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We study incentive designs for a class of stochastic Stackelberg games with one leader and a large number of (finite as well as infinite population of) followers. We investigate whether the leader can craft a strategy under a dynamic information structure that induces a desired behavior among the followers. For the finite population setting, under sufficient conditions, we show that there exist symmetric incentive strategies for the leader that attain approximately optimal performance from the leader's viewpoint and lead to an approximate symmetric (pure) Nash best response among the followers. Driving the follower population to infinity, we arrive at the interesting result that in this infinite-population regime the leader cannot design a smooth "finite-energy" incentive strategy, namely, a mean-field limit for such games is not well-defined. As a way around this, we introduce a class of stochastic Stackelberg games with a leader, a major follower, and a finite or infinite population of minor followers, where the leader provides an incentive only for the major follower, who in turn influences the rest of the followers through her strategy. For this class of problems, we are able to establish the existence of an incentive strategy with finitely many minor followers. We also show that if the leader's strategy with finitely many minor followers converges as their population size grows, then the limit defines an incentive strategy for the corresponding mean-field Stackelberg game. Examples of quadratic Gaussian games are provided to illustrate both positive and negative results. In addition, as a byproduct of our analysis, we establish existence of a randomized incentive strategy for the class mean-field Stackelberg games, which in turn provides an approximation for an incentive strategy of the corresponding finite population Stackelberg game.

In this work we are interested in general linear inverse problems where the corresponding forward problem is solved iteratively using fixed point methods. Then one-shot methods, which iterate at the same time on the forward problem solution and on the inverse problem unknown, can be applied. We analyze two variants of the so-called multi-step one-shot methods and establish sufficient conditions on the descent step for their convergence, by studying the eigenvalues of the block matrix of the coupled iterations. Several numerical experiments are provided to illustrate the convergence of these methods in comparison with the classical usual and shifted gradient descent. In particular, we observe that very few inner iterations on the forward problem are enough to guarantee good convergence of the inversion algorithm.

With the aid of hardware and software developments, there has been a surge of interests in solving partial differential equations by deep learning techniques, and the integration with domain decomposition strategies has recently attracted considerable attention due to its enhanced representation and parallelization capacity of the network solution. While there are already several works that substitute the numerical solver of overlapping Schwarz methods with the deep learning approach, the non-overlapping counterpart has not been thoroughly studied yet because of the inevitable interface overfitting problem that would propagate the errors to neighbouring subdomains and eventually hamper the convergence of outer iteration. In this work, a novel learning approach, i.e., the compensated deep Ritz method, is proposed to enable the flux transmission across subregion interfaces with guaranteed accuracy, thereby allowing us to construct effective learning algorithms for realizing the more general non-overlapping domain decomposition methods in the presence of overfitted interface conditions. Numerical experiments on a series of elliptic boundary value problems including the regular and irregular interfaces, low and high dimensions, smooth and high-contrast coefficients on multidomains are carried out to validate the effectiveness of our proposed domain decomposition learning algorithms.

Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.

This paper concerns the numerical solution of the two-dimensional time-dependent partial integro-differential equation (PIDE) that holds for the values of European-style options under the two-asset Kou jump-diffusion model. A main feature of this equation is the presence of a nonlocal double integral term. For its numerical evaluation, we extend a highly efficient algorithm derived by Toivanen (2008) in the case of the one-dimensional Kou integral. The acquired algorithm for the two-dimensional Kou integral has optimal computational cost: the number of basic arithmetic operations is directly proportional to the number of spatial grid points in the semidiscretization. For the effective discretization in time, we study seven contemporary operator splitting schemes of the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind. All these schemes allow for a convenient, explicit treatment of the integral term. By ample numerical experiments for put-on-the-average option values, the stability and convergence behaviour as well as the mutual performance of the seven operator splitting schemes are investigated. Moreover, the Greeks Delta and Gamma are considered.

Strategy iteration is a technique frequently used for two-player games in order to determine the winner or compute payoffs, but to the best of our knowledge no general framework for strategy iteration has been considered. Inspired by previous work on simple stochastic games, we propose a general formalisation of strategy iteration for solving least fixpoint equations over a suitable class of complete lattices, based on MV-chains. We devise algorithms that can be used for non-expansive fixpoint functions represented as so-called min-, respectively, max-decompositions. Correspondingly, we develop two different techniques: strategy iteration from above, which has to solve the problem that iteration might reach a fixpoint that is not the least, and from below, which is algorithmically simpler, but requires a more involved correctness argument. We apply our method to solve energy games and compute behavioural metrics for probabilistic automata.

An interactive mechanism is an algorithm that stores a data set and answers adaptively chosen queries to it. The mechanism is called differentially private, if any adversary cannot distinguish whether a specific individual is in the data set by interacting with the mechanism. We study composition properties of differential privacy in concurrent compositions. In this setting, an adversary interacts with k interactive mechanisms in parallel and can interleave its queries to the mechanisms arbitrarily. Previously, [Vadhan and Wang, TCC 2021] proved an optimal concurrent composition theorem for pure-differential privacy. We significantly generalize and extend their results. Namely, we prove optimal parallel composition properties for several major notions of differential privacy in the literature, including approximate DP, R\'enyi DP, and zero-concentrated DP. Our results demonstrate that the adversary gains no advantage by interleaving its queries to independently running mechanisms. Hence, interactivity is a feature that differential privacy grants us for free.

We consider a singularly perturbed convection-diffusion problem that has in addition a shift term. We show a solution decomposition using asymptotic expansions and a stability result. Based upon this we provide a numerical analysis of high order finite element method on layer adapted meshes. We also apply a new idea of using a coarser mesh in places where weak layers appear. Numerical experiments confirm our theoretical results.

A new decomposition method for nonstationary signals, named Adaptive Local Iterative Filtering (ALIF), has been recently proposed in the literature. Given its similarity with the Empirical Mode Decomposition (EMD) and its more rigorous mathematical structure, which makes feasible to study its convergence compared to EMD, ALIF has really good potentiality to become a reference method in the analysis of signals containing strong nonstationary components, like chirps, multipaths and whistles, in many applications, like Physics, Engineering, Medicine and Finance, to name a few. In [11], the authors analyzed the spectral properties of the matrices produced by the ALIF method, in order to study its stability. Various results are achieved in that work through the use of Generalized Locally Toeplitz (GLT) sequences theory, a powerful tool originally designed to extract information on the asymptotic behavior of the spectra for PDE discretization matrices. In this manuscript we focus on answering some of the open questions contained in [11], and in doing so, we also develop new theory and results for the GLT sequences.

We identify and demonstrate a weakness of Petri Nets (PN) in specifying composite behavior of reactive systems. Specifically, we show how, when specifying multiple requirements in one PN model, modelers are obliged to specify mechanisms for combining these requirements. This yields, in many cases, over-specification and incorrect models. We demonstrate how some execution paths are missed, and some are generated unintentionally. To support this claim, we analyze PN models from the literature, identify the combination mechanisms, and demonstrate their effect on the correctness of the model. To address this problem, we propose to model the system behavior using behavioral programming (BP), a software development and modeling paradigm designed for seamless integration of independent requirements. Specifically, we demonstrate how the semantics of BP, which define how to interweave scenarios into a single model, allow avoiding the over-specification. Additionally, while BP maintains the same mathematical properties as PN, it provides means for changing the model dynamically, thus increasing the agility of the specification. We compare BP and PN in quantitative and qualitative measures by analyzing the models, their generated execution paths, and the specification process. Finally, while BP is supported by tools that allow for applying formal methods and reasoning techniques to the model, it lacks the legacy of PN tools and algorithms. To address this issue, we propose semantics and a tool for translating BP models to PN and vice versa.