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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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An algorithm for coupling multibranch in vitro experiment to numerical physiology simulation for a hybrid cardiovascular model
Abstract The hybrid cardiovascular modeling approach integrates an in vitro experiment with a computational lumped‐parameter simulation, enabling direct physical testing of medical devices in the context of closed‐loop physiology. The interface between the in vitro and computational domains is essential for properly capturing the dynamic interactions of the two. To this end, we developed an iterative algorithm capable of coupling an in vitro experiment containing multiple branches to a lumped‐parameter physiology simulation. This algorithm identifies the unique flow waveform solution for each branch of the experiment using an iterative Broyden's approach. For the purpose of algorithm testing, we first used mathematical surrogates to represent the in vitro experiments and demonstrated five scenarios where the in vitro surrogates are coupled to the computational physiology of a Fontan patient. This testing approach allows validation of the coupling result accuracy as the mathematical surrogates can be directly integrated into the computational simulation to obtain the “true solution” of the coupled system. Our algorithm successfully identified the solution flow waveforms in all test scenarios with results matching the true solutions with high accuracy. In all test cases, the number of iterations to achieve the desired convergence criteria was less than 130. To emulate realistic in vitro experiments in which noise contaminates the measurements, we perturbed the surrogate models by adding random noise. The convergence tolerance achievable with the coupling algorithm remained below the magnitudes of the added noise in all cases. Finally, we used this algorithm to couple a physical experiment to the computational physiology model to demonstrate its real‐world applicability.
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- Award ID(s):
- 1749017
- PAR ID:
- 10453415
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Biomedical Engineering
- Volume:
- 36
- Issue:
- 3
- ISSN:
- 2040-7939
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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