We consider a population dynamics model in investigating data from controlled experiments with aphids in broccoli patches surrounded by different margin types (bare or weedy ground) and three levels of insecticide spray (no, light, or heavy spray). The experimental data is clearly aggregate in nature. In previous efforts [1], the aggregate nature of the data was ignored. In this paper, we embrace this aspect of the experiment and correctly model the data as aggregate data, comparing the results to the previous approach. We discuss cases in which the approach may provide similar results as well as cases in which there is a clear difference in the resulting fit to the data.

A dense plasma focus (DPF) nuclear fusion device is an attractive pulsed neutron source in many applications, due to its relatively large neutron yield, produced in a short time duration. To design a DPF that generates neutrons within a specified time profile, or that generates a neutron energy spectrum with specific properties, it is necessary to characterize and model the results of the fusion process releasing the neutrons. The time–energy spectrum of fusion neutrons is an ideal quantity used in validating multiphysics codes that simulate the pinch and fusion processes, because it is a quantity that requires physics of each of the stages leading up to and ending in the fusion reaction to be simulated correctly. In particular, since DPF fusion neutrons are not monoenergetic—and there can often be several fusion pinches creating neutrons—a computer simulation matching high quality neutron spectrum measurements provides confidence in the fidelity of the simulation. In order to make such a comparison, it is first necessary to have quality measurements from which to infer the spectrum. In this work, we project neutron spectroscopy as the classical tomographic inverse problem from neutron time-of-flight data at multiple distances, but enhanced by using an additional measurement of the time profile of the fusion pinch near its source and by using detector pairs set up in a geometry that allows for scatter background subtraction. The detector pairs enhance the quality of the time-of-flight measurements, and the additional constraint posed by the measured time profile allows for reconstructions discretized as finely as the time measurements and in energy as finely as 100 keV, without the problem being underdetermined. We present results from a deuterium-fueled DPF at the U.S. Department of Energy’s Nevada National Security Site and show that we can infer the time–energy spectrum from our measurements for both single and multi-pinch fusion reactions with equal fidelity.

In biomedical/physiological/ecological experiments, it is common for measurements in time series data to be collected from multiple subjects. Often it is the case that a subject cannot be measured or identified at multiple time points (often referred to as aggregate population data). Due to a lack of alternative methods, this form of data is typically treated as if it is collected from a single individual. As we show by examples, this assumption leads to an overconfidence in model parameter (means, variances) values and model based predictions. We discuss these issues in the context of a mathematical model to determine T-cell behavior with cancer chimeric antigen receptor (CAR) therapies where during the collection of data cancerous mice are sacrificed at each measurement time.

Broadband Laser Ranging (BLR) is a recently developed instrument for measuring the position of rapidly moving surfaces. Two gas gun experiments were performed to characterize a 16-channel Broadband Laser Ranging system. The precision of the measured position data exceeded expectations and has revealed systematic errors of up to 12 microns. These errors appear to be caused by the interaction of the data analysis routines with the recording hardware's frequency-dependent signal response.

We consider a population dynamics model in investigating data from controlled experiments with aphids in broccoli patches surrounded by different margin types (bare or weedy ground) and three levels of insecticide spray (no, light, or heavy spray). The experimental data is clearly aggregate in nature. We compare two approaches, one of which ignores this aggregate nature and one which embraces this aspect of the experiments, to carry out parameter estimation computations along with statistical analysis to compare the two approaches using several model dynamics.

We consider models and methods for the determination of reflective properties that describes the microscopic properties of complex composite materials. Specifically we compare methods for the determination of reflectivity of complex composite materials including the methods based on Efimov’s model for permittivity, homogenization approximating techniques and the Prohorov Metric Framework in which one determines a probability distribution for the permittivity. Our ultimate goal is to be able to ascertain material changes (e. g., oxidation) when the materials are subjected to high stress and high temperatures.

This report compiles multiple articles describing activities developed and performed under the Site-Directed Research and Development Program for the benefit the Nation during Fiscal Year 2018. Sustained investment and ongoing core innovation are the keys to successful research and development programs. These elements drive the Site-Directed Research and Development (SDRD) Program and provide solutions to some of the most challenging problems our nation and our allies face. Our Nevada enterprise, consisting of both our management and operating entity and our NNSA field office, is aggressively investing in SDRD and injecting innovation through strategic partnerships. Our partnerships with universities, industry, and our sister institutions within the National Nuclear Security Administration (NNSA) build upon our core capabilities and allow us to create innovative solutions to support our mission requirements. In 2018, we raised the investment level of SDRD for only the second time in the history of the program. Nearly at our congressionally authorized limit, SDRD has substantial resources to successfully address numerous issues. Investment is only one part of the equation; innovation is generated through collaborations that bring discovery and provide the “technical differentiation” and the return on investment we seek. This report demonstrates an enduring theme of how partnerships help us drive the best outcomes and provide the maximum impact possible, while using our resources efficiently.

Broadband laser ranging (BLR) is essentially a spectral interferometer used to infer distance to a moving target. The light source is a mode-locked fiber laser, and chromatic dispersion maps the spectral interference pattern into the time domain, yielding chirped beat signals at the detector. A BLR record is a sequence of these chirped signals, representing consecutive target positions. To infer distance to a target, each underlying pulse envelope must be consistently registered and subtracted despite environmentally-induced variability. Then, nonlinear transformation of the phase is applied to remove the chirp, an FFT is performed to determine the peak frequency of the de-chirped signal, and a calibration factor relating de-chirped frequency to distance results in target position. Here, these analysis steps are discussed in detail.

In our previous work, a reduced order model (ROM) for a stochastic system was made, where noisy data was projected onto principal component analysis (PCA)-derived basis vectors to obtain an accurate reconstruction of the noise-free data. That work used techniques designed for deterministic data, PCA was used for the basis function generation and $L_2$ projection was used to create the reconstructions. In this work, probabilistic approaches are used. The probabilistic PCA (PPCA) is used to generate the basis, which then allows the noise in the training data to be estimated. PPCA has also been improved so that the derived basis vectors are orthonormal and the variance of the basis expansion coefficients over the training data set can be estimated. The standard approach assumes a unit variance for these coefficients. Based on the results of the PPCA, model selection criteria are applied to automatically choose the dimension of the ROM. In our previous work, a heuristic approach was used to pick the dimension. Lastly, a new statistical approach is used for the projection step where the variance information obtained from the improved PPCA is used as a prior to improve the projection. This gives improved accuracy over $L_2$ projection when the projected data is noisy. In addition, the noise statistics for the projected data are not assumed to be the same as that of the training data, but are estimated in the projection process. The entire approach gives a fully stochastic method for computing a ROM from noisy training data, determining ideal model selection, and projecting noisy test data, thus enabling accurate predictions of noise-free data from data that is dominated by noise.

In physiological experiments, it is common for measurements to be collected from multiple subjects. Often it is the case that a subject cannot be measured or identified at multiple time points (referred to as unidentified individual data in this work but often referred to as aggregate population data [5, Chapter 5]). Due to a lack of alternative methods, this form of data is typically treated as if it is collected from a single individual. This assumption leads to an overconfidence in model parameter values and model based predictions. We propose a novel method which accounts for inter-individual variability in experiments where only unidentified individual data is available. Both parametric and nonparametric methods for estimating the distribution of parameters which vary among individuals are developed. These methods are illustrated using both simulated data, and data taken from a physiological experiment. Taking the approach outlined in this paper results in more accurate quantification of the uncertainty attributed to inter-individual variability.

We discuss efficient methods for computing gradients in inverse problems for estimation of distributions for individual parameters in models where only aggregate or population level data is available. The ideas are illustrated with two examples arising in applications.

Normalized differences of several adjacent observations, referred to as pseudo-measurement errors in this paper, are used in so-called difference-based estimation methods as building blocks for the variance estimate of measurement errors. Numerical results demonstrate that pseudo-measurement errors can be used to serve the role of measurement errors. Based on this information, we propose the use of pseudo-measurement errors to determine an appropriate statistical model and then to subsequently investigate whether there is a mathematical model misspecification or error. We also propose to use the information provided by pseudo-measurement errors to quantify uncertainty in parameter estimation by bootstrapping methods. A number of numerical examples are given to illustrate the effectiveness of these proposed methods.

Reflectance spectroscopy obtained from thermally treated silicon nitride carbon based ceramic matrix composites is used to quantity the oxidation products SiO2 and SiN. The data collection is described in detail in order to point out the potential biasing present in the data processing. A probability distribution is imposed on selected model parameters, and then non-parametrically estimated. A non-parametric estimation is chosen since the exact composition of the material is unknown due to the inherent heterogeneity of ceramic composites. The probability distribution is estimated using the Prohorov metric framework in which the infinite dimensional optimization is reduced to a finite dimensional optimization using an approximating space composed of linear splines. A weighted least squares estimation is carried out, and uncertainty quantification is performed on the model parameters, including a piecewise asymptotic confidence band for the estimated probability density. Our estimation results indicate a distinguishable increase in the SiO2 present in the samples which were heat treated for 100 hours compared to those treated for 10 hours.

We analyze a quasi-chemical model for bacterial growth in the context of a parameter estimation problem using available experimental data sets. For many of the data sets the dynamics are simple and we show that the quasi-chemical model (QCM) is over parameterized in these cases. This results in larger uncertainty in the estimated parameters and is some cases instability in the inverse problem. We illustrate methods for reducing the uncertainty present in the estimated model parameters. We first consider a model reduction technique where subsets of the QCM parameters are fixed at nominal values and hypothesis testing is used to compare the nested models. An orthogonal decomposition of the sensitivity matrix is used to guide the choice of which parameters are fixed. Additionally, surrogate models are developed to compare to the QCM. In most cases one of the surrogate models is able to fit the data nearly as well as the QCM model while reducing the uncertainty in the model parameters.

We consider probability measure estimation in a nonparametric model using a least-squares approach under the Prohorov metric framework. We summarize the computational methods and related convergence results that were recently developed by our group. New results are presented on the bias and the variance due to the approximation and new pointwise asymptotic normality of the approximated probability measure estimator. We propose the use of a model selection criterion to balance the bias and the variance, and compare the new pointwise confidence bands constructed using the asymptotic normality results with those obtained by Monte Carlo simulations.