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Inference, optimization, and inverse problems are but three examples of mathematical operations that require the repeated solution of a complex system of mathematical equations. To this end, surrogates are often used to approximate the output of these large computer simulations, providing fast and cheap approximation solutions. Statistical emulators are surrogates that, in addition to predicting the mean behavior of the system, provide an estimate of the error in that prediction. Classical Gaussian stochastic process emulators predict scalar outputs based on a modest number of input parameters. Making predictions across a space-time field of input variables is not feasible using classical Gaussian process methods. Parallel partial emulation is a new statistical emulator methodology that predicts a field of outputs based on the input parameters. Parallel partial emulation is constructed as a Gaussian process in parameter space, but no correlation among space or time points is assumed. Thus the computational work of parallel partial emulation scales as the cube of the number of input parameters (as traditional Gaussian Process emulation) and linearly with a space-time grid. The numerical methods used in numerical simulations are often designed to exploit properties of the equations tobe solved. For example, modern solvers for hyperbolic conservation laws satisfy conservation at each time step, insuring overall conservation of the physical variables. Similarly, symplectic methods are used to solve Hamiltonian problems in physics. It is of interest, then, to study whether parallel partial emulation predictions inherit properties possessed by the simulation outputs. Does an emulated solution of a conservation law preserve the conserved quantities? Does an emulator of a Hamiltonian system preserve the energy? This paper investigates the properties of emulator predictions, in the context of systems of partial differential equations. We study conservation properties for three different kinds ofequations-conservation laws, reaction-diffusion systems, and a Hamiltonian system.We also investigate the effective convergence, in parameter space, of the predicted solution of a highly nonlinear system modeling shape memory alloys.
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Novel statistical emulator construction for volcanic ash transport model Ash3d with physically motivated measures
Statistical emulators are a key tool for rapidly producing probabilistic hazard analysis of geophysical processes. Given output data computed for a relatively small number of parameter inputs, an emulator interpolates the data, providing the expected value of the output at untried inputs and an estimate of error at that point. In this work, we propose to fit Gaussian Process emulators to the output from a volcanic ash transport model, Ash3d. Our goal is to predict the simulated volcanic ash thickness from Ash3d at a location of interest using the emulator. Our approach is motivated by two challenges to fitting emulators—characterizing the input wind field and interactions between that wind field and variable grain sizes. We resolve these challenges by using physical knowledge on tephra dispersal. We propose new physically motivated variables as inputs and use normalized output as the response for fitting the emulator. Subsetting based on the initial conditions is also critical in our emulator construction. Simulation studies characterize the accuracy and efficiency of our emulator construction and also reveal its current limitations. Our work represents the first emulator construction for volcanic ash transport models with considerations of the simulated physical process.
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- PAR ID:
- 10284891
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 476
- Issue:
- 2242
- ISSN:
- 1364-5021
- Page Range / eLocation ID:
- 20200161
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rspa.2020.0161
