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An existence and uniqueness theorem for a class of stochastic delay differential equations is presented, and the convergence of Euler approximations for these equations is proved under general conditions. Moreover, the rate of almost sure convergence is obtained under local Lipschitz and also under monotonicity conditions.
Complex engineering systems have to be carefully monitored to meet demanding performance requirements, including detecting anomalies in their operations. There are two major monitoring challenges for these systems. The first challenge is that information collected from the monitored system is often partial and/or unreliable, in the sense that some occurred events may not be reported and/or may be reported incorrectly (e.g., reported as another event). The second is that anomalies often consist of sequences of event patterns separated in space and time. This paper introduces and analyzes a diagnoser algorithm that meets these challenges for detecting and counting occurrences of anomalies in engineering systems. The proposed diagnoser algorithm assumes that models are available for characterizing plant operations (via stochastic automata) and sensors (via probabilistic mappings) used for reporting partial and unreliable information. Methods for analyzing the effects of model uncertainties on the diagnoser performance are also discussed. In order to select configurations that reduce sensor costs, while satisfying diagnoser performance requirements, a sensor configuration selection algorithm developed in previous work is then extended for the proposed diagnoser algorithm. The proposed algorithms and methods are then applied to a multi-unit-operation system, which is derived from an actual facility application. Results show that the proposed diagnoser algorithm is able to detect and count occurrences of anomalies accurately and that its performance is robust to model uncertainties. Furthermore, the sensor configuration selection algorithm is able to suggest optimal sensor configurations with significantly reduced costs, while still yielding acceptable performance for counting the occurrences of anomalies.
We present a new algorithm for detecting transiting extrasolar planets in time-series photometry. The Quasiperiodic Automated Transit Search (QATS) algorithm relaxes the usual assumption of strictly periodic transits by permitting a variable, but bounded, interval between successive transits. We show that this method is capable of detecting transiting planets with significant transit timing variations without any loss of significance-{sup s}mearing{sup -}as would be incurred with traditional algorithms; however, this is at the cost of a slightly increased stochastic background. The approximate times of transit are standard products of the QATS search. Despite the increased flexibility, we show that QATS has a run-time complexity that is comparable to traditional search codes and is comparably easy to implement. QATS is applicable to data having a nearly uninterrupted, uniform cadence and is therefore well suited to the modern class of space-based transit searches (e.g., Kepler, CoRoT). Applications of QATS include transiting planets in dynamically active multi-planet systems and transiting planets in stellar binary systems.
We present a novel software framework for modeling large-scale engineered systems as mathematical optimization problems. A key motivating feature in such systems is their hierarchical, highly structured topology. Existing mathematical optimization modeling environments do not facilitate the natural expression and manipulation of hierarchically structured systems. Rather, the modeler is forced to "flatten" the system description, hiding structure that may be exploited by solvers, and obfuscating the system that the modeling environment is attempting to represent. To correct this deficiency, we propose a Python-based "block-oriented" modeling approach for representing the discrete components within the system. Our approach is an extension of the Pyomo library for specifying mathematical optimization problems. Through the use of a modeling components library, the block-oriented approach facilitates a clean separation of system superstructure from the details of individual components. This approach also naturally lends itself to expressing design and operational decisions as disjunctive expressions over the component blocks. By expressing a mathematical optimization problem in a block-oriented manner, inherent structure (e.g., multiple scenarios) is preserved for potential exploitation by solvers. In particular, we show that block-structured mathematical optimization problems can be straightforwardly manipulated by decomposition-based multi-scenario algorithmic strategies, specifically in the context of the PySP stochastic programming library. We illustrate our block-oriented modeling approach using a case study drawn from the electricity grid operations domain: unit commitment with transmission switching and N - 1 reliability constraints. Finally, we demonstrate that the overhead associated with block-oriented modeling only minimally increases model instantiation times, and need not adversely impact solver behavior. (C) 2013 Elsevier Ltd. All rights reserved.
Future advances in large-scale machine learning depend heavily on the availability of new and more powerful nonlinear optimization methods. The goal of this project is to address this challenge by designing new optimization algorithms that can deal with the scale, nonlinearity and stochasticity arising in machine learning models. Our research will provide the theoretical foundations for the new methods, and will involve intensive numerical testing on state-of-the-art machine learning systems. The project is organized into three interrelated projects that constitute a broad research agenda in the area of large-scale stochastic optimization for data science.
This report summarizes work completed under the Laboratory Directed Research and Development (LDRD) project "Uncertainty Quantification of Geophysical Inversion Using Stochastic Differential Equations." Geophysical inversions often require computationally expensive algorithms to find even one solution, let alone propagating uncertainties through to the solution domain. The primary purpose of this project was to find more computationally efficient means to approximate solution uncertainty in geophysical inversions. We found multiple computationally efficient methods of propagating Earth model uncertainty into uncertainties in solutions of full waveform seismic moment tensor inversions. However, the optimum method of approximating the uncertainty in these seismic source solutions was to use the Karhunen-Love theorem with data misfit residuals. This method was orders of magnitude more computationally efficient than traditional Monte Carlo methods and yielded estimates of uncertainty that closely approximated those of Monte Carlo. We will summarize the various methods we evaluated for estimating uncertainty in seismic source inversions as well as work toward this goal in the realm of 3-D seismic tomographic inversion uncertainty.
This SolarExPert project has developed a Sustainable Grid Platform (SGP) with scalable architecture of distributed control and optimization. The SGP consists of the following major functions: 1) an advanced grid architecture with hierarchical and distributed communication and control, combined with the OpenFMB standard and implemented on the Multi-Agent OpenDSS (MA-OpenDSS) platform; 2) an online distributed stochastic optimal power flow; 3) an online distributed system state estimation algorithm; 4) the distributed Volt/VAR optimization and frequency control algorithms; 5) the distributed distribution system restoration strategy. The developed SGP together with advanced functions are tested in 1 million (1M)-node distribution system on the MA-OpenDSS platform. Furthermore, the models and algorithms are tested in 100,000-node system HiL simulation, and also in P-HiL implementation with 100 physical devices. The developed functions haven been validated and tested on the selected actual distribution feeder with the data collected from the field of Maui Meadows in Hawaii. The distributed PV hosting capacities with cooperative Volt/VAR and Volt/VAR/Watt control are estimated and compared to provide recommendations for customers and the utility company.
In this paper we propose a new class of coupling methods for the sensitivity analysis of high dimensional stochastic systems and in particular for lattice Kinetic Monte Carlo (KMC). Sensitivity analysis for stochastic systems is typically based on approximating continuous derivatives with respect to model parameters by the mean value of samples from a finite difference scheme. Instead of using independent samples the proposed algorithm reduces the variance of the estimator by developing a strongly correlated-“coupled”- stochastic process for both the perturbed and unperturbed stochastic processes, defined in a common state space. The novelty of our construction is that the new coupled process depends on the targeted observables, e.g., coverage, Hamiltonian, spatial correlations, surface roughness, etc., hence we refer to the proposed method as goal-oriented sensitivity analysis. In particular, the rates of the coupled Continuous Time Markov Chain are obtained as solutions to a goal-oriented optimization problem, depending on the observable of interest, by considering the minimization functional of the corresponding variance. We show that this functional can be used as a diagnostic tool for the design and evaluation of different classes of couplings. Furthermore, the resulting KMC sensitivity algorithm has an easy implementation that is based on the Bortz–Kalos–Lebowitz algorithm's philosophy, where events are divided in classes depending on level sets of the observable of interest. Finally, we demonstrate in several examples including adsorption, desorption, and diffusion Kinetic Monte Carlo that for the same confidence interval and observable, the proposed goal-oriented algorithm can be two orders of magnitude faster than existing coupling algorithms for spatial KMC such as the Common Random Number approach. We also provide a complete implementation of the proposed sensitivity analysis algorithms, including various spatial KMC examples, in a supplementary MATLAB source code.
Resilient observer design for Cyber-Physical Systems (CPS) in the presence of adversarial false data injection attacks (FDIA) is an active area of research. The existing state-of-the-art algorithms tend to break down as more and more knowledge of the system is built into the attack model; also as the percentage of attacked nodes increases. From the view of optimization theory, the problem is often cast as a classical error correction problem for which a theoretical limit of has been established as the maximum percentage attacked nodes for which state recovery is guaranteed. Beyond this limit, the performance of -minimization based schemes, for instance, deteriorates rapidly. Similar performance degradation occurs for other types of resilient observers beyond certain percentages of attacked nodes. In order to increase the corresponding percentage of attacked nodes for which state recoveries can be guaranteed, researchers have begun to incorporate prior information into the underlying resilient observer design framework. For the most pragmatic cases, this prior information is often obtained through a data-driven machine learning process. Existing results have shown a strong positive correlation between the maximum attacked percentages that can be tolerated and the accuracy of the data-driven model. Motivated by these results, this chapter examines the case for pruning algorithms designed to improve the Positive Prediction Value (PPV) of the resulting prior information, given stochastic uncertainty characteristics of the underlying machine learning model. Theoretical quantification of the achievable improvement is given. Simulation results show that the pruning algorithm significantly increases the maximum correctable percentage of attacked nodes, even for machine learning model whose prediction power is comparable to the random flip of a coin.
Incorporating experimental data is essential for increasing the credibility of simulation-aided decision making and design. This paper presents a method which uses a computational model to guide the optimal acquisition of experimental data to produce data-informed predictions of quantities of interest (QoI). Many strategies for optimal experimental design (OED) select data that maximize some utility that measures the reduction in uncertainty of uncertain model parameters, for example the expected information gain between prior and posterior distributions of these parameters. In this paper, we seek to maximize the expected information gained from the pushforward of an initial (prior) density to the push-forward of the updated (posterior) density through the parameter-to-prediction map. The formulation presented is based upon the solution of a specific class of stochastic inverse problems which seeks a probability density that is consistent with the model and the data in the sense that the push-forward of this density through the parameter-to-observable map matches a target density on the observable data. While this stochastic inverse problem forms the mathematical basis for our approach, we develop a one-step algorithm, focused on push-forward probability measures, that leverages inference-for-prediction to bypass constructing the solution to the stochastic inverse problem. A number of numerical results are presented to demonstrate the utility of this optimal experimental design for prediction and facilitate comparison of our approach with traditional OED.
In this work, we develop a Reinforcement Learning (RL) approach to the supervisory control problem for advanced energy systems, such as novel nuclear reactors and other demand-driven, mission-critical, and component-health-sensitive energy plants. The inclusive problem landscape considered captures the stochastic confluence of plant performance, component health evolution, power demand from the grid, diverse maintenance actions, and operator-defined goals and constraints, all considered over meaningfully long-enough reasoning horizons. Key aspects of the proposed approach are a receding horizon control-inspired technique dictating time- or event-triggered supervisory policy (re-)constructions, as well as additional capability-enabling contributions such as timescale compression, to handle long reasoning horizons and uncertainty in parts of the problem, and practical yet demonstrably-effective handling of hybrid action spaces with continuous and discrete decision variables. The resulting algorithm consists of a simulation-based RL agent constructing stochastic supervisory control policies over nontrivial action spaces and for long horizons, applying the learned policy to the system for a much shorter interval, and perpetually repeating, to construct the next long-horizon policy. That next policy will only be applied, again, for a short interval, yet originally far-in-time events move progressively closer, their associated uncertainty decreases, and new events and aspects enter the reasoning horizon. The proposed methodology bridges fundamental receding horizon concepts with the unequivocally stronger and more scalable reasoning of contemporary RL. Numerical examples using Soft Actor–Critic Deep RL illustrate the operation and efficacy of the proposed technique for a power plant tasked with health-aware load following missions in a dynamic electricity market landscape.
An optimization algorithm for a group of nonsmooth nonconvex problems inspired by two-stage stochastic programming problems is proposed. The main challenges for these problems include (1) the problems lack the popular lower-type properties such as prox-regularity assumed in many nonsmooth nonconvex optimization algorithms, (2) the objective can not be analytically expressed and (3) the evaluation of function values and subgradients are computationally expensive. To address these challenges, this report first examines the properties that exist in many two-stage problems, specifically upper-C 2 objectives. Then, we show that quadratic penalty method for securityconstrained alternating current optimal power flow (SCACOPF) contingency problems can make the contingency solution functions upper-C 2 . Based on these observations, a simplified bundle algorithm that bears similarity to sequential quadratic programming (SQP) method is proposed. It is more efficient in implementation and computation compared to conventional bundle methods. Global convergence analysis of the algorithm is presented under novel and reasonable assumptions. The proposed algorithm therefore fills the gap of theoretical convergence for smoothed SCACOPF problems. The inconsistency that might arise in our treatment of the constraints are addressed through a penalty algorithm whose convergence analysis is also provided. Finally, theoretical capabilities and numerical performance of the algorithm are demonstrated through numerical examples.
Here, the accurate mapping of potential energy surfaces (PESs) is crucial to our understanding of the numerous physical and chemical processes mediated by atomic rearrangements, such as conformational changes and chemical reactions, and the thermodynamic and kinetic feasibility of these processes. Stochastic electronic structure theories, e.g., Quantum Monte Carlo (QMC) methods, enable highly accurate total energy calculations that in principle can be used to construct the PES. However, their stochastic nature poses a challenge to the computation and use of forces and Hessians, which are typically required in algorithms for minimum-energy pathway (MEP) and transition state (TS) identification, such as the nudged elastic band (NEB) algorithm and its climbing image formulation. Here, we present strategies that utilize the surrogate Hessian line-search method, previously developed for QMC structural optimization, to efficiently identify MEP and TS structures without requiring force calculations at the level of the stochastic electronic structure theory. By modifying the surrogate Hessian algorithm to operate in path-orthogonal subspaces and at saddle points, we show that it is possible to identify MEPs and TSs by using a force-free QMC approach. We demonstrate these strategies via two examples, the inversion of the ammonia (NH 3 ) molecule and the nucleophilic substitution (S N 2) reaction F – + CH 3 F → FCH 3 + F – . We validate our results using Density Functional Theory (DFT)- and Coupled Cluster (CCSD, CCSD(T))-based NEB calculations. We then introduce a hybrid DFT-QMC approach to compute thermodynamic and kinetic quantities, free energy differences, rate constants, and equilibrium constants that incorporates stochastically optimized structures and their energies, and show that this scheme improves upon DFT accuracy. Our methods generalize straightforwardly to other systems and other high-accuracy theories that similarly face challenges computing energy gradients, paving the way for highly accurate PES mapping, transition state determination, and thermodynamic and kinetic calculations at significantly reduced computational expense.
This study presents an evaluation method for resistive random access memory retention reliability based on the Metropolis Monte Carlo algorithm and Gibbs free energy. The method, which does not rely on a time evolution, provides an extremely efficient way to compare the relative retention properties of metal-insulator-metal structures. It requires a small number of iterations and may be used for statistical analysis. The presented approach is used to compare the relative robustness of a single layer ZrO{sub 2} device with a double layer ZnO/ZrO{sub 2} one, and obtain results which are in good agreement with experimental data.
We study random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration, which is based on the stochastic Galerkin finite element method, and we compare its accuracy with that of Monte Carlo and stochastic collocation methods. The coefficients of the eigenvalue expansions are computed from a stochastic Rayleigh quotient. Our approach allows the computation of interior eigenvalues by deflation methods, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. The effectiveness of the methods is illustrated by numerical experiments on benchmark problems arising from vibration analysis.
This paper proposes an innovative algorithm for predicting short-term electrical maximum demand by using historical demand data. The ability to recognize in peak demand pattern for commercial or industrial customers would propose numerous direct and indirect benefits to the customers and utility providers in terms of demand reduction, cost control, and system stability. Prior works in electrical maximum demand forecasting have been mainly focused on seasonal effects, which is not a feasible approach for industrial manufacturing facilities in short-term load forecasting. The proposed algorithm, denoted as the Adaptive Rate of Change (ARC), determines the logarithmic rate-of-change in load profile prior to a peak by postulating the demand curve as a stochastic, mean-reverting process. The rationale behind this analysis, is that the energy efficient program requires not only demand estimation but also to warn the user of imminent maximum peak occurrence. This paper analyzes demand trend data and incorporates stochastic model and mean reverting half-life to develop an electrical maximum demand forecasting algorithm, which is statistically evaluated by cross-table and F-score for three different manufacturing facilities. Lastly, the aggregate results show an overall accuracy of 0.91 and a F-score of 0.43, which indicates that the algorithm is effective predicting peak demand in predicting peak demand.
While application of clustering algorithms to atom probe tomography data have enabled quantification of solute clusters in terms of number density, size, and subcomposition there exist other properties (e.g., volume, surface area, and composition) that are better determined by defining an interface between the cluster and the surrounding matrix. The limitation in composition results from an ion selection step where the expected matrix ion types are omitted from the cluster search algorithm to enhance the contrast between the matrix and cluster and to reduce the complexity of the search. Previously, composition determination within solute clusters has utilized a secondary envelopment and erosion step on top of conventional methods such as maximum separation. In this work, we present a novel stochastic method that combines the particle identification fidelity of a conventional clustering algorithm with the analytical flexibility of mesh-based approaches through the generation of alpha shapes for each identified cluster. Furthermore, the corresponding mesh accounts for concave components of the clusters and determines the volume and surface area of the clusters; additionally, the mesh boundary is utilized to update the total composition according to the internal ions.
A stabilization problem of the equilibrium of stochastically forced nonlinear discrete-time system with incomplete information is considered. Our approach uses a regulator which synthesizes the required stochastic sensitivity of the equilibrium. Mathematically, this problem is reduced to the solution of some quadratic matrix equations. A description of attainability sets and algorithms for regulators design is given. The general results are applied to the suppression of unwanted large-amplitude oscillations around the equilibria of the stochastically forced Verhulst model with noisy observations.