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Adding GPU Support to the Markov Chain Monte Carlo Code Catmip

In geophysics, we are confronted with many under-determined inverse problems. For example, all of our observations of earthquakes are made at the Earth’s surface. So, when we try to infer how slip during an earthquake evolves in space and time, we find that there are many potential slip histories that are consistent with our limited observations and our understanding of earthquake physics. One way to approach these problems is with Bayesian analysis which allows us to infer the ensemble of all potential slip models that satisfy the observations and our prior knowledge of earthquake physics. In Bayesian analysis, our prior knowledge is known as the prior probability density function or prior PDF, the fit to the data is known as the data likelihood, and the target PDF that satisfies both the prior PDF and data likelihood is known as the posterior PDF. However, simulating the posterior PDF typically requires using Markov Chain Monte Carlo (MCMC) to draw tens of billions of random realizations of earthquake slip models, which may not be computationally feasible. To make this and similar geophysical inversions computationally tractable, we developed the Cascading Adaptive Transitional Metropolis In Parallel (CATMIP) algorithm. CATMIP is an efficient parallel Markov Chain Monte Carlo (MCMC) sampler that is used for model fitting and uncertainty quantification in geophysics. Example use cases are earthquake rupture modeling, determining mineral composition on Mars, reconstructing the history of ocean salinity, and historical earthquake relocation. CATMIP employs many parallel instances of the Metropolis algorithm for sampling in a transitioning framework. Transitioning is a process in which a set of random samples at equilibrium with a known probability density function (PDF) are used as seeds for the Markov chains to sample successive target PDFs that incrementally move the distribution from the starting seeds to the final desired PDF that describes the relative plausibility of potential values for the model parameters. The algorithm is implemented as a Master-Worker model employing MPI for communication. The worker processes are loosely coupled with global parameters periodically optimized by the master process. This provides a very high amount of parallelism with little communication between updates. During the presentation we will discuss the history of the algorithm and elaborate the earthquake rupture modeling use case for the CATMIP package. Our first step toward GPU optimization was to optimize the code for the CPU. CPU profiling revealed that most of the compute time is spent in calls to level 2 BLAS routines and calls to GSL random number generators. We revised the algorithm to employ level 3 BLAS routines instead. In our presentation we will describe how this was accomplished. Adding GPU support to CATMIP consisted mostly of replacing the calls to GSL with calls to GPU vendor-provided library routines. A small number of loops were directly implemented in CUDA. In the presentation will provide implementation details. Finally, we will discuss methods for profiling and opportunities for further optimizing GPU execution. By creating a code with the flexibility to run on either a CPU or GPU architecture, CATMIP can be used on systems ranging from large CPU-based HPC environments to single servers with GPU acceleration and everything in between.

HECC↗

NASA NTRS1990

Order/disorder and phase diagram of H on Pd(100)

A phase boundary for H-Pd(100) was calculated using the Metropolis (1953) algorithm and the embedded atom method (EAM) described by Daw and Foiles (1987). The calculated phase boundary agreed with an experimentally determined phase boundary in its curvature and the coverage at which maximum Tc appeared, but was about 125 K lower than the experimental phase boundary.

Tibbits, P.↗

NASA NTRS2023

Prototype Global Analysis of LISA Data with Multiple Source Types

The novel data analysis challenges posed by the Laser Interferometer Space Antenna (LISA) arise from the overwhelmingly large number of astrophysical sources in the measurement band and the density with which they are found in the data. Robust detection and characterization of the numerous gravitational wave sources in LISA data can not be done sequentially, but rather through a simultaneous global fit of a data model containing the full suite of astrophysical and instrumental features present in the data. While previous analyses have focused on individual source types in isolation, here we present the first demonstration of a LISA global fit analysis containing combined astrophysical populations. The prototype pipeline uses a blocked Metropolis Hastings algorithm to alternatingly fit to a population of ultra compact galactic binaries, known “verification binaries” already identified by electromagnetic observations, a population of massive black hole mergers, and an instrument noise model. The Global LISA Analysis Software Suite (GLASS) is assembled from independently developed samplers for the different model components. The modular design enables flexibility to future development by defining standard interfaces for adding new, or updating additional, components to the global fit without being overly prescriptive for how those modules must be internally designed. The GLASS pipeline is demonstrated on data simulated for the LISA Data Challenge 2b. Results of the analysis and a road-map for continued development are described in detail.

Gravitational Waves↗

NASA NTRS2005

Adaptive Metropolis Sampling with Product Distributions

The Metropolis-Hastings (MH) algorithm is a way to sample a provided target distribution pi(z). It works by repeatedly sampling a separate proposal distribution T(x,x') to generate a random walk {x(t)}. We consider a modification of the MH algorithm in which T is dynamically updated during the walk. The update at time t uses the {x(t' less than t)} to estimate the product distribution that has the least Kullback-Leibler distance to pi. That estimate is the information-theoretically optimal mean-field approximation to pi. We demonstrate through computer experiments that our algorithm produces samples that are superior to those of the conventional MH algorithm.

Wolpert, David H.↗

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.↗

NASA NTRS2005

New Algorithms for Estimating Spacecraft Position Using Scanning Techniques for Deep Space Network Antennas

As more and more nonlinear estimation techniques become available, our interest is in finding out what performance improvement, if any, they can provide for practical nonlinear problems that have been traditionally solved using linear methods. In this paper we examine the problem of estimating spacecraft position using conical scan (conscan) for NASA's Deep Space Network antennas. We show that for additive disturbances on antenna power measurement, the problem can be transformed into a linear one, and we present a general solution to this problem, with the least square solution reported in literature as a special case. We also show that for additive disturbances on antenna position, the problem is a truly nonlinear one, and we present two approximate solutions based on linearization and Unscented Transformation respectively, and one 'exact' solution based on Markov Chain Monte Carlo (MCMC) method. Simulations show that, with the amount of data collected in practice, linear methods perform almost the same as MCMC methods. It is only when we artificially reduce the amount of collected data and increase the level of noise that nonlinear methods show significantly better accuracy than that achieved by linear methods, at the expense of more computation.

Metropolis-Hastings (MH)↗

NASA NTRS2004

Application of Simulated Annealing and Related Algorithms to TWTA Design

Simulated Annealing (SA) is a stochastic optimization algorithm used to search for global minima in complex design surfaces where exhaustive searches are not computationally feasible. The algorithm is derived by simulating the annealing process, whereby a solid is heated to a liquid state and then cooled slowly to reach thermodynamic equilibrium at each temperature. The idea is that atoms in the solid continually bond and re-bond at various quantum energy levels, and with sufficient cooling time they will rearrange at the minimum energy state to form a perfect crystal. The distribution of energy levels is given by the Boltzmann distribution: as temperature drops, the probability of the presence of high-energy bonds decreases. In searching for an optimal design, local minima and discontinuities are often present in a design surface. SA presents a distinct advantage over other optimization algorithms in its ability to escape from these local minima. Just as high-energy atomic configurations are visited in the actual annealing process in order to eventually reach the minimum energy state, in SA highly non-optimal configurations are visited in order to find otherwise inaccessible global minima. The SA algorithm produces a Markov chain of points in the design space at each temperature, with a monotonically decreasing temperature. A random point is started upon, and the objective function is evaluated at that point. A stochastic perturbation is then made to the parameters of the point to arrive at a proposed new point in the design space, at which the objection function is evaluated as well. If the change in objective function values (Delta)E is negative, the proposed new point is accepted. If (Delta)E is positive, the proposed new point is accepted according to the Metropolis criterion: rho((Delta)f) = exp((-Delta)E/T), where T is the temperature for the current Markov chain. The process then repeats for the remainder of the Markov chain, after which the temperature is decremented and the process repeats. Eventually (and hopefully), a near-globally optimal solution is attained as T approaches zero. Several exciting variants of SA have recently emerged, including Discrete-State Simulated Annealing (DSSA) and Simulated Tempering (ST). The DSSA algorithm takes the thermodynamic analogy one step further by categorizing objective function evaluations into discrete states. In doing so, many of the case-specific problems associated with fine-tuning the SA algorithm can be avoided; for example, theoretical approximations for the initial and final temperature can be derived independently of the case. In this manner, DSSA provides a scheme that is more robust with respect to widely differing design surfaces. ST differs from SA in that the temperature T becomes an additional random variable in the optimization. The system is also kept in equilibrium as the temperature changes, as opposed to the system being driven out of equilibrium as temperature changes in SA. ST is designed to overcome obstacles in design surfaces where numerous local minima are separated by high barriers. These algorithms are incorporated into the optimal design of the traveling-wave tube amplifier (TWTA). The area under scrutiny is the collector, in which it would be ideal to use negative potential to decelerate the spent electron beam to zero kinetic energy just as it reaches the collector surface. In reality this is not plausible due to a number of physical limitations, including repulsion and differing levels of kinetic energy among individual electrons. Instead, the collector is designed with multiple stages depressed below ground potential. The design of this multiple-stage collector is the optimization problem of interest. One remaining problem in SA and DSSA is the difficulty in determining when equilibrium has been reached so that the current Markov chain can be terminated. It has been suggested in recent literature that simulating the thermodynamic properties opecific heat, entropy, and internal energy from the Boltzmann distribution can provide good indicators of having reached equilibrium at a certain temperature. These properties are tested for their efficacy and implemented in SA and DSSA code with respect to TWTA collector optimization.

Radke, Eric M.↗

NASA NTRS2011

Markov Chain Monte Carlo Bayesian Learning for Neural Networks

Conventional training methods for neural networks involve starting al a random location in the solution space of the network weights, navigating an error hyper surface to reach a minimum, and sometime stochastic based techniques (e.g., genetic algorithms) to avoid entrapment in a local minimum. It is further typically necessary to preprocess the data (e.g., normalization) to keep the training algorithm on course. Conversely, Bayesian based learning is an epistemological approach concerned with formally updating the plausibility of competing candidate hypotheses thereby obtaining a posterior distribution for the network weights conditioned on the available data and a prior distribution. In this paper, we developed a powerful methodology for estimating the full residual uncertainty in network weights and therefore network predictions by using a modified Jeffery's prior combined with a Metropolis Markov Chain Monte Carlo method.

Goodrich, Michael S.↗