arXiv subjects

David Higdon

Publications and source records attributed to David Higdon.

At least 19 recordsLinked to original sources

arXiv2026

Rational approximation and intrinsic Gaussian processes

Gaussian processes (GPs) defined through intrinsic random fields provide a flexible framework for modeling spatial phenomena, and have been advocated in a variety of applications over the past several decades. Nevertheless, their adoption has lagged behind traditional, covariance-based approaches, in part because the intrinsic formulation has lacked an accompanying toolkit of computational methods and dependence specifications that facilitate fitting and prediction. We develop here a systematic framework for modeling intrinsic GPs and introduce practical algorithms and dependence/variogram models for modeling, inference and computation that parallel those of traditional, stationary GPs. We explore a close connection between intrinsic GP models and rational approximation, which clarifies the underlying problem structure. Numerical examples illustrate how the new tools can be deployed in practice, highlighting the advantages of intrinsic-field modeling in terms of robustness, interpretability, and computational efficiency.

math.NA↗

arXiv2025

Bayesian Deep Gaussian Processes for Correlated Functional Data: A Case Study in Cosmological Matter Power Spectra

Understanding the structure of our universe and the distribution of matter is an area of active research. As cosmological surveys grow in complexity, the development of emulators to efficiently and effectively predict matter power spectra is essential. We are particularly motivated by the Mira-Titan Universe simulation suite that, for a specified cosmological parameterization (termed a "cosmology"), provides multiple response curves of various fidelities, including correlated functional realizations. Our objective is two-fold. First, we estimate the underlying matter power spectra, with appropriate uncertainty quantification (UQ), from all of the provided curves. To this end, we propose a novel Bayesian deep Gaussian process (DGP) hierarchical model which synthesizes all the simulation information to estimate the underlying matter power spectra while providing effective UQ. Our model extends previous work on Bayesian DGPs from scalar responses to correlated functional outputs. Second, we leverage our predicted power spectra from various cosmologies in order to accurately predict the entire matter power spectra for an unobserved cosmology. For this task, we use basis function representations of the functional spectra to train a separate Gaussian process emulator. Our method performs well in synthetic exercises and against the benchmark cosmological emulator (Cosmic Emu).

stat.AP↗

arXiv2024

Bayesian "Deep" Process Convolutions: An Application in Cosmology

The nonlinear matter power spectrum in cosmology describes how matter density fluctuations vary with scale in the universe, providing critical insights into large-scale structure formation. The matter power spectrum includes both smooth regions and highly oscillatory features. Cosmologists rely on noisy, multi-resolution realizations of large N-body simulations to study these phenomena, which require appropriate smoothing techniques to learn about underlying structures. We introduce a Bayesian Deep Process Convolution (DPC) model that flexibly adapts its smoothness parameter across the input space, enabling it to capture both smooth and variable structure within a single framework. The DPC model leverages common patterns across related functions to improve estimation in regions with sparse data. Compared to existing methods, the DPC model offers superior accuracy and uncertainty quantification in simulated data, and qualitatively superior performance with the cosmological data. This methodology will be useful in cosmology and other fields requiring flexible modeling of smooth nonstationary surfaces.

astro-ph.CO↗

arXiv2024

Towards Improved Uncertainty Quantification of Stochastic Epidemic Models Using Sequential Monte Carlo

Sequential Monte Carlo (SMC) algorithms represent a suite of robust computational methodologies utilized for state estimation and parameter inference within dynamical systems, particularly in real-time or online environments where data arrives sequentially over time. In this research endeavor, we propose an integrated framework that combines a stochastic epidemic simulator with a sequential importance sampling (SIS) scheme to dynamically infer model parameters, which evolve due to social as well as biological processes throughout the progression of an epidemic outbreak and are also influenced by evolving data measurement bias. Through iterative updates of a set of weighted simulated trajectories based on observed data, this framework enables the estimation of posterior distributions for these parameters, thereby capturing their temporal variability and associated uncertainties. Through simulation studies, we showcase the efficacy of SMC in accurately tracking the evolving dynamics of epidemics while appropriately accounting for uncertainties. Moreover, we delve into practical considerations and challenges inherent in implementing SMC for parameter estimation within dynamic epidemiological settings, areas where the substantial computational capabilities of high-performance computing resources can be usefully brought to bear.

stat.AP↗

arXiv2022

The Mira-Titan Universe IV. High Precision Power Spectrum Emulation

Modern cosmological surveys are delivering datasets characterized by unprecedented quality and statistical completeness; this trend is expected to continue into the future as new ground- and space-based surveys come online. In order to maximally extract cosmological information from these observations, matching theoretical predictions are needed. At low redshifts, the surveys probe the nonlinear regime of structure formation where cosmological simulations are the primary means of obtaining the required information. The computational cost of sufficiently resolved large-volume simulations makes it prohibitive to run very large ensembles. Nevertheless, precision emulators built on a tractable number of high-quality simulations can be used to build very fast prediction schemes to enable a variety of cosmological inference studies. We have recently introduced the Mira-Titan Universe simulation suite designed to construct emulators for a range of cosmological probes. The suite covers the standard six cosmological parameters $\{\omega_m,\omega_b, \sigma_8, h, n_s, w_0\}$ and, in addition, includes massive neutrinos and a dynamical dark energy equation of state, $\{\omega_{\nu}, w_a\}$. In this paper we present the final emulator for the matter power spectrum based on 111 cosmological simulations, each covering a (2.1Gpc)$^3$ volume and evolving 3200$^3$ particles. An additional set of 1776 lower-resolution simulations and TimeRG perturbation theory results for the power spectrum are used to cover scales straddling the linear to mildly nonlinear regimes. The emulator provides predictions at the two to three percent level of accuracy over a wide range of cosmological parameters and is publicly released as part of this paper.

astro-ph.CO↗

arXiv2020

Active Learning for Deep Gaussian Process Surrogates

Deep Gaussian processes (DGPs) are increasingly popular as predictive models in machine learning (ML) for their non-stationary flexibility and ability to cope with abrupt regime changes in training data. Here we explore DGPs as surrogates for computer simulation experiments whose response surfaces exhibit similar characteristics. In particular, we transport a DGP's automatic warping of the input space and full uncertainty quantification (UQ), via a novel elliptical slice sampling (ESS) Bayesian posterior inferential scheme, through to active learning (AL) strategies that distribute runs non-uniformly in the input space -- something an ordinary (stationary) GP could not do. Building up the design sequentially in this way allows smaller training sets, limiting both expensive evaluation of the simulator code and mitigating cubic costs of DGP inference. When training data sizes are kept small through careful acquisition, and with parsimonious layout of latent layers, the framework can be both effective and computationally tractable. Our methods are illustrated on simulation data and two real computer experiments of varying input dimensionality. We provide an open source implementation in the "deepgp" package on CRAN.

stat.ME↗

arXiv2020

Analyzing Stochastic Computer Models: A Review with Opportunities

In modern science, computer models are often used to understand complex phenomena, and a thriving statistical community has grown around analyzing them. This review aims to bring a spotlight to the growing prevalence of stochastic computer models -- providing a catalogue of statistical methods for practitioners, an introductory view for statisticians (whether familiar with deterministic computer models or not), and an emphasis on open questions of relevance to practitioners and statisticians. Gaussian process surrogate models take center stage in this review, and these, along with several extensions needed for stochastic settings, are explained. The basic issues of designing a stochastic computer experiment and calibrating a stochastic computer model are prominent in the discussion. Instructive examples, with data and code, are used to describe the implementation of, and results from, various methods.

stat.ME↗

arXiv2017

The Mira-Titan Universe II: Matter Power Spectrum Emulation

We introduce a new cosmic emulator for the matter power spectrum covering eight cosmological parameters. Targeted at optical surveys, the emulator provides accurate predictions out to a wavenumber k~5/Mpc and redshift z<=2. Besides covering the standard set of LCDM parameters, massive neutrinos and a dynamical dark energy of state are included. The emulator is built on a sample set of 36 cosmological models, carefully chosen to provide accurate predictions over the wide and large parameter space. For each model, we have performed a high-resolution simulation, augmented with sixteen medium-resolution simulations and TimeRG perturbation theory results to provide accurate coverage of a wide k-range; the dataset generated as part of this project is more than 1.2Pbyte. With the current set of simulated models, we achieve an accuracy of approximately 4%. Because the sampling approach used here has established convergence and error-control properties, follow-on results with more than a hundred cosmological models will soon achieve ~1% accuracy. We compare our approach with other prediction schemes that are based on halo model ideas and remapping approaches. The new emulator code is publicly available.

astro-ph.CO↗

arXiv2015

The Mira-Titan Universe: Precision Predictions for Dark Energy Surveys

Ground and space-based sky surveys enable powerful cosmological probes based on measurements of galaxy properties and the distribution of galaxies in the Universe. These probes include weak lensing, baryon acoustic oscillations, abundance of galaxy clusters, and redshift space distortions; they are essential to improving our knowledge of the nature of dark energy. On the theory and modeling front, large-scale simulations of cosmic structure formation play an important role in interpreting the observations and in the challenging task of extracting cosmological physics at the needed precision. These simulations must cover a parameter range beyond the standard six cosmological parameters and need to be run at high mass and force resolution. One key simulation-based task is the generation of accurate theoretical predictions for observables, via the method of emulation. Using a new sampling technique, we explore an 8-dimensional parameter space including massive neutrinos and a variable dark energy equation of state. We construct trial emulators using two surrogate models (the linear power spectrum and an approximate halo mass function). The new sampling method allows us to build precision emulators from just 26 cosmological models and to increase the emulator accuracy by adding new sets of simulations in a prescribed way. This allows emulator fidelity to be systematically improved as new observational data becomes available and higher accuracy is required. Finally, using one LCDM cosmology as an example, we study the demands imposed on a simulation campaign to achieve the required statistics and accuracy when building emulators for dark energy investigations.

astro-ph.CO↗

arXiv2013

The Coyote Universe III: Simulation Suite and Precision Emulator for the Nonlinear Matter Power Spectrum

Many of the most exciting questions in astrophysics and cosmology, including the majority of observational probes of dark energy, rely on an understanding of the nonlinear regime of structure formation. In order to fully exploit the information available from this regime and to extract cosmological constraints, accurate theoretical predictions are needed. Currently such predictions can only be obtained from costly, precision numerical simulations. This paper is the third in a series aimed at constructing an accurate calibration of the nonlinear mass power spectrum on Mpc scales for a wide range of currently viable cosmological models, including dark energy. The first two papers addressed the numerical challenges, and the scheme by which an interpolator was built from a carefully chosen set of cosmological models. In this paper we introduce the "Coyote Univers"' simulation suite which comprises nearly 1,000 N-body simulations at different force and mass resolutions, spanning 38 wCDM cosmologies. This large simulation suite enables us to construct a prediction scheme, or emulator, for the nonlinear matter power spectrum accurate at the percent level out to k~1 h/Mpc. We describe the construction of the emulator, explain the tests performed to ensure its accuracy, and discuss how the central ideas may be extended to a wider range of cosmological models and applications. A power spectrum emulator code is released publicly as part of this paper.

astro-ph.CO↗

arXiv2013

The Coyote Universe Extended: Precision Emulation of the Matter Power Spectrum

Modern sky surveys are returning precision measurements of cosmological statistics such as weak lensing shear correlations, the distribution of galaxies, and cluster abundance. To fully exploit these observations, theorists must provide predictions that are at least as accurate as the measurements, as well as robust estimates of systematic errors that are inherent to the modeling process. In the nonlinear regime of structure formation, this challenge can only be overcome by developing a large-scale, multi-physics simulation capability covering a range of cosmological models and astrophysical processes. As a first step to achieving this goal, we have recently developed a prediction scheme for the matter power spectrum (a so-called emulator), accurate at the 1% level out to k~1/Mpc and z=1 for wCDM cosmologies based on a set of high-accuracy N-body simulations. It is highly desirable to increase the range in both redshift and wavenumber and to extend the reach in cosmological parameter space. To make progress in this direction, while minimizing computational cost, we present a strategy that maximally re-uses the original simulations. We demonstrate improvement over the original spatial dynamic range by an order of magnitude, reaching k~10 h/Mpc, a four-fold increase in redshift coverage, to z=4, and now include the Hubble parameter as a new independent variable. To further the range in k and z, a new set of nested simulations run at modest cost is added to the original set. The extension in h is performed by including perturbation theory results within a multi-scale procedure for building the emulator. This economical methodology still gives excellent error control, ~5% near the edges of the domain of applicability of the emulator. A public domain code for the new emulator is released as part of the work presented in this paper.

astro-ph.CO↗

arXiv2011

Nonparametric Reconstruction of the Dark Energy Equation of State from Diverse Data Sets

The cause of the accelerated expansion of the Universe poses one of the most fundamental questions in physics today. In the absence of a compelling theory to explain the observations, a first task is to develop a robust phenomenology. If the acceleration is driven by some form of dark energy, then, the phenomenology is determined by the dark energy equation of state w. A major aim of ongoing and upcoming cosmological surveys is to measure w and its time dependence at high accuracy. Since w(z) is not directly accessible to measurement, powerful reconstruction methods are needed to extract it reliably from observations. We have recently introduced a new reconstruction method for w(z) based on Gaussian process modeling. This method can capture nontrivial time-dependences in w(z) and, most importantly, it yields controlled and unbaised error estimates. In this paper we extend the method to include a diverse set of measurements: baryon acoustic oscillations, cosmic microwave background measurements, and supernova data. We analyze currently available data sets and present the resulting constraints on w(z), finding that current observations are in very good agreement with a cosmological constant. In addition we explore how well our method captures nontrivial behavior of w(z) by analyzing simulated data assuming high-quality observations from future surveys. We find that the baryon acoustic oscillation measurements by themselves already lead to remarkably good reconstruction results and that the combination of different high-quality probes allows us to reconstruct w(z) very reliably with small error bounds.

astro-ph.CO↗

arXiv2010

Nonparametric Dark Energy Reconstruction from Supernova Data

Understanding the origin of the accelerated expansion of the Universe poses one of the greatest challenges in physics today. Lacking a compelling fundamental theory to test, observational efforts are targeted at a better characterization of the underlying cause. If a new form of mass-energy, dark energy, is driving the acceleration, the redshift evolution of the equation of state parameter w(z) will hold essential clues as to its origin. To best exploit data from observations it is necessary to develop a robust and accurate reconstruction approach, with controlled errors, for w(z). We introduce a new, nonparametric method for solving the associated statistical inverse problem based on Gaussian Process modeling and Markov chain Monte Carlo sampling. Applying this method to recent supernova measurements, we reconstruct the continuous history of w out to redshift z=1.5.

astro-ph.CO↗

arXiv2010

Nonparametric Reconstruction of the Dark Energy Equation of State

A basic aim of ongoing and upcoming cosmological surveys is to unravel the mystery of dark energy. In the absence of a compelling theory to test, a natural approach is to better characterize the properties of dark energy in search of clues that can lead to a more fundamental understanding. One way to view this characterization is the improved determination of the redshift-dependence of the dark energy equation of state parameter, w(z). To do this requires a robust and bias-free method for reconstructing w(z) from data that does not rely on restrictive expansion schemes or assumed functional forms for w(z). We present a new nonparametric reconstruction method that solves for w(z) as a statistical inverse problem, based on a Gaussian Process representation. This method reliably captures nontrivial behavior of w(z) and provides controlled error bounds. We demonstrate the power of the method on different sets of simulated supernova data; the approach can be easily extended to include diverse cosmological probes.

astro-ph.CO↗

arXiv2010

The Coyote Universe II: Cosmological Models and Precision Emulation of the Nonlinear Matter Power Spectrum

The power spectrum of density fluctuations is a foundational source of cosmological information. Precision cosmological probes targeted primarily at investigations of dark energy require accurate theoretical determinations of the power spectrum in the nonlinear regime. To exploit the observational power of future cosmological surveys, accuracy demands on the theory are at the one percent level or better. Numerical simulations are currently the only way to produce sufficiently error-controlled predictions for the power spectrum. The very high computational cost of (precision) N-body simulations is a major obstacle to obtaining predictions in the nonlinear regime, while scanning over cosmological parameters. Near-future observations, however, are likely to provide a meaningful constraint only on constant dark energy equation of state 'wCDM' cosmologies. In this paper we demonstrate that a limited set of only 37 cosmological models -- the "Coyote Universe" suite -- can be used to predict the nonlinear matter power spectrum at the required accuracy over a prior parameter range set by cosmic microwave background observations. This paper is the second in a series of three, with the final aim to provide a high-accuracy prediction scheme for the nonlinear matter power spectrum for wCDM cosmologies.

astro-ph.CO↗

arXiv2010

The Coyote Universe I: Precision Determination of the Nonlinear Matter Power Spectrum

Near-future cosmological observations targeted at investigations of dark energy pose stringent requirements on the accuracy of theoretical predictions for the clustering of matter. Currently, N-body simulations comprise the only viable approach to this problem. In this paper we demonstrate that N-body simulations can indeed be sufficiently controlled to fulfill these requirements for the needs of ongoing and near-future weak lensing surveys. By performing a large suite of cosmological simulation comparison and convergence tests we show that results for the nonlinear matter power spectrum can be obtained at 1% accuracy out to k~1 h/Mpc. The key components of these high accuracy simulations are: precise initial conditions, very large simulation volumes, sufficient mass resolution, and accurate time stepping. This paper is the first in a series of three, with the final aim to provide a high-accuracy prediction scheme for the nonlinear matter power spectrum.

astro-ph↗

arXiv2008

Simulations and cosmological inference: A statistical model for power spectra means and covariances

We describe an approximate statistical model for the sample variance distribution of the non-linear matter power spectrum that can be calibrated from limited numbers of simulations. Our model retains the common assumption of a multivariate Normal distribution for the power spectrum band powers, but takes full account of the (parameter dependent) power spectrum covariance. The model is calibrated using an extension of the framework in Habib et al. (2007) to train Gaussian processes for the power spectrum mean and covariance given a set of simulation runs over a hypercube in parameter space. We demonstrate the performance of this machinery by estimating the parameters of a power-law model for the power spectrum. Within this framework, our calibrated sample variance distribution is robust to errors in the estimated covariance and shows rapid convergence of the posterior parameter constraints with the number of training simulations.

astro-ph↗

arXiv2007

Cosmic Calibration: Constraints from the Matter Power Spectrum and the Cosmic Microwave Background

Several cosmological measurements have attained significant levels of maturity and accuracy over the last decade. Continuing this trend, future observations promise measurements of the statistics of the cosmic mass distribution at an accuracy level of one percent out to spatial scales with k~10 h/Mpc and even smaller, entering highly nonlinear regimes of gravitational instability. In order to interpret these observations and extract useful cosmological information from them, such as the equation of state of dark energy, very costly high precision, multi-physics simulations must be performed. We have recently implemented a new statistical framework with the aim of obtaining accurate parameter constraints from combining observations with a limited number of simulations. The key idea is the replacement of the full simulator by a fast emulator with controlled error bounds. In this paper, we provide a detailed description of the methodology and extend the framework to include joint analysis of cosmic microwave background and large scale structure measurements. Our framework is especially well-suited for upcoming large scale structure probes of dark energy such as baryon acoustic oscillations and, especially, weak lensing, where percent level accuracy on nonlinear scales is needed.

astro-ph↗