Search NASA⌕ Search

SEARCH · Search NASA

Results for “Mathematical analysis”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

Bistability analysis in mathematical model of M-phase control in Xenopus oocyte extracts

To gain more insight into the molecular mechanism that regulates eukaryotic cell cycle, we propose a modification to an existing Mathematical model of mitosis-promoting factor control in Xenopus oocyte extract. To achieve this, we redirect how cell exits from mitosis using ubiquitin ligase enzyme APCP:Cdc20 which is acting as a modifier in cyclin degradation. We present a modified wiring map using combination of mass action kinetics and Michaelis–Menten kinetics to represent the biochemical processes. This results in a system of non-linear ordinary differential equations. The equilibria of the system were analyzed analytically. Using MPF as a bifurcation parameter, we analytically derived conditions under which bistability can occur in the MPF activation module. We find that for bistability to occur, it is necessary that the rate of phosphorylation of Wee1 must be greater than the rate of its dephosphorylation and that the concentration of MPF must lie in certain range of parameter values. Numerical simulations conducted show that the time taken to exit the cell cycle is shorter than those published in some reports and the sensitivity analysis results suggest that the rate of synthesis of cyclin is the most sensitive parameter in the dynamics of cell cycle. Through numerical experiments, we found new parameter sets that will give rise to bistability in the MPF activation module.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Energy Saving by Insulating a Cooking Pot

Sandia National Laboratories is one of the most technologically advanced and resourceful national laboratories in the United States. Their mission is to use science and technology, innovative research, and global engagement to counter threats, reduce dangers, and respond to disasters. To achieve this mission, Sandia is divided into ten divisions that are overlooked by Dr. Stephen Younger, the Laboratories Director. Amongst the ten divisions, the internship will take place in Division 8000, which is Integrated Security Solutions. Inside this division, the internship will be a Structural and Thermal Analysis R&D. Apart from being strictly focused on security, the division performs research for small companies that do not count with the resources to perform it themselves. During the summer internship, one of these cases is being attended. A small company invented an insulating cover that is supposed to be placed over the top of a cooking pot lid. They claim that the insulating cover helps in the saving of energy. As an intern and researcher, the task is to determine how much energy is effectively saved by placing the insulating cover over the cooking pot. To achieve the goal, mathematical analyses, experiments and simulations will be done. Then, if appropriate, a better design will be made to increase the energy saved by the insulating cover. For the mathematical analysis, the thermal resistance method was used, resulting in energy saved at a rate of 14.2795 W. After the mathematical analysis was done, a test plan was developed and presented to the supervisor for approval. After some recommendations, the test plan will be modified accordingly.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2017

Transient dynamics of terrestrial carbon storage: Mathematical foundation and its applications

Terrestrial ecosystems have absorbed roughly 30% of anthropogenic CO 2 emissions over the past decades, but it is unclear whether this carbon (C) sink will endure into the future. Despite extensive modeling and experimental and observational studies, what fundamentally determines transient dynamics of terrestrial C storage under global change is still not very clear. Here we develop a new framework for understanding transient dynamics of terrestrial C storage through mathematical analysis and numerical experiments. Our analysis indicates that the ultimate force driving ecosystem C storage change is the C storage capacity, which is jointly determined by ecosystem C input (e.g., net primary production, NPP) and residence time. Since both C input and residence time vary with time, the C storage capacity is time-dependent and acts as a moving attractor that actual C storage chases. The rate of change in C storage is proportional to the C storage potential, which is the difference between the current storage and the storage capacity. The C storage capacity represents instantaneous responses of the land C cycle to external forcing, whereas the C storage potential represents the internal capability of the land C cycle to influence the C change trajectory in the next time step. The influence happens through redistribution of net C pool changes in a network of pools with different residence times. Moreover, this and our other studies have demonstrated that one matrix equation can replicate simulations of most land C cycle models (i.e., physical emulators). As a result, simulation outputs of those models can be placed into a three-dimensional (3-D) parameter space to measure their differences. The latter can be decomposed into traceable components to track the origins of model uncertainty. In addition, the physical emulators make data assimilation computationally feasible so that both C flux- and pool-related datasets can be used to better constrain model predictions of land C sequestration. Altogether, this new mathematical framework offers new approaches to understanding, evaluating, diagnosing, and improving land C cycle models.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Searching for fat tails in CRISPR-Cas systems: Data analysis and mathematical modeling

Understanding CRISPR-Cas systems—the adaptive defence mechanism that about half of bacterial species and most of archaea use to neutralise viral attacks—is important for explaining the biodiversity observed in the microbial world as well as for editing animal and plant genomes effectively. The CRISPR-Cas system learns from previous viral infections and integrates small pieces from phage genomes called spacers into the microbial genome. The resulting library of spacers collected in CRISPR arrays is then compared with the DNA of potential invaders. One of the most intriguing and least well understood questions about CRISPR-Cas systems is the distribution of spacers across the microbial population. Here, using empirical data, we show that the global distribution of spacer numbers in CRISPR arrays across multiple biomes worldwide typically exhibits scale-invariant power law behaviour, and the standard deviation is greater than the sample mean. We develop a mathematical model of spacer loss and acquisition dynamics which fits observed data from almost four thousand metagenomes well. In analogy to the classical ‘rich-get-richer’ mechanism of power law emergence, the rate of spacer acquisition is proportional to the CRISPR array size, which allows a small proportion of CRISPRs within the population to possess a significant number of spacers. Our study provides an alternative explanation for the rarity of all-resistant super microbes in nature and why proliferation of phages can be highly successful despite the effectiveness of CRISPR-Cas systems.

59 BASIC BIOLOGICAL SCIENCES↗
DOE OSTI2016

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods. Further, in the event of a hard fault occurring, a decision has to be made as to what remedial action should be taken in order to resume the execution of the algorithm. The action that is chosen can have a dramatic effect on the performance and characteristics of the scheme. Ideally, the resulting algorithm should be subjected to the same kind of mathematical analysis that was applied to the original, deterministic variant. The purpose of this work is to provide an analysis of the behaviour of the multigrid algorithm in the presence of faults. Multigrid is arguably the method of choice for the solution of large-scale linear algebra problems arising from discretization of partial differential equations and it is of considerable importance to anticipate its behaviour on an exascale machine. The analysis of resilience of algorithms is in its infancy and the current work is perhaps the first to provide a mathematical model for faults and analyse the behaviour of a state-of-the-art algorithm under the model. It is shown that the Two Grid Method fails to be resilient to faults. Attention is then turned to identifying the minimal necessary remedial action required to restore the rate of convergence to that enjoyed by the ideal fault-free method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Is the Multigrid Method Fault Tolerant? The Two-Grid Case

The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on numerical methods. Further, in the event of a hard fault occurring, a decision has to be made as to what remedial action should be taken in order to resume the execution of the algorithm. The action that is chosen can have a dramatic effect on the performance and characteristics of the scheme. Ideally, the resulting algorithm should be subjected to the same kind of mathematical analysis that was applied to the original, deterministic variant. The purpose of this work is to provide the first rigorous analysis of the behavior of the multigrid algorithm in the presence of faults. Specifically, we prove estimates on the behavior of the Two Grid Method similar to the classical asymptotic results. Multigrid is arguably the method of choice for the solution of large-scale linear algebra problems arising from discretization of partial differential equations, and it is of considerable importance to anticipate its behavior on an exascale machine. The analysis of resilience of algorithms is in its infancy, and the current work is perhaps the first to provide a mathematical model for faults and analyze the behavior of a state-of-the-art algorithm under the model. It is shown that the Two Grid Method fails to be resilient to faults. Attention is then turned to identifying the minimal necessary remedial action required to restore the rate of convergence to that enjoyed by the ideal fault-free method.

Mathematics↗
DOE OSTI2014

SU-E-J-235: Varian Portal Dosimetry Accuracy at Detecting Simulated Delivery Errors

Purpose: To use receiver operating characteristic (ROC) analysis to quantify the Varian Portal Dosimetry (VPD) application's ability to detect delivery errors in IMRT fields. Methods: EPID and VPD were calibrated/commissioned using vendor-recommended procedures. Five clinical plans comprising 56 modulated fields were analyzed using VPD. Treatment sites were: pelvis, prostate, brain, orbit, and base of tongue. Delivery was on a Varian Trilogy linear accelerator at 6MV using a Millenium120 multi-leaf collimator. Image pairs (VPD-predicted and measured) were exported in dicom format. Each detection test imported an image pair into Matlab, optionally inserted a simulated error (rectangular region with intensity raised or lowered) into the measured image, performed 3%/3mm gamma analysis, and saved the gamma distribution. For a given error, 56 negative tests (without error) were performed, one per 56 image pairs. Also, 560 positive tests (with error) with randomly selected image pairs and randomly selected in-field error location. Images were classified as errored (or error-free) if percent pixels with γ<κ was < (or ≥) τ. (Conventionally, κ=1 and τ=90%.) A ROC curve was generated from the 616 tests by varying τ. For a range of κ and τ, true/false positive/negative rates were calculated. This procedure was repeated for inserted errors of different sizes. VPD was considered to reliably detect an error if images were correctly classified as errored or error-free at least 95% of the time, for some κ+τ combination. Results: 20mm{sup 2} errors with intensity altered by ≥20% could be reliably detected, as could 10mm{sup 2} errors with intensity was altered by ≥50%. Errors with smaller size or intensity change could not be reliably detected. Conclusion: Varian Portal Dosimetry using 3%/3mm gamma analysis is capable of reliably detecting only those fluence errors that exceed the stated sizes. Images containing smaller errors can pass mathematical analysis, though may be detected by visual inspection. This work was not funded by Varian Oncology Systems. Some authors have other work partly funded by Varian Oncology Systems.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2018

In Situ Monitoring of the Growth of Nickel, Manganese, and Cobalt Hydroxide Precursors during Co-Precipitation Synthesis of Li-Ion Cathode Materials

The electrochemical performance of cathode materials for Li-ion batteries depends on the morphology of the material, which, in turn, depends on the synthesis conditions. Very few studies have focused on the impact of process conditions on the final morphology of the cathode particles and analyzed the growth during synthesis. In this paper, the evolution of nickel, manganese, and cobalt hydroxide precursor, Ni 1/3 Mn 1/3 Co 1/3 (OH) 2 , is investigated using a combination of in situ and ex situ techniques during the commonly-used coprecipitation process. These include in situ wide angle X-ray scattering, in-situ ultra-small angle X-ray scattering and ex situ particle size analysis. The growth rate of crystalline Ni 1/3 Mn 1/3 Co 1/3 (OH) 2 primary particle is found to be almost constant, consistent with a mathematical analysis of process. The growth of the Ni 1/3 Mn 1/3 Co 1/3 (OH) 2 secondary particle and its particle size distribution revealed different growth stages for samples prepared at different pH. These techniques are complimented with scanning electron microscopy and electrochemical testing to track the morphology and performance of the hydroxide particle and the subsequent calcined LiNi 1/3 Mn 1/3 Co 1/3 O 2 cathode active material. This study presents insights into the synthesis process and provides a deeper understanding to aid in process optimization.

Cathode precursor↗
DOE OSTI2013

Topology for Statistical Modeling of Petascale Data

Many commonly used algorithms for mathematical analysis do not scale well enough to accommodate the size or complexity of petascale data produced by computational simulations. The primary goal of this project is to develop new mathematical tools that address both the petascale size and uncertain nature of current data. At a high level, the approach of the entire team involving all three institutions is based on the complementary techniques of combinatorial topology and statistical modelling. In particular, we use combinatorial topology to filter out spurious data that would otherwise skew statistical modelling techniques, and we employ advanced algorithms from algebraic statistics to efficiently find globally optimal fits to statistical models. The overall technical contributions can be divided loosely into three categories: (1) advances in the field of combinatorial topology, (2) advances in statistical modelling, and (3) new integrated topological and statistical methods. Roughly speaking, the division of labor between our 3 groups (Sandia Labs in Livermore, Texas A&M in College Station, and U Utah in Salt Lake City) is as follows: the Sandia group focuses on statistical methods and their formulation in algebraic terms, and finds the application problems (and data sets) most relevant to this project, the Texas A&M Group develops new algebraic geometry algorithms, in particular with fewnomial theory, and the Utah group develops new algorithms in computational topology via Discrete Morse Theory. However, we hasten to point out that our three groups stay in tight contact via videconference every 2 weeks, so there is much synergy of ideas between the groups. The following of this document is focused on the contributions that had grater direct involvement from the team at the University of Utah in Salt Lake City.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

A novel mathematical model for controllable near-field electrospinning

Near-field electrospinning (NFES) had better controllability than conventional electrospinning. However, due to the lack of guidance of theoretical model, precise deposition of micro/nano fibers could only accomplished by experience. To analyze the behavior of charged jet in NFES using mathematical model, the momentum balance equation was simplified and a new expression between jet cross-sectional radius and axial position was derived. Using this new expression and mass conservation equation, expressions for jet cross-sectional radius and velocity were derived in terms of axial position and initial jet acceleration in the form of exponential functions. Based on Slender-body theory and Giesekus model, a quadratic equation for initial jet acceleration was acquired. With the proposed model, it was able to accurately predict the diameter and velocity of polymer fibers in NFES, and mathematical analysis rather than experimental methods could be applied to study the effects of the process parameters in NFES. Moreover, the movement velocity of the collector stage can be regulated by mathematical model rather than experience. Therefore, the model proposed in this paper had important guiding significance to precise deposition of polymer fibers.

36 MATERIALS SCIENCE↗
DOE OSTI2016

A Theoretical Framework for Calibration in Computer Models: Parametrization, Estimation and Convergence Properties

Calibration parameters in deterministic computer experiments are those attributes that cannot be measured or available in physical experiments. Here, an approach to estimate them by using data from physical experiments and computer simulations. A theoretical framework is given which allows us to study the issues of parameter identifiability and estimation. We define the L 2 -consistency for calibration as a justification for calibration methods. It is shown that a simplified version of the original KO method leads to asymptotically L 2 -inconsistent calibration. This L 2 -inconsistency can be remedied by modifying the original estimation procedure. A novel calibration method, called the L 2 calibration, is proposed and proven to be L 2 -consistent and enjoys optimal convergence rate. Furthermore a numerical example and some mathematical analysis are used to illustrate the source of the L 2 -inconsistency problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Reliable and Efficient Machine Learning (Final Technical Report)

Modern scientific experiments generate massive amounts of data at a pace much faster than humans can manually analyze. While machine learning has revolutionized commercial data analysis (such as recommending movies or recognizing faces), applying these tools to complex scientific discovery is challenging because scientific answers must be precise, interpretable, and adhere to physical laws. The research under this project aims to develop new mathematical tools and computer algorithms specifically designed for scientific applications. Major progress has been made in automatically cleaning and deconstructing messy experimental data, analyzing the visual information of physical phenomena, determining the underlying physical variables, and providing rig orous mathematical analysis of interesting algorithms and concepts widely used in machine learning. This project addressed the critical gap between our ability to generate massive scientific data and our ability to extract interpretable information from it. We established mathematical foundations for Scientific Machine Learning (SciML) aimed at effective data analytics and automated discovery. Our work focused on three core objectives: (1) developing reliable feature extraction methods for dynamic high-dimensional data, (2) establishing mathematical foundations for discovering dynamics via neural networks, and (3) creating rigorous optimization techniques for these models. Key outcomes come from two fronts. On the practical side, they include the development of algorithms that significantly enhance the extraction of signals from field data, as well as the capability to handle situations that exhibit smooth variations or physical stretching due to temperature changes. They also include the creation of an automated framework for discovering fundamental state variables from raw experimental data, demonstrating the ability to identify intrinsic physical dimensions without prior knowledge of the governing laws. On the theoretical front, the research results in theoretical advances in Optimal Transport, a widely used notion in SciML, specifically regarding functions with fixed-size nodal sets, provide sharp bounds relevant to uncertainty quantification. Meanwhile, the outcomes also include the establishment of convergence theories for nonlocal gradient descent methods, enabling robust optimization with noisy data in high-dimensional settings commonly encountered in scientific modeling. The project also helps creating opportunities to train the next generation of researchers, equipping them with the necessary technical skills for today’s workplace and preparing them for future advances.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

The DPG Method for the Convection-Reaction Problem Revisited

We study both conforming and non-conforming versions of the practical DPG method for the convection-reaction problem. We determine that the most common approach for DPG stability analysis (construction of a local Fortin operator) is infeasible for the convection-reaction problem. We then develop a line of argument based on the direct construction of a global Fortin operator; we find that employing a polynomial enrichment for the test space does not suffice for this purpose, motivating the introduction of a (two-element) subgrid mesh. The argument combines mathematical analysis with numerical experiments

42 ENGINEERING↗
DOE OSTI2022

The DPG Method for the Convection-Reaction Problem, Revisited

In this work, we study both conforming and non-conforming versions of the practical DPG method for the convection-reaction problem. We determine that the most common approach for DPG stability analysis – construction of a local Fortin operator – is infeasible for the convection-reaction problem. We then develop a line of argument based on a direct proof of discrete stability; we find that employing a polynomial enrichment for the test space does not suffice for this purpose, motivating the introduction of a (two-element) subgrid mesh. The argument combines mathematical analysis with numerical experiments.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Towards sharp error analysis of extended Lagrangian molecular dynamics

The extended Lagrangian molecular dynamics (XLMD) method provides a useful framework for reducing the computational cost of a class of molecular dynamics simulations with constrained latent variables. The XLMD method relaxes the constraints by introducing a fictitious mass ε for the latent variables and solving a set of singularly perturbed ordinary differential equations. While favorable numerical performance of XLMD has been demonstrated in several different contexts in the past decade, mathematical analysis of the method remains scarce. Here, we propose the first error analysis of the XLMD method in the context of a classical polarizable force field model. While the dynamics with respect to the atomic degrees of freedom are general and nonlinear, the key mathematical simplification of the polarizable force field model is that the constraints on the latent variables are given by a linear system of equations. We prove that when the initial value of the latent variables is compatible in a sense that we define, XLMD converges as the fictitious mass ε is made small with $\mathscr{O}$(ε) error for the atomic degrees of freedom and with $\mathscr{O}$($\sqrt{ε}$) error for the latent variables, when the dimension of the latent variable d' is 1. Furthermore, when the initial value of the latent variables is improved to be optimally compatible in a certain sense, we prove that the convergence rate can be improved to $\mathscr{O}$(ε) for the latent variables as well. Numerical results verify that both estimates are sharp not only for d'=1, but also for arbitrary d'. In the setting of general d', we do obtain convergence, but with the non-sharp rate of $\mathscr{O}$($\sqrt{ε}$) for both the atomic and latent variables.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2014

AUTOMATED, HIGHLY ACCURATE VERIFICATION OF RELAP5-3D

Computer programs that analyze light water reactor safety solve complex systems of governing, closure and special process equations to model the underlying physics. In addition, these programs incorporate many other features and are quite large. RELAP5-3D[1] has over 300,000 lines of coding for physics, input, output, data management, user-interaction, and post-processing. For software quality assurance, the code must be verified and validated before being released to users. Verification ensures that a program is built right by checking that it meets its design specifications. Recently, there has been an increased importance on the development of automated verification processes that compare coding against its documented algorithms and equations and compares its calculations against analytical solutions and the method of manufactured solutions[2]. For the first time, the ability exists to ensure that the data transfer operations associated with timestep advancement/repeating and writing/reading a solution to a file have no unintended consequences. To ensure that the code performs as intended over its extensive list of applications, an automated and highly accurate verification method has been modified and applied to RELAP5-3D. Furthermore, mathematical analysis of the adequacy of the checks used in the comparisons is provided.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2025

Advantages of imperfect dice rolls over coin flips for random number generation

With an eye toward neural-inspired probabilistic computation, recent work has examined the development of true random number generators via stochastic devices. Typically, these devices are operated in a two-state regime to produce a sequence of binary outcomes (i.e., coin flips). However, there is no guarantee that stochastic devices will infallibly produce fair outputs and small deviations from a uniform distribution may have unwanted complications in applications. Using mathematical analysis, we contend that opting instead for a multi-state device (i.e., a dice roll) has benefits in these unfair paradigms. To demonstrate these benefits, we apply this framework to the analysis of a tunnel diode operated in a stochastic regime. In particular, interpreting the binary stochastic output of the tunnel diode as a multi-state die roll output also sees advantages in remaining closer to uniform. Overall, our approach provides a compelling argument for mathematical driven co-design and development of novel probabilistic computing devices and hardware.

applied mathematics↗
DOE OSTI2018

Analysis of Experimental Results by the Havriliak–Negami Model in Dielectric Spectroscopy

The problem of identifying experimental results on studying frequency dependences of the real and imaginary parts of dielectric permittivity concerning the correspondence to one of accepted frequency dispersion models: Debye, Cole–Cole, Davidson–Cole, and Havriliak–Negami models is considered. Based on expressions for components of complex dielectric permittivity with the use of mathematical analysis, a sequence of steps for the determination of the following characteristic parameters of the generalized Havriliak–Negami model is developed: static ε{sub S} and high-frequency ε{sub ∞} dielectric permittivity, frequency dispersion indices α and β, and relaxation time τ. As an example, the parameters ε{sub S}, ε{sub ∞}, α, β, and τ are calculated for a sample of a frozen disperse medium based on fine-grained quartz powder at a humidity of 13% in a temperature range from –140 to 0°C.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗