowledge on the uncertain system are recursively updated. In the last section, we will show a numerical test that confirms the theoretical results.
Dieter Armbruster, Michael Herty, Giuseppe Visconti
The Ensemble Kalman Filter (EnKF) belongs to the class of iterative particle filtering methods and can be used for solving control--to--observable inverse problems. In this context, the EnKF is known as Ensemble Kalman Inversion (EKI). In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one--dimensional linear stability analysis reveals possible drawbacks in the phase space of moments provided by the continuous limits of the EKI, but observed also in the multi--dimensional setting. In this work we address this issue by introducing a stabilization of the dynamics which leads to a method with globally asymptotically stable solutions. We illustrate the performance of the stabilized version by using test inverse problems from the literature and comparing it with the classical continuous limit formulation of the method.
Xiaoqian Gong, Benedetto Piccoli, Giuseppe Visconti
This article aims to study coupled mean-field equation and ODEs with discrete events motivated by vehicular traffic flow. Precisely, multi-lane traffic flow in presence of human-driven and autonomous vehicles is considered, with the autonomous vehicles possibly influenced by external policy makers. First a finite-dimensional hybrid system is developed based on the continuous Bando-Follow-the-Leader dynamics coupled with discrete events due to lane-change maneuvers. Then the mean-field limit of the finite-dimensional hybrid system is rigorously derived for the dynamics of the human-driven vehicles. The microscopic lane-change maneuvers of the human-driven vehicles generates a source term to the mean-field PDE. This leads to an infinite-dimensional hybrid system, which is described by coupled Vlasov-type PDE, ODEs and discrete events.
We prove existence of radially symmetric solutions and validity of Euler-Lagrange necessary conditions for a class of variational problems such that neither direct methods nor indirect methods of Calculus of Variations apply. We obtain existence and qualitative properties of the solutions by means of ad-hoc superlinear perturbations of the functional having the same minimizers of the original one.
This paper investigates the convergence of learning dynamics in Stackelberg games. In the class of games we consider, there is a hierarchical game being played between a leader and a follower with continuous action spaces. We establish a number of connections between the Nash and Stackelberg equilibrium concepts and characterize conditions under which attracting critical points of simultaneous gradient descent are Stackelberg equilibria in zero-sum games. Moreover, we show that the only stable critical points of the Stackelberg gradient dynamics are Stackelberg equilibria in zero-sum games. Using this insight, we develop a gradient-based update for the leader while the follower employs a best response strategy for which each stable critical point is guaranteed to be a Stackelberg equilibrium in zero-sum games. As a result, the learning rule provably converges to a Stackelberg equilibria given an initialization in the region of attraction of a stable critical point. We then consider a follower employing a gradient-play update rule instead of a best response strategy and propose a two-timescale algorithm with similar asymptotic convergence guarantees. For this algorithm, we also provide finite-time high probability bounds for local convergence to a neighborhood of a stable Stackelberg equilibrium in general-sum games. Finally, we present extensive numerical results that validate our theory, provide insights into the optimization landscape of generative adversarial networks, and demonstrate that the learning dynamics we propose can effectively train generative adversarial networks.
Julio Backhoff-Veraguas, Joaquin Fontbona, Gonzalo Rios
et al.
We introduce and study a novel model-selection strategy for Bayesian learning, based on optimal transport, along with its associated predictive posterior law: the Wasserstein population barycenter of the posterior law over models. We first show how this estimator, termed Bayesian Wasserstein barycenter (BWB), arises naturally in a general, parameter-free Bayesian model-selection framework, when the considered Bayesian risk is the Wasserstein distance. Examples are given, illustrating how the BWB extends some classic parametric and non-parametric selection strategies. Furthermore, we also provide explicit conditions granting the existence and statistical consistency of the BWB, and discuss some of its general and specific properties, providing insights into its advantages compared to usual choices, such as the model average estimator. Finally, we illustrate how this estimator can be computed using the stochastic gradient descent (SGD) algorithm in Wasserstein space introduced in a companion paper arXiv:2201.04232v2 [math.OC], and provide a numerical example for experimental validation of the proposed method.
Many systems of interest in science and engineering are made up of interacting subsystems. These subsystems, in turn, could be made up of collections of smaller interacting subsystems and so on. In a series of papers David Spivak with collaborators formalized these kinds of structures (systems of systems) as algebras over presentable colored operads. It is also very useful to consider maps between dynamical systems. This amounts to viewing dynamical systems as objects in an appropriate category. This is the point taken by DeVille and Lerman in the study of dynamics on networks. The goal of this paper is to describe an algebraic structure that encompasses both approaches to systems of systems. To this end we replace the monoidal category of wiring diagrams by a monoidal double category whose objects are surjective submersions. This allows us, on one hand, build new large open systems out of collections of smaller open subsystems and on the other keep track of maps between open systems. As a special case we recover the results of DeVille and Lerman on fibrations of networks of manifolds.
Abstract. Elemental carbon (EC), due to its exclusive origin in primary combustion sources, has been widely used as a tracer to track the portion of co-emitted primary organic carbon (OC) and, by extension, to estimate secondary OC (SOC) from ambient observations of EC and OC. Key to this EC tracer method is to determine an appropriate OC/EC ratio that represents primary combustion emission sources (i.e., (OC/EC)pri) at the observation site. The conventional approaches include regressing OC against EC within a fixed percentile of the lowest (OC/EC) ratio data (usually 5-20%) or relying on a subset of sampling days with low photochemical activity and dominated by local emissions. The drawback of these approaches is rooted in its empirical nature, i.e., a lack of clear quantitative criteria in the selection of data subsets for the (OC/EC)pri determination. We examine here a method that derives (OC/EC)pri through calculating a hypothetical set of (OC/EC)pri and SOC followed by seeking the minimum of the coefficient of correlation (R2) between SOC and EC. The hypothetical (OC/EC)pri that generates the minimum R2(SOC,EC) then represents the actual (OC/EC)pri ratio if variations of EC and SOC are independent. This Minimum R Squared (MRS) method has a clear quantitative criterion for the (OC/EC)pri calculation. The general concept embodied in the MRS method was initially proposed by Miller et al (2005), but has not been evaluated for accuracy or utility since its debut. This work uses numerically simulated data to evaluate the accuracy of SOC estimation by the MRS method and to compare with two commonly used methods: minimum OC/EC (OC/ECmin) and OC/EC percentile (OC/EC10%). Log-normally distributed EC and OC concentrations with known proportion of SOC are numerically produced through a Mersenne twister pseudorandom number generator. Three scenarios are considered, including a single primary source, two independent primary sources, and two correlated primary sources. Among the three SOC estimation methods, the MRS method consistently yields the most accurate SOC estimation. Unbiased SOC estimation by OC/ECmin and OC/EC10% only occur when the left tail of OC/EC distribution is aligned with the peak of the (OC/EC)pri distribution, which is fortuitous rather than norm. In contrast, MRS provides an unbiased SOC estimation since it is insensitive to the position of (OC/EC)pri relative to the (OC/EC) distribution. Sensitivity tests of OC and EC measurement uncertainty on SOC estimation demonstrate the superior accuracy of MRS over the other two approaches.
We consider upper and lower bounds for maxmin allocations of a completely divisible good in both competitive and cooperative strategic contexts. We then derive a subgradient algorithm to compute the exact value up to any fixed degree of precision.
Giordano Pola, Pierdomenico Pepe, Maria Domenica Di Benedetto
Time-delay systems are an important class of dynamical systems that provide a solid mathematical framework to deal with many application domains of interest. In this paper we focus on nonlinear control systems with unknown and time-varying delay signals and we propose one approach to the control design of such systems, which is based on the construction of symbolic models. Symbolic models are abstract descriptions of dynamical systems where one symbolic state and one symbolic input correspond to an aggregate of states and an aggregate of inputs. We first introduce the notion of incremental input-delay-to-state stability and characterize it by means of Lyapunov-Krasovskii functionals. We then derive sufficient conditions for the existence of symbolic models that are shown to be alternating approximately bisimilar to the original system. Further results are also derived which prove the computability of the proposed symbolic models in a finite number of steps.