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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling-based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time-consuming if the simulation-based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. Finally, an algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1999

A Spherical Earth Solution for TOA Lightning Location Retrieval

With the advent of high-speed digital computers, the method of chi square minimization is a highly practical means for analyzing a wide variety of (otherwise intractable) nonlinear inversion problems in applied mathematical physics. Little thought or effort is required to apply the chi square method to obtain quick and reasonable estimates of a solution, and the method offers a means to assess retrieval errors. Because the method is simple and practical it is sometimes hastily applied to problems that can be solved by formal analytic or quasi-analytic means. Presently, Global Atmospherics Inc. (GAI) finds the minimum of a chi square function to analyze time-of-arrival (TOA) and magnetic bearing data derived from the National Lightning Detection Network (NLDN); ellipsoidal Earth geometry is assumed. An analytic solution to this problem has not yet been attained, but the consideration and solving of less general problem statements might eventually lead to a final solution. In the present study, the problem of retrieving lightning ground-strike location on a spherical Earth surface using a network of 4 or more time-of-arrival (TOA) sensors is considered. It is shown that this problem has an analytic solution and therefore does not require the use of nonlinear estimation theory (such as the chi square method mentioned above). The mathematical robustness of the analytic solution is tested using computer-generated lightning sources and simulated TOA measurement errors. A quasi-analytic extension of the spherical Earth solution for an oblate spheroidal Earth geometry is considered in a related study. The incorporation of magnetic bearing information into these analytic solutions would lead to a general and elegant analytic retrieval scheme that would most likely replace the chi square estimation theory currently employed by Global Atmospherics Inc. (GAI).

Koshak, William J,↗
DOE OSTI2019

A Method for Three-Dimensional Deformation Predictions in the Rolling of Steel Strips

In order to achieve a better accuracy-speed balance in the three-dimensional (3-D) deformation calculations of strip rolling, especially for cold rolling of comparatively thin and wide steel strips, a new integrative method is developed. The basic and related mathematic equations and the coupling algorithm are given in detail. To test the approach and prove its effectiveness, four examples are calculated by the proposed method and the models based on the traditional influence function method, taking a six-high strip finishing mill and the rolled piece of a certain specification as research objects. The lateral distributions of front tensile stress that is closely related to the 3-D plastic deformation of the rolled metal and the flatness in the four cases are presented and discussed. The comparison of computational expense and analysis of the results preliminarily confirms that it is workable for this purpose.

36 MATERIALS SCIENCE↗
DOE OSTI2017

NPTFit: A Code Package for Non-Poissonian Template Fitting

We present NPTFit, an open-source code package, written in Python and Cython, for performing non-Poissonian template fits (NPTFs). The NPTF is a recently developed statistical procedure for characterizing the contribution of unresolved point sources (PSs) to astrophysical data sets. The NPTF was first applied to Fermi gamma-ray data to provide evidence that the excess of ∼GeV gamma-rays observed in the inner regions of the Milky Way likely arises from a population of sub-threshold point sources, and the NPTF has since found additional applications studying sub-threshold extragalactic sources at high Galactic latitudes. The NPTF generalizes traditional astrophysical template fits to allow for the ability to search for populations of unresolved PSs that may follow a given spatial distribution. NPTFit builds upon the framework of the fluctuation analyses developed in X-ray astronomy, thus it likely has applications beyond those demonstrated with gamma-ray data. The NPTFit package utilizes novel computational methods to perform the NPTF efficiently. The code is available at http://github.com/bsafdi/NPTFit and up-to-date and extensive documentation may be found at http://nptfit.readthedocs.io.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2018

A Method of Parametric Solution of Convolution Equations

A method of moments is developed for the parametric form the solution of convolution equations of the first kind. Two models, the gamma model and the shifted gamma model, are studied in details and a non-parametric approach based on regularization is presented.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Level-set topology optimization with PDE generated conformal meshes

This paper presents a level-set topology optimization approach that uses conformal meshes for the analysis of the displacement field. The structure’s boundary is represented by the iso-contour of a level-set field discretized on a fixed background design mesh. The conformal mesh is updated for each design iteration via a PDE based mesh morphing process that identifies the set of facets in the background mesh that are homeomorphic to the boundary and relaxes the homeomorphic mesh to conform to the structure’s boundary and ensure high element quality. The conformal mesh allows for a more accurate computation of the response versus density and some level-set based methods which interpolate material properties using the volume fraction. Numerical examples illustrate the proposed approach by optimizing linear-elastic two- and three-dimensional structures, wherein insight into the performance of the mesh morphing process is provided. The examples also highlight the scalability of the approach.

42 ENGINEERING↗
DOE OSTI2016

Stability and accuracy of 3D neutron transport simulations using the 2D/1D method in MPACT

A consistent “2D/1D” neutron transport method is derived from the 3D Boltzmann transport equation, to calculate fuel-pin-resolved neutron fluxes for realistic full-core Pressurized Water Reactor (PWR) problems. The 2D/1D method employs the Method of Characteristics to discretize the radial variables and a lower order transport solution to discretize the axial variable. This paper describes the theory of the 2D/1D method and its implementation in the MPACT code, which has become the whole-core deterministic neutron transport solver for the Consortium for Advanced Simulations of Light Water Reactors (CASL) core simulator VERA-CS. Several applications have been performed on both leadership-class and industry-class computing clusters. Results are presented for whole-core solutions of the Watts Bar Nuclear Power Station Unit 1 and compared to both continuous-energy Monte Carlo results and plant data.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2017

Annotations on the virtual element method for second-order elliptic problems

This document contains working annotations on the Virtual Element Method (VEM) for the approximate solution of diffusion problems with variable coefficients. To read this document you are assumed to have familiarity with concepts from the numerical discretization of Partial Differential Equations (PDEs) and, in particular, the Finite Element Method (FEM). This document is not an introduction to the FEM, for which many textbooks (also free on the internet) are available. Eventually, this document is intended to evolve into a tutorial introduction to the VEM (but this is really a long-term goal).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Multiple element isotope probes, NanoSIMS, and the functional genomics of microbial carbon cycling in soils in response to chronic climatic change

In this project, we developed an innovative and ground-breaking technique, quantitative stable isotope probing, a technique that uses density separation of nucleic acids as a quantitative measurement technique. This work is substantial because it advances SIP beyond the qualitative technique that has dominate the field for years. The first methods paper was published in Applied and Environmental Microbiology (Hungate et al. 2015), and this paper describes the mathematical model underlying the quantitative interpretation. A second methods paper (Schwartz et al. 2015) provides a conceptual overview of the method and its application to research problems. A third methods paper was just published (Koch et al. 2018), in which we develop the quantitative model combining sequencing and isotope data to estimate actual rates of microbial growth and death in natural populations. This work has met much enthusiasm in scientific presentations around the world. It has met with equally enthusiastic resistance in the peer-review process, though our record of publication to date argues that people are accepting the merits of the approach. The skepticism and resistance are also potentially signs that this technique is pushing the field forward, albeit with some of the discomfort that accompanies extrapolation. Part of this is a cultural element in the field – the field of microbiology is not accustomed to the assumptions of ecosystem science. Research conducted in this project has pushed the philosophical perspective that major advances can occur when we advocate a sound merger between the traditions of strong inference in microbiology with those of grounded scaling in ecosystem science.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2018

Multiple element isotope probes, NanoSIMS, and the functional genomics of microbial carbon cycling in soils in response to chronic climatic change

In this project, we developed an innovative and groundbreaking technique, quantitative stable isotope probing, a technique that uses density separation of nucleic acids as a quantitative measurement technique. This work is substantial because it advances SIP beyond the qualitative technique that has dominate the field for years. The first methods paper was published in Applied and Environmental Microbiology (Hungate et al. 2015), and this paper describes the mathematical model underlying the quantitative interpretation. A second methods paper (Schwartz et al. 2015) provides a conceptual overview of the method and its application to research problems. A third methods paper was just published (Koch et al. 2018), in which we develop the quantitative model combining sequencing and isotope data to estimate actual rates of microbial growth and death in natural populations. This work has met much enthusiasm in scientific presentations around the world. It has met with equally enthusiastic resistance in the peerreview process, though our record of publication to date argues that people are accepting the merits of the approach. The skepticism and resistance are also potentially signs that this technique is pushing the field forward, albeit with some of the discomfort that accompanies extrapolation. Part of this is a cultural element in the field – the field of microbiology is not accustomed to the assumptions of ecosystem science. Research conducted in this project has pushed the philosophical perspective that major advances can occur when we advocate a sound merger between the traditions of strong inference in microbiology with those of grounded scaling in ecosystem science.

quantitative stable isotope probing↗
NASA NTRS1978

Model verification of mixed dynamic systems

A parameter-estimation method is described for verifying the mathematical model of mixed (combined interactive components from various engineering fields) dynamic systems against pertinent experimental data. The model verification problem is divided into two separate parts: defining a proper model and evaluating the parameters of that model. The main idea is to use differences between measured and predicted behavior (response) to adjust automatically the key parameters of a model so as to minimize response differences. To achieve the goal of modeling flexibility, the method combines the convenience of automated matrix generation with the generality of direct matrix input. The equations of motion are treated in first-order form, allowing for nonsymmetric matrices, modeling of general networks, and complex-mode analysis. The effectiveness of the method is demonstrated for an example problem involving a complex hydraulic-mechanical system.

Chrostowski, J. D.↗
DOE OSTI2015

Peridynamic Multiscale Finite Element Methods

The problem of computing quantum-accurate design-scale solutions to mechanics problems is rich with applications and serves as the background to modern multiscale science research. The prob- lem can be broken into component problems comprised of communicating across adjacent scales, which when strung together create a pipeline for information to travel from quantum scales to design scales. Traditionally, this involves connections between a) quantum electronic structure calculations and molecular dynamics and between b) molecular dynamics and local partial differ- ential equation models at the design scale. The second step, b), is particularly challenging since the appropriate scales of molecular dynamic and local partial differential equation models do not overlap. The peridynamic model for continuum mechanics provides an advantage in this endeavor, as the basic equations of peridynamics are valid at a wide range of scales limiting from the classical partial differential equation models valid at the design scale to the scale of molecular dynamics. In this work we focus on the development of multiscale finite element methods for the peridynamic model, in an effort to create a mathematically consistent channel for microscale information to travel from the upper limits of the molecular dynamics scale to the design scale. In particular, we first develop a Nonlocal Multiscale Finite Element Method which solves the peridynamic model at multiple scales to include microscale information at the coarse-scale. We then consider a method that solves a fine-scale peridynamic model to build element-support basis functions for a coarse- scale local partial differential equation model, called the Mixed Locality Multiscale Finite Element Method. Given decades of research and development into finite element codes for the local partial differential equation models of continuum mechanics there is a strong desire to couple local and nonlocal models to leverage the speed and state of the art of local models with the flexibility and accuracy of the nonlocal peridynamic model. In the mixed locality method this coupling occurs across scales, so that the nonlocal model can be used to communicate material heterogeneity at scales inappropriate to local partial differential equation models. Additionally, the computational burden of the weak form of the peridynamic model is reduced dramatically by only requiring that the model be solved on local patches of the simulation domain which may be computed in parallel, taking advantage of the heterogeneous nature of next generation computing platforms. Addition- ally, we present a novel Galerkin framework, the 'Ambulant Galerkin Method', which represents a first step towards a unified mathematical analysis of local and nonlocal multiscale finite element methods, and whose future extension will allow the analysis of multiscale finite element methods that mix models across scales under certain assumptions of the consistency of those models.

42 ENGINEERING↗
DOE OSTI2021

Implicit-Explicit Multirate Infinitesimal GARK Methods

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called infinitesimal formulation for the fast time scale through definition of a sequence of modified “fast" initial-value problems that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of GARK methods and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1974

Cardiac mechanics: Physiological, clinical, and mathematical considerations

Recent studies concerning the basic physiological and biochemical principles underlying cardiac muscle contraction, methods for the assessment of cardiac function in the clinical situation, and mathematical approaches to cardiac mechanics are presented. Some of the topics covered include: cardiac ultrastructure and function in the normal and failing heart, myocardial energetics, clinical applications of angiocardiography, use of echocardiography for evaluating cardiac performance, systolic time intervals in the noninvasive assessment of left ventricular performance in man, evaluation of passive elastic stiffness for the left ventricle and isolated heart muscle, a conceptual model of myocardial infarction and cardiogenic shock, application of Huxley's sliding-filament theory to the mechanics of normal and hypertrophied cardiac muscle, and a rheological modeling of the intact left ventricle. Individual items are announced in this issue.

Mirsky, I.↗