Model reduction for dynamical systems is a promising approach for reducing the computational cost of large-scale physics-based simulations to enable high-fidelity models to be used in many- query (e.g., Bayesian inference) and near-real-time (e.g., fast-turnaround simulation) contexts. While model reduction works well for specialized problems such as linear time-invariant systems, it is much more difficult to obtain accurate, stable, and efficient reduced-order models (ROMs) for systems with general nonlinearities. This report describes several advances that enable nonlinear reduced-order models (ROMs) to be deployed in a variety of time-critical settings. First, we present an error bound for the Gauss-Newton with Approximated Tensors (GNAT) nonlinear model reduction technique. This bound allows the state-space error for the GNAT method to be quantified when applied with the backward Euler time-integration scheme. Second, we present a methodology for preserving classical Lagrangian structure in nonlinear model reduction. This technique guarantees that important properties--such as energy conservation and symplectic time-evolution maps--are preserved when performing model reduction for models described by a Lagrangian formalism (e.g., molecular dynamics, structural dynamics). Third, we present a novel technique for decreasing the temporal complexity --defined as the number of Newton-like iterations performed over the course of the simulation--by exploiting time-domain data. Fourth, we describe a novel method for refining projection-based reduced-order models a posteriori using a goal-oriented framework similar to mesh-adaptive h -refinement in finite elements. The technique allows the ROM to generate arbitrarily accurate solutions, thereby providing the ROM with a 'failsafe' mechanism in the event of insufficient training data. Finally, we present the reduced-order model error surrogate (ROMES) method for statistically quantifying reduced- order-model errors. This enables ROMs to be rigorously incorporated in uncertainty-quantification settings, as the error model can be treated as a source of epistemic uncertainty. This work was completed as part of a Truman Fellowship appointment. We note that much additional work was performed as part of the Fellowship. One salient project is the development of the Trilinos-based model-reduction software module Razor , which is currently bundled with the Albany PDE code and currently allows nonlinear reduced-order models to be constructed for any application supported in Albany. Other important projects include the following: 1. ROMES-equipped ROMs for Bayesian inference: K. Carlberg, M. Drohmann, F. Lu (Lawrence Berkeley National Laboratory), M. Morzfeld (Lawrence Berkeley National Laboratory). 2. ROM-enabled Krylov-subspace recycling: K. Carlberg, V. Forstall (University of Maryland), P. Tsuji, R. Tuminaro. 3. A pseudo balanced POD method using only dual snapshots: K. Carlberg, M. Sarovar. 4. An analysis of discrete v. continuous optimality in nonlinear model reduction: K. Carlberg, M. Barone, H. Antil (George Mason University). Journal articles for these projects are in progress at the time of this writing.
Consider a classic hierarchy tree as a basic model of a 'system-of-systems' network, where each node represents a component system (which may itself consist of a set of sub-systems). For this general composite system, we present a technique for computing the optimal testing strategy, which is based on Bayesian decision analysis. In previous work, we developed a Bayesian approach for computing the distribution of the reliability of a system-of-systems structure that uses test data and prior information. This allows for the determination of both an estimate of the reliability and a quantification of confidence in the estimate. Improving the accuracy of the reliability estimate and increasing the corresponding confidence require the collection of additional data. However, testing all possible sub-systems may not be cost-effective, feasible, or even necessary to achieve an improvement in the reliability estimate. To address this sampling issue, we formulate a Bayesian methodology that systematically determines the optimal sampling strategy under specified constraints and costs that will maximally improve the reliability estimate of the composite system, e.g., by reducing the variance of the reliability distribution. This methodology involves calculating the 'Bayes risk of a decision rule' for each available sampling strategy, where risk quantifies the relative effect that each sampling strategy could have on the reliability estimate. A general numerical algorithm is developed and tested using an example multicomponent system. The results show that the procedure scales linearly with the number of components available for testing.
Complex systems are made up of multiple interdependent parts, and the behavior of the entire system cannot always be directly inferred from the behavior of the individual parts. They are nonlinear and system responses are not necessarily additive. Examples of complex systems include energy, cyber and telecommunication infrastructures, human and animal social structures, and biological structures such as cells. To meet the goals of infrastructure development, maintenance, and protection for cyber-related complex systems, novel modeling and simulation technology is needed. Sandia has shown success using M&S in the nuclear weapons (NW) program. However, complex systems represent a significant challenge and relative departure from the classical M&S exercises, and many of the scientific and mathematical M&S processes must be re-envisioned. Specifically, in the NW program, requirements and acceptable margins for performance, resilience, and security are well-defined and given quantitatively from the start. The Quantification of Margins and Uncertainties (QMU) process helps to assess whether or not these safety, reliability and performance requirements have been met after a system has been developed. In this sense, QMU is used as a sort of check that requirements have been met once the development process is completed. In contrast, performance requirements and margins may not have been defined a priori for many complex systems, (i.e. the Internet, electrical distribution grids, etc.), particularly not in quantitative terms. This project addresses this fundamental difference by investigating the use of QMU at the start of the design process for complex systems. Three major tasks were completed. First, the characteristics of the cyber infrastructure problem were collected and considered in the context of QMU-based tools. Second, UQ methodologies for the quantification of model discrepancies were considered in the context of statistical models of cyber activity. Third, Bayesian methods for optimal testing in the QMU framework were developed. This completion of this project represent an increased understanding of how to apply and use the QMU process as a means for improving model predictions of the behavior of complex systems. 4
Staggered bioterrorist attacks with aerosolized pathogens on population centers present a formidable challenge to resource allocation and response planning. The response and planning will commence immediately after the detection of the first attack and with no or little information of the second attack. In this report, we outline a method by which resource allocation may be performed. It involves probabilistic reconstruction of the bioterrorist attack from partial observations of the outbreak, followed by an optimization-under-uncertainty approach to perform resource allocations. We consider both single-site and time-staggered multi-site attacks (i.e., a reload scenario) under conditions when resources (personnel and equipment which are difficult to gather and transport) are insufficient. Both communicable (plague) and non-communicable diseases (anthrax) are addressed, and we also consider cases when the data, the time-series of people reporting with symptoms, are confounded with a reporting delay. We demonstrate how our approach develops allocations profiles that have the potential to reduce the probability of an extremely adverse outcome in exchange for a more certain, but less adverse outcome. We explore the effect of placing limits on daily allocations. Further, since our method is data-driven, the resource allocation progressively improves as more data becomes available.
The Design-through-Analysis Realization Team (DART) is chartered with reducing the time Sandia analysts require to complete the engineering analysis process. The DART system analysis team studied the engineering analysis processes employed by analysts in Centers 9100 and 8700 at Sandia to identify opportunities for reducing overall design-through-analysis process time. The team created and implemented a rigorous analysis methodology based on a generic process flow model parameterized by information obtained from analysts. They also collected data from analysis department managers to quantify the problem type and complexity distribution throughout Sandia's analyst community. They then used this information to develop a community model, which enables a simple characterization of processes that span the analyst community. The results indicate that equal opportunity for reducing analysis process time is available both by reducing the ''once-through'' time required to complete a process step and by reducing the probability of backward iteration. In addition, reducing the rework fraction (i.e., improving the engineering efficiency of subsequent iterations) offers approximately 40% to 80% of the benefit of reducing the ''once-through'' time or iteration probability, depending upon the process step being considered. Further, the results indicate that geometry manipulation and meshing is the largest portion of an analyst's effort, especially for structural problems, and offers significant opportunity for overall time reduction. Iteration loops initiated late in the process are more costly than others because they increase ''inner loop'' iterations. Identifying and correcting problems as early as possible in the process offers significant opportunity for time savings.
Three years of large-scale PDE-constrained optimization research and development are summarized in this report. We have developed an optimization framework for 3 levels of SAND optimization and developed a powerful PDE prototyping tool. The optimization algorithms have been interfaced and tested on CVD problems using a chemically reacting fluid flow simulator resulting in an order of magnitude reduction in compute time over a black box method. Sandia's simulation environment is reviewed by characterizing each discipline and identifying a possible target level of optimization. Because SAND algorithms are difficult to test on actual production codes, a symbolic simulator (Sundance) was developed and interfaced with a reduced-space sequential quadratic programming framework (rSQP++) to provide a PDE prototyping environment. The power of Sundance/rSQP++ is demonstrated by applying optimization to a series of different PDE-based problems. In addition, we show the merits of SAND methods by comparing seven levels of optimization for a source-inversion problem using Sundance and rSQP++. Algorithmic results are discussed for hierarchical control methods. The design of an interior point quadratic programming solver is presented.