Ensemble Kalman Methods: A Mean Field Perspective

This paper provides a unifying mean field based framework for the derivation and analysis of ensemble Kalman methods. Both state estimation and parameter estimation problems are considered, and formulations in both discrete and continuous time are employed. For state estimation problems both the control and filtering approaches are studied; analogously, for parameter estimation (inverse) problems the optimization and Bayesian perspectives are both studied. The approach taken unifies a wide-ranging literature in the field, provides a framework for analysis of ensemble Kalman methods, and suggests open problems

A new approach to optimal designs for correlated observations

This paper presents a new and efficient method for the construction of optimal designs for regression models with dependent error processes. In contrast to most of the work in this field, which starts with a model for a finite number of observations and considers the asymptotic properties of estimators and designs as the sample size converges to infinity, our approach is based on a continuous time model. We use results from stochastic anal- ysis to identify the best linear unbiased estimator (BLUE) in this model. Based on the BLUE, we construct an efficient linear estimator and corresponding optimal designs in the model for finite sample size by minimizing the mean squared error between the opti- mal solution in the continuous time model and its discrete approximation with respect to the weights (of the linear estimator) and the optimal design points, in particular in the multi-parameter case. In contrast to previous work on the subject the resulting estimators and corresponding optimal designs are very efficient and easy to implement. This means that they are practi- cally not distinguishable from the weighted least squares estimator and the corresponding optimal designs, which have to be found numerically by non-convex discrete optimization. The advantages of the new approach are illustrated in several numerical examples.Comment: Keywords and Phrases: linear regression, correlated observations, optimal design, Gaussian white mouse model, Doob representation, quadrature formulas AMS Subject classification: Primary 62K05; Secondary: 62M0

A Comparison of Discrete and Continuous Neural Network Approaches to Solve the Class/Teacher Timetabling Problem

This study explores the application of neural network-based heuristics to the class/teacher timetabling problem (CTTP). The paper begins by presenting the basic CTTP characteristics in terms of hard and soft constraints and proposing a formulation for the energy function required to map the problem within the artificial neural network model. There follow two distinct approaches to simulating neural network evolution. The first uses a Potts mean-field annealing simulation based on continuous Potts neurons, which has obtained favorable results in various combi¬natorial optimization problems. Afterwards, a discrete neural network simulation, based on discrete winner-take-all neurons, is proposed. The paper concludes with a comparison of the computational results taken from the application of both heuris¬tics to hard hypothetical and real CTTP instances. This experiment demonstrates that the discrete approach performs better, in terms of solution quality as well as execution time

Fresh Multiple Access: A Unified Framework Based on Large Models and Mean-Field Approximations

Information freshness has attracted increasingly attention in the past decade as it plays a critical role in the emerging real-time applications. Age of information (AoI) holds the promise of effectively characterizing the information freshness, hence widely considered as a fundamental performance metric. However, in multiple-device scenarios, most existing works focus on the analysis and optimization of AoI based on queueing systems. The study for a unified approach for general multiple access control scheme in freshness-oriented scenarios remains open. In this paper, we take into consideration the combination of the fundamental freshness metric AoI and multiple access control schemes to achieve efficient cross-layer analysis and optimization in freshness-oriented scenarios, which is referred to as fresh multiple access. To this end, we build a unified framework with a discrete-time tandem queue model for fresh multiple access. The unified framework enables the analysis and optimization for general multiple access protocols in fresh multiple access. To handle the high dimension framework embedded in fresh multiple access, we introduce large model approaches for the Markov chain formulation in AoI oriented scenarios. Two typical AoI-based metric are studied including age of incorrect information (AoII) and peak AoII. Moreover, to address the computational complexity of the large model, we present mean-field approximations which significantly reduces the dimension of the Markov chain model by approximating the integral affect of massive devices in fresh multiple access.Comment: accepted by Journal of Communications and Network

Variational assimilation of sparse time-averaged data for efficient adjoint-based optimization of unsteady RANS simulations

Data assimilation (DA) plays a crucial role in extracting valuable information from flow measurements in fluid dynamics problems. Often only time-averaged data is available, which poses challenges for DA in the context of unsteady flow problems. Recent works have shown promising results in optimizing Reynolds-averaged Navier-Stokes (RANS) simulations of stationary flows using sparse data through variational data assimilation, enabling the reconstruction of mean flow profiles. In this study we perform three-dimensional variational data assimilation of sparse time-averaged data into an unsteady RANS (URANS) simulation by means of a stationary divergence-free forcing term in the URANS equations. Efficiency and speed of our method are enhanced by employing coarse URANS simulations and leveraging the stationary discrete adjoint method for the time-averaged URANS equations. The data assimilation codes were developed in-house using OpenFOAM for the URANS simulations as well as for the solution of the adjoint problem, and Python for the gradient-based optimization. Our results demonstrate that data assimilation of sparse time-averaged velocity measurements not only enables accurate mean flow reconstruction, but also improves the flow dynamics, specifically the vortex shedding frequency. To validate the efficacy of our approach, we applied it to turbulent flows around cylinders of various shapes at Reynolds numbers ranging from 3000 to 22000. Our findings indicate that data points near the cylinder play a crucial role in improving the vortex shedding frequency, while additional data points further downstream are necessary to also reconstruct the time-averaged velocity field in the wake region

Continuous time mean-variance portfolio optimization through the mean field approach

A simple mean-variance portfolio optimization problem in continuous time is solved using the mean field approach. In this approach, the original optimal control problem, which is time inconsistent, is viewed as the McKean\u2013Vlasov limit of a family of controlled many-component weakly interacting systems. The prelimit problems are solved by dynamic programming, and the solution to the original problem is obtained by passage to the limit