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NASA NTRS2008

Black Hole Mergers as Probes of Structure Formation

Intense structure formation and reionization occur at high redshift, yet there is currently little observational information about this very important epoch. Observations of gravitational waves from massive black hole (MBH) mergers can provide us with important clues about the formation of structures in the early universe. Past efforts have been limited to calculating merger rates using different models in which many assumptions are made about the specific values of physical parameters of the mergers, resulting in merger rate estimates that span a very wide range (0.1 - 104 mergers/year). Here we develop a semi-analytical, phenomenological model of MBH mergers that includes plausible combinations of several physical parameters, which we then turn around to determine how well observations with the Laser Interferometer Space Antenna (LISA) will be able to enhance our understanding of the universe during the critical z 5 - 30 structure formation era. We do this by generating synthetic LISA observable data (total BH mass, BH mass ratio, redshift, merger rates), which are then analyzed using a Markov Chain Monte Carlo method. This allows us to constrain the physical parameters of the mergers. We find that our methodology works well at estimating merger parameters, consistently giving results within 1- of the input parameter values. We also discover that the number of merger events is a key discriminant among models. This helps our method be robust against observational uncertainties. Our approach, which at this stage constitutes a proof of principle, can be readily extended to physical models and to more general problems in cosmology and gravitational wave astrophysics.

Alicea-Munoz, E.↗

NASA NTRS2008

Using Black Hole Mergers to Explore Structure Formation

Observations of gravitational waves from massive black hole mergers will open a new window into the era of structure formation in the early universe. Past efforts have concentrated on calculating merger rates using different physical assumptions, resulting in merger rate estimates that span a wide range (0.1 - 1 0A4 mergers/year). We develop a semi-analytical, phenomenological model of massive black hole mergers that includes plausible combinations of several physical parameters, which we then turn around to determine how well observations with the Laser Interferometer Space Antenna (LISA) will be able to enhance our understanding of the universe during the critical z approx. 5 - 30 epoch. Our approach involves generating synthetic LISA observable data (total BH masses, BH mass ratios, redshifts, merger rates), which are then analyzed using a Markov Chain Monte Carlo method, thus finding constraints for the physical parameters of the mergers. We find that our method works well at estimating merger parameters and that the number of merger events is a key discriminant among models, therefore making our method robust against observational uncertainties. Our approach can also be extended to more physically-driven models and more general problems in cosmology.

Alicea-Munoz, E.↗

NASA NTRS2009

Using Black Hole Mergers to Explore Structure Formation

Observations of gravitational waves from massive black hole mergers will open a new window into the era of structure formation in the early universe. Past efforts have concentrated on calculating merger rates using different physical assumptions, resulting in merger rate estimates that span a wide range (0.1 - 10(exp 4) mergers/year). We develop a semi-analytical, phenomenological model of massive black hole mergers that includes plausible combinations of several physical parameters, which we then turn around to determine how well observations with the Laser Interferometer Space Antenna (LISA) will be able to enhance our understanding of the universe during the critical z approximately equal to 5-30 epoch. Our approach involves generating synthetic LISA observable data (total BH masses, BH mass ratios, redshifts, merger rates), which are then analyzed using a Markov Chain Monte Carlo method, thus finding constraints for the physical parameters of the mergers. We find that our method works well at estimating merger parameters and that the number of merger events is a key discriminant among models, therefore making our method robust against observational uncertainties. Our approach can also be extended to more physically-driven models and more general problems in cosmology. This work is supported in part by the Cooperative Education Program at NASA/GSFC.

Alicea-Munoz, E.↗

NASA NTRS2008

Black Hole Mergers as Probes of Structure Formation

Observations of gravitational waves from massive black hole (MBH) mergers can provide us with important clues about the era of structure formation in the early universe. Previous research in this field has been limited to calculating merger rates of MBHs using different models where many assumptions are made about the specific values of physical parameters of the mergers, resulting in merger rate estimates that span 5 to 6 orders of magnitude. We develop a semi-analytical, phenomenological model that includes plausible combinations of several physical parameters involved in the mergers. which we then turn around to determine how well LISA observations will be able to enhance our understanding of the universe during the critical z approximately equal to 5-30 structure formation era. We do this by generating synthetic LISA observable data (masses, redshifts, merger rates), which are then analyzed using a Markov Chain Monte Carlo (MCMC) method. This allows us to constrain the physical parameters of the mergers.

Alicea-Munoz, Emily↗

NASA NTRS2008

Precision Measurement of Black Hole Binary Dynamics: Analyzing the LISA Data Stream

One of the richest potential sources of insight into fundamental physics that LISA will be capable of observing is the inspiral of supermassive black hole binaries (BHBs). However, the data analysis challenge presented by the LISA data stream is quite unlike the situation for present day gravitational wave detectors. In order to make the precision measurements necessary to achieve LISA's science goals, the BHB signal must be distinguished from a data stream that not only contains instrumental noise, but potentially thousands of other signals as well, so that the "background" we wish to separate out to focus on the BHB signal is likely to be highly nonstationary and nongaussian, as well as being of scientific interest in its own right. In addition, whereas the theoretical templates that we calculate in order to ultimately estimate the parameters can afford to be somewhat inaccurate and still be effective for present day and near future detectors, this is not the case for LISA, and extremely high fidelity of the theoretical templates for high signal-to-noise signals will be required to prevent theoretical errors from dominating the parameter estimates. NVe, will describe efforts in the community of LISA data analysts to address the challenges regarding the specific issue of BHB signals. These efforts include using a Markov Chain Monte Carlo approach with the freedom to model the BHB and the other signals present in the data stream simultaneously, rather than trying to remove other signals and risk biasing the remaining data. The Mock LISA Data Challenge is a community of LISA scientists who generate rounds of simulated LISA noise with increasingly difficult signal content, and invite the LISA data analysis community to exercise their methods, or develop new methods, in an attempt to extract the parameters for the signals embedded in the mock data. In addition to practical approaches such ,is this to assess the level of parameter accuracy, one can apply the Fisher matrix formalism to assess both the statistical errors from noise and the theoretical errors

McWilliams, Sean T.↗

NASA NTRS2010

Separating Gravitational Wave Signals from Instrument Artifacts

Central to the gravitational wave detection problem is the challenge of separating features in the data produced by astrophysical sources from features produced by the detector. Matched filtering provides an optimal solution for Gaussian noise, but in practice, transient noise excursions or "glitches" complicate the analysis. Detector diagnostics and coincidence tests can be used to veto many glitches which may otherwise be misinterpreted as gravitational wave signals. The glitches that remain can lead to long tails in the matched filter search statistics and drive up the detection threshold. Here we describe a Bayesian approach that incorporates a more realistic model for the instrument noise allowing for fluctuating noise levels that vary independently across frequency bands, and deterministic "glitch fitting" using wavelets as "glitch templates", the number of which is determined by a trans-dimensional Markov chain Monte Carlo algorithm. We demonstrate the method's effectiveness on simulated data containing low amplitude gravitational wave signals from inspiraling binary black hole systems, and simulated non-stationary and non-Gaussian noise comprised of a Gaussian component with the standard LIGO/Virgo spectrum, and injected glitches of various amplitude, prevalence, and variety. Glitch fitting allows us to detect significantly weaker signals than standard techniques.

Littenberg, Tyson B.↗

NASA NTRS2012

Modeling Phase-Aligned Gamma-Ray and Radio Millisecond Pulsar Light Curves

Since the discovery of the first eight gamma-ray millisecond pulsars (MSPs) by the Fermi Large Area Telescope, this population has been steadily expanding. Four of the more recent detections, PSR J00340534, PSR J1939+2134 (B1937+21; the first MSP ever discovered), PSR J1959+2048 (B1957+20; the first discovery of a black widow system), and PSR J2214+3000, exhibit a phenomenon not present in the original discoveries: nearly phase-aligned radio and gamma-ray light curves (LCs). To account for the phase alignment, we explore models where both the radio and gamma-ray emission originate either in the outer magnetosphere near the light cylinder or near the polar caps. Using a Markov Chain Monte Carlo technique to search for best-fit model parameters, we obtain reasonable LC fits for the first three of these MSPs in the context of altitude-limited outer gap (alOG) and two-pole caustic (alTPC) geometries (for both gamma-ray and radio emission). These models differ from the standard outer gap (OG)/two-pole caustic (TPC) models in two respects: the radio emission originates in caustics at relatively high altitudes compared to the usual conal radio beams, and we allow both the minimum and maximum altitudes of the gamma-ray and radio emission regions to vary within a limited range (excluding the minimum gamma-ray altitude of the alTPC model, which is kept constant at the stellar radius, and that of the alOG model, which is set to the position-dependent null charge surface altitude). Alternatively, phase-aligned solutions also exist for emission originating near the stellar surface in a slot gap scenario (low-altitude slot gap (laSG) models). We find that the alTPC models provide slightly better LC fits than the alOG models, and both of these give better fits than the laSG models (for the limited range of parameters considered in the case of the laSG models). Thus, our fits imply that the phase-aligned LCs are likely of caustic origin, produced in the outer magnetosphere, and that the radio emission for these pulsars may come from close to the light cylinder. In addition, we were able to constrain the minimum and maximum emission altitudes with typical uncertainties of 30% of the light cylinder radius. Our results therefore describe a third gamma-ray MSP subclass, in addition to the two previously found by Venter et al.: those with LCs fit by standard OG/TPC models and those with LCs fit by pair-starved polar cap models.

Venter, C.↗

NASA NTRS2012

Pulsar Emission Geometry and Accelerating Field Strength

The high-quality Fermi LAT observations of gamma-ray pulsars have opened a new window to understanding the generation mechanisms of high-energy emission from these systems, The high statistics allow for careful modeling of the light curve features as well as for phase resolved spectral modeling. We modeled the LAT light curves of the Vela and CTA I pulsars with simulated high-energy light curves generated from geometrical representations of the outer gap and slot gap emission models. within the vacuum retarded dipole and force-free fields. A Markov Chain Monte Carlo maximum likelihood method was used to explore the phase space of the magnetic inclination angle, viewing angle. maximum emission radius, and gap width. We also used the measured spectral cutoff energies to estimate the accelerating parallel electric field dependence on radius. under the assumptions that the high-energy emission is dominated by curvature radiation and the geometry (radius of emission and minimum radius of curvature of the magnetic field lines) is determined by the best fitting light curves for each model. We find that light curves from the vacuum field more closely match the observed light curves and multiwavelength constraints, and that the calculated parallel electric field can place additional constraints on the emission geometry

DeCesar, Megan E.↗

NASA NTRS2012

Modeling and Maximum Likelihood Fitting of Gamma-Ray and Radio Light Curves of Millisecond Pulsars Detected with Fermi

Pulsed gamma rays have been detected with the Fermi Large Area Telescope (LAT) from more than 20 millisecond pulsars (MSPs), some of which were discovered in radio observations of bright, unassociated LAT sources. We have fit the radio and gamma-ray light curves of 19 LAT-detected MSPs in the context of geometric, outermagnetospheric emission models assuming the retarded vacuum dipole magnetic field using a Markov chain Monte Carlo maximum likelihood technique. We find that, in many cases, the models are able to reproduce the observed light curves well and provide constraints on the viewing geometries that are in agreement with those from radio polarization measurements. Additionally, for some MSPs we constrain the altitudes of both the gamma-ray and radio emission regions. The best-fit magnetic inclination angles are found to cover a broader range than those of non-recycled gamma-ray pulsars.

Johnson, T. J.↗

NASA NTRS2012

A Bayesian Framework for Reliability Analysis of Spacecraft Deployments

Deployable subsystems are essential to mission success of most spacecraft. These subsystems enable critical functions including power, communications and thermal control. The loss of any of these functions will generally result in loss of the mission. These subsystems and their components often consist of unique designs and applications for which various standardized data sources are not applicable for estimating reliability and for assessing risks. In this study, a two stage sequential Bayesian framework for reliability estimation of spacecraft deployment was developed for this purpose. This process was then applied to the James Webb Space Telescope (JWST) Sunshield subsystem, a unique design intended for thermal control of the Optical Telescope Element. Initially, detailed studies of NASA deployment history, "heritage information", were conducted, extending over 45 years of spacecraft launches. This information was then coupled to a non-informative prior and a binomial likelihood function to create a posterior distribution for deployments of various subsystems uSing Monte Carlo Markov Chain sampling. Select distributions were then coupled to a subsequent analysis, using test data and anomaly occurrences on successive ground test deployments of scale model test articles of JWST hardware, to update the NASA heritage data. This allowed for a realistic prediction for the reliability of the complex Sunshield deployment, with credibility limits, within this two stage Bayesian framework.

Evans, John W.↗

NASA NTRS2012

Mapping the Milky Way Galaxy with LISA

Gravitational wave detectors in the mHz band (such as the Laser Interferometer Space Antenna, or LISA) will observe thousands of compact binaries in the galaxy which can be used to better understand the structure of the Milky Way. To test the effectiveness of LISA to measure the distribution of the galaxy, we simulated the Close White Dwarf Binary (CWDB) gravitational wave sky using different models for the Milky Way. To do so, we have developed a galaxy density distribution modeling code based on the Markov Chain Monte Carlo method. The code uses different distributions to construct realizations of the galaxy. We then use the Fisher Information Matrix to estimate the variance and covariance of the recovered parameters for each detected CWDB. This is the first step toward characterizing the capabilities of space-based gravitational wave detectors to constrain models for galactic structure, such as the size and orientation of the bar in the center of the Milky Way

McKinnon, Jose A.↗

NASA NTRS2012

Metallicities of Emission-Line Galaxies from HST ACS PEARS and HST WFC3 ERS Grism Spectroscopy at 0.6 is less than z is less than 2.4

Galaxies selected on the basis of their emission line strength. show low metallicities, regardless of their redshifts. We conclude this from a sample of faint galaxies at redshifts between 0.6 < z < 2.4, selected by their prominent emission lines in low resolution grism spectra in the optiCa.i with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST) and in the near-infrared using Wide-Field Camera 3 (WFC3). Using a sample of 11 emission line galaxies (ELGs) at 0.6 < z < 2.4 with luminosities of -22 approx < MB approx -19 which have [OII], H-Beta, and [OIII] line flux measurements from the combination of two grism spectral surveys, we use the R23 method to derive the gas-phase oxygen abundances: 7.5 <12+log(0/H)<8.5. The galaxy stellar masses are derived using Bayesian based Markov Chain Monte Carlo (pi MC(exp 2)) fitting of their Spectral Energy Distribution (SED), and span the mass range 8.1 < log(M(stellar)/M(solar)) < 10.1. These galaxies show a mass-metal1icity (M-L) and Luminosity-Metallicity (LZ) relation, which is offset by -<J.6 dex in meta1licity at given absolute magnitude and stellar mass relative to the local SDSS galaxies, as well as continuum selected DEEP2 samples at similar redshifts. The emission-line selected galaxies most resemble the local "green peas" galaxies and Lyman-alpha galaxies at z approx = 0.3 and z approx = 2.3 in the M-Z and L-Z relations and their morphologies. The G - M(sub 20) morphology analysis shows that 10 out of 11 show disturbed morphology, even as the star-forming regions are compact. These galaxies may be intrinsically metal poor, being at early stages of formation, or the low metallicities may be due to gas infall and accretion due to mergers.

Xia, Lifang↗

NASA NTRS2013

Estimating the Properties of Hard X-Ray Solar Flares by Constraining Model Parameters

We wish to better constrain the properties of solar flares by exploring how parameterized models of solar flares interact with uncertainty estimation methods. We compare four different methods of calculating uncertainty estimates in fitting parameterized models to Ramaty High Energy Solar Spectroscopic Imager X-ray spectra, considering only statistical sources of error. Three of the four methods are based on estimating the scale-size of the minimum in a hypersurface formed by the weighted sum of the squares of the differences between the model fit and the data as a function of the fit parameters, and are implemented as commonly practiced. The fourth method is also based on the difference between the data and the model, but instead uses Bayesian data analysis and Markov chain Monte Carlo (MCMC) techniques to calculate an uncertainty estimate. Two flare spectra are modeled: one from the Geostationary Operational Environmental Satellite X1.3 class flare of 2005 January 19, and the other from the X4.8 flare of 2002 July 23.We find that the four methods give approximately the same uncertainty estimates for the 2005 January 19 spectral fit parameters, but lead to very different uncertainty estimates for the 2002 July 23 spectral fit. This is because each method implements different analyses of the hypersurface, yielding method-dependent results that can differ greatly depending on the shape of the hypersurface. The hypersurface arising from the 2005 January 19 analysis is consistent with a normal distribution; therefore, the assumptions behind the three non- Bayesian uncertainty estimation methods are satisfied and similar estimates are found. The 2002 July 23 analysis shows that the hypersurface is not consistent with a normal distribution, indicating that the assumptions behind the three non-Bayesian uncertainty estimation methods are not satisfied, leading to differing estimates of the uncertainty. We find that the shape of the hypersurface is crucial in understanding the output from each uncertainty estimation technique, and that a crucial factor determining the shape of hypersurface is the location of the low-energy cutoff relative to energies where the thermal emission dominates. The Bayesian/MCMC approach also allows us to provide detailed information on probable values of the low-energy cutoff, Ec, a crucial parameter in defining the energy content of the flare-accelerated electrons. We show that for the 2002 July 23 flare data, there is a 95% probability that Ec lies below approximately 40 keV, and a 68% probability that it lies in the range 7-36 keV. Further, the low-energy cutoff is more likely to be in the range 25-35 keV than in any other 10 keV wide energy range. The low-energy cutoff for the 2005 January 19 flare is more tightly constrained to 107 +/- 4 keV with 68% probability.

X-rays↗

NASA NTRS2013

Uncertainty Quantification of GEOS-5 L-band Radiative Transfer Model Parameters Using Bayesian Inference and SMOS Observations

Uncertainties in L-band (1.4 GHz) radiative transfer modeling (RTM) affect the simulation of brightness temperatures (Tb) over land and the inversion of satellite-observed Tb into soil moisture retrievals. In particular, accurate estimates of the microwave soil roughness, vegetation opacity and scattering albedo for large-scale applications are difficult to obtain from field studies and often lack an uncertainty estimate. Here, a Markov Chain Monte Carlo (MCMC) simulation method is used to determine satellite-scale estimates of RTM parameters and their posterior uncertainty by minimizing the misfit between long-term averages and standard deviations of simulated and observed Tb at a range of incidence angles, at horizontal and vertical polarization, and for morning and evening overpasses. Tb simulations are generated with the Goddard Earth Observing System (GEOS-5) and confronted with Tb observations from the Soil Moisture Ocean Salinity (SMOS) mission. The MCMC algorithm suggests that the relative uncertainty of the RTM parameter estimates is typically less than 25 of the maximum a posteriori density (MAP) parameter value. Furthermore, the actual root-mean-square-differences in long-term Tb averages and standard deviations are found consistent with the respective estimated total simulation and observation error standard deviations of m3.1K and s2.4K. It is also shown that the MAP parameter values estimated through MCMC simulation are in close agreement with those obtained with Particle Swarm Optimization (PSO).

MCMC↗

NASA NTRS2013

STIS Coronagraphic Imaging of Fomalhaut: Main Belt Structure and the Orbit of Fomalhaut b

We present new optical coronagraphic data of Fomalhaut obtained with HST/STIS in 2010 and 2012. Fomalhaut b is recovered at both epochs to high significance. The observations include the discoveries of tenuous nebulosity beyond the main dust belt detected to at least 209AU projected radius, and a approx. 50AU wide azimuthal gap in the belt northward of Fomalhaut b. The two epochs of Space Telescope Imaging Spectrograph (STIS) photometry exclude optical variability greater than 35%. A Markov chain Monte Carlo analysis demonstrates that the orbit of Fomalhaut b is highly eccentric, with e = 0.8 +/- 0.1, a = 177 +/- 68AU, and q = 32 +/- 24AU. Fomalhaut b is apsidally aligned with the belt and 90% of allowed orbits have mutual inclination <=36 deg. Fomalhaut b's orbit is belt crossing in the sky plane projection, but only 12% of possible orbits have ascending or descending nodes within a 25AU wide belt annulus. The high eccentricity invokes a dynamical history where Fomalhaut b may have experienced a significant dynamical interaction with a hypothetical planet Fomalhaut c, and the current orbital configuration may be relatively short-lived. The Tisserand parameter with respect to a hypothetical Fomalhaut planet at 30AU or 120AU lies in the range 2-3, similar to highly eccentric dwarf planets in our solar system. We argue that Fomalhaut b's minimum mass is that of a dwarf planet in order for a circumplanetary satellite system to remain bound to a sufficient radius from the planet to be consistent with the dust scattered light hypothesis. In the coplanar case, Fomalhaut b will collide with the main belt around 2032, and the subsequent emergent phenomena may help determine its physical nature.

Tisserand parameter↗

NASA NTRS2014

The Evolution of the Galaxy Rest-Frame Ultraviolet Luminosity Function Over the First Two Billion Years

We present a robust measurement and analysis of the rest-frame ultraviolet (UV) luminosity function at z = 4 to 8. We use deep Hubble Space Telescope imaging over the CANDELS/GOODS fields, the Hubble Ultra Deep Field and the Hubble Frontier Field deep parallel observations near the Abell 2744 and MACS J0416.1- 2403 clusters. The combination of these surveys provides an effective volume of 0.6-1.2 ×10(exp 6) Mpc(exp 3) over this epoch, allowing us to perform a robust search for bright (M(sub UV) less than −21) and faint (M(sub UV) = −18) galaxies. We select galaxies using a well-tested photometric redshift technique with careful screening of contaminants, finding a sample of 7446 galaxies at 3.5 less than z less than 8.5, with more than 1000 galaxies at z of approximately 6 - 8. We measure both a stepwise luminosity function for galaxies in our redshift samples, as well as a Schechter function, using a Markov Chain Monte Carlo analysis to measure robust uncertainties. At the faint end our UV luminosity functions agree with previous studies, yet we find a higher abundance of UV-bright galaxies at z of greater than or equal to 6. Our bestfit value of the characteristic magnitude M* is consistent with −21 at z of greater than or equal to 5, different than that inferred based on previous trends at lower redshift. At z = 8, a single power-law provides an equally good fit to the UV luminosity function, while at z = 6 and 7, an exponential cutoff at the bright-end is moderately preferred. We compare our luminosity functions to semi-analytical models, and find that the lack of evolution in M* is consistent with models where the impact of dust attenuation on the bright-end of the luminosity function decreases at higher redshift, though a decreasing impact of feedback may also be possible. We measure the evolution of the cosmic star-formation rate (SFR) density by integrating our observed luminosity functions to M(sub UV) = −17, correcting for dust attenuation, and find that the SFR density declines proportionally to (1 + z)((exp −4.3)(+/-)(0.5)) at z greater than 4, consistent with observations at z greater than or equal to 9. Our observed luminosity functions are consistent with a reionization history that starts at redshift of approximately greater than 10, completes at z greater than 6, and reaches a midpoint (x(sub HII) = 0.5) at 6.7 less than z less than 9.4. Finally, using a constant cumulative number density selection and an empirically derived rising star-formation history, our observations predict that the abundance of bright z = 9 galaxies is likely higher than previous constraints, though consistent with recent estimates of bright z similar to 10 galaxies.

Galaxy↗

NASA NTRS2014

The Gemini NICI Planet-Finding Campaign: The Offset Ring of HR 4796 A

We present J, H, CH4 short (1.578 micrometers), CH4 long (1.652 micrometers) and K(sub s)-band images of the dust ring around the 10 Myr old star HR 4796 A obtained using the Near Infrared Coronagraphic Imager (NICI) on the Gemini-South 8.1 m Telescope. Our images clearly show for the first time the position of the star relative to its circumstellar ring thanks to NICI's translucent focal plane occulting mask. We employ a Bayesian Markov chain Monte Carlo method to constrain the offset vector between the two. The resulting probability distribution shows that the ring center is offset from the star by 16.7 +/- 1.3 milliarcseconds along a position angle of 26 +/- 3deg, along the PA of the ring, 26.47 +/- 0.04deg. We find that the size of this offset is not large enough to explain the brightness asymmetry of the ring. The ring is measured to have mostly red reflectivity across the JHKs filters, which seems to indicate micron-sized grains. Just like Neptune's 3:2 and 2:1 mean-motion resonances delineate the inner and outer edges of the classical Kuiper belt, we find that the radial extent of the HR 4796 A and the Fomalhaut rings could correspond to the 3:2 and 2:1 mean-motion resonances of hypothetical planets at 54.7 AU and 97.7 AU in the two systems, respectively. A planet orbiting HR 4796 A at 54.7 AU would have to be less massive than 1.6 Jup mass so as not to widen the ring too much by stirring.

Near Infrared Coronagraphic Imager (NICI)↗

NASA NTRS2016

Probabilistic Damage Characterization Using the Computationally-Efficient Bayesian Approach

This work presents a computationally-ecient approach for damage determination that quanti es uncertainty in the provided diagnosis. Given strain sensor data that are polluted with measurement errors, Bayesian inference is used to estimate the location, size, and orientation of damage. This approach uses Bayes' Theorem to combine any prior knowledge an analyst may have about the nature of the damage with information provided implicitly by the strain sensor data to form a posterior probability distribution over possible damage states. The unknown damage parameters are then estimated based on samples drawn numerically from this distribution using a Markov Chain Monte Carlo (MCMC) sampling algorithm. Several modi cations are made to the traditional Bayesian inference approach to provide signi cant computational speedup. First, an ecient surrogate model is constructed using sparse grid interpolation to replace a costly nite element model that must otherwise be evaluated for each sample drawn with MCMC. Next, the standard Bayesian posterior distribution is modi ed using a weighted likelihood formulation, which is shown to improve the convergence of the sampling process. Finally, a robust MCMC algorithm, Delayed Rejection Adaptive Metropolis (DRAM), is adopted to sample the probability distribution more eciently. Numerical examples demonstrate that the proposed framework e ectively provides damage estimates with uncertainty quanti cation and can yield orders of magnitude speedup over standard Bayesian approaches.

Warner, James E.↗