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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 199 records · Page 11
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DOE OSTI2022

Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models

We derive an asymptotic expansion for the log-likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal-to-noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation-Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone towards the analysis of EM and maximum likelihood estimation in a wide range of models. A motivating application for the study of low SNR mixture models is cryo-electron microscopy data, which can be modeled as a GMM with algebraic constraints imposed on the mixture centers. We discuss the application of our expansion to algebraically constrained GMMs, among other example models of interest. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Estimating posterior quantity of interest expectations in a multilevel scalable framework

Scalable approaches for uncertainty quantification are necessary for characterizing prediction confidence in large-scale subsurface flow simulations with uncertain permeability. To this end we explore a multilevel Monte Carlo approach for estimating posterior moments of a particular quantity of interest, where we employ an element-agglomerated algebraic multigrid (AMG) technique to generate the hierarchy of coarse spaces with guaranteed approximation properties for both the generation of spatially correlated random fields and the forward simulation of Darcy's law to model subsurface flow. In both these components (sampling and forward solves), we exploit solvers that rely on state-of-the-art scalable AMG. To illustrate the applicability of this approach, numerical tests are performed on two 3D examples-a unit cube and an egg-shaped domain with an irregular boundary-where the scalability of each simulation as well as the scalability of the overall algorithm are demonstrated.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Convergence analysis of single rate and multirate fixed stress split iterative coupling schemes in heterogeneous poroelastic media

Recently, the accurate modeling of flow–structure interactions has gained more attention and importance for both petroleum and environmental engineering applications. Of particular interest is the coupling between subsurface flow and reservoir geomechanics. Different single rate and multirate iterative and explicit coupling schemes have been proposed and analyzed in the past. In addition, Banach fixed point contraction results were obtained for iterative coupling schemes, and conditionally stable results were obtained for explicit coupling schemes. In this work, we will consider the mathematical analysis of the single rate and multirate fixed stress split iterative coupling schemes for spatially heterogeneous poroelastic media. We will re–establish the contractivity for both schemes in the localized case, and we will show that heterogeneities come at the expense of imposing more restricted conditions on the number of fine flow time steps that can be taken within one coarse mechanics time step in the multirate case. Our mathematical analysis is supplemented by numerical simulations validating our derived upper bounds. Finally, to the best of our knowledge, this is the first rigorous mathematical analysis of the multirate fixed–stress split iterative coupling scheme in heterogeneous poroelastic media.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Graphical methods for evaluating covering arrays

Covering arrays relax the condition of orthogonal arrays by only requiring that all combination of levels be covered but not requiring that the appearance of all combination of levels be balanced. This allows for a much larger number of factors to be simultaneously considered but at the cost of poorer estimation of the factor effects. To better understand patterns between sets of columns and evaluate the degree of coverage to compare and select between alternative arrays, we suggest several new graphical methods that show some of the patterns of coverage for different designs. As a result, these graphical methods for evaluating covering arrays are illustrated with a few examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Selecting the Best Wild Card Entries in a Covering Array

For some covering arrays, there are some wild card positions in the array where there is flexibility about what factor levels can be chosen with no impact on the basic properties of covering array, because all of the required pairs have been covered. The choice of how to fill these wild card positions can influence other properties, such as the degrees of orthogonality or three-way coverage of the array. Here in this article, some criteria are proposed for identifying the best choices for the wild card positions to create covering arrays with highly desirable properties. Accompanying graphical summaries are also described to highlight differing performance for several examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Uncertainty analysis of trade-offs between multiple responses using hypervolume

When multiple responses are considered in process optimization, the degree to which they can be simultaneously optimized depends on the optimization objectives and the amount of trade-offs between the responses. The normalized hypervolume of the Pareto front is a useful summary to quantify the amount of trade-offs required to balance performance across the multiple responses. In order to quantify the impact of uncertainty of the estimated response surfaces and add realism to what future data to expect, 2 versions of the scaled normalized hypervolume of the Pareto front are presented. To demonstrate the variation of the hypervolume distributions, we explore a case study for a chemical process involving 3 responses, each with a different type of optimization goal. Our results show that the global normalized hypervolume characterizes the proximity to the ideal results possible, while the instance-specific summary considers the richness of the front and the severity of trade-offs between alternatives. Furthermore, the 2 scaling schemes complement each other and highlight different features of the Pareto front and hence are useful to quantify what solutions are possible for simultaneous optimization of multiple responses.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Selecting a near-optimal design for multiple criteria with improved robustness to different user priorities

In a decision-making process, relying on only one objective can often lead to oversimplified decisions that ignore important considerations. Incorporating multiple, and likely competing, objectives is critical for balancing trade-offs on different aspects of performance. When multiple objectives are considered, it is often hard to make a precise decision on how to weight the different objectives when combining their performance for ranking and selecting designs. We show that there are situations when selecting a design with near-optimality for a broad range of weight combinations of the criteria is a better test selection strategy compared with choosing a design that is strictly optimal under very restricted conditions. Here, we propose a new design selection strategy that identifies several top-ranked solutions across broad weight combinations using layered Pareto fronts and then selects the final design that offers the best robustness to different user priorities. This method involves identifying multiple leading solutions based on the primary objectives and comparing the alternatives using secondary objectives to make the final decision. We focus on the selection of screening designs because they are widely used both in industrial research, development, and operational testing. The method is illustrated with an example of selecting a single design from a catalog of designs of a fixed size. However, the method can be adapted to more general designed experiment selection problems that involve searching through a large design space.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Kink dynamics in a parametric Φ 6 system: a model with controllably many internal modes

We explore a variant of the Φ 6 model originally proposed in Phys. Rev.D 12 (1975) 1606 as a prototypical, so-called, “bag” model in which domain walls play the role of quarks within hadrons. We examine the steady state of the model, namely an apparent bound state of two kink structures. We explore its linearization, and we find that, as a function of a parameter controlling the curvature of the potential, an effectively arbitrary number of internal modes may arise in the point spectrum of the linearization about the domain wall profile. We explore some of the key characteristics of kink-antikink collisions, such as the critical velocity and the multi-bounce windows, and how they depend on the principal parameter of the model. We find that the critical velocity exhibits a non-monotonic dependence on the parameter controlling the curvature of the potential. For the multi-bounce windows, we find that their range and complexity decrease as the relevant parameter decreases (and as the number of internal modes in the model increases). We use a modified collective coordinates method [in the spirit of recent works such as Phys. Rev.D 94 (2016) 085008] in order to capture the relevant phenomenology in a semi-analytical manner.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

Abstract M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U (1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U (1)-action. When the limiting rotation is non-resonant, these maps admit formal U (1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U (1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbed U (1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

On Properties of Adjoint Systems for Evolutionary PDEs

We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the adjoint system associated with an evolutionary partial differential equation has an infinite-dimensional Hamiltonian structure, which is useful for connecting the fully continuous, semi-discrete, and fully discrete levels. We subsequently address the question of discretize-then-optimize versus optimize-then-discrete for both semi-discretization and time integration, by characterizing the commutativity of discretize-then-optimize methods versus optimize-then-discretize methods uniquely in terms of an adjoint-variational quadratic conservation law. For Galerkin semi-discretizations and one-step time integration methods in particular, we explicitly construct these commuting methods by using structure-preserving discretization techniques.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A Type II Hamiltonian Variational Principle and Adjoint Systems for Lie Groups

We present a novel Type II variational principle on the cotangent bundle of a Lie group which enforces Type II boundary conditions, i.e., fixed initial position and final momentum. In general, such Type II variational principles are only globally defined on vector spaces or locally defined on general manifolds; however, by left translation, we are able to define this variational principle globally on cotangent bundles of Lie groups. Type II boundary conditions are particularly important for adjoint sensitivity analysis, which is our motivating application. As such, we additionally discuss adjoint systems on Lie groups, their properties, and how they can be used to solve optimization problems subject to dynamics on Lie groups.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

On Bayes Equality and Related Issues

To make calculations in the Bayesian analysis, the formalism of which is based on the layering of a probability measure defined on the product of measurable spaces, it is useful to have a summary of the properties of this layering. In this paper we formulate and prove those of them that are used in calculations more often than others. Particularly, we prove Hoeffding-type inequalities using direct elementary techniques.

97 MATHEMATICS AND COMPUTING↗