Stochastic optimization is an effective approach to solve inverse problems, especially when traditional deterministic optimization methods fail or do not perform well. Important stochastic optimization methods include stochastic gradient descent, Bayesian inference, particle-based Monte Carlo sampling, and many more. With modern data collection techniques, a large amount of data is available as the input in inverse problems, which creates great needs of data driven optimization methods. In this mini-symposium, we focus on discussions of numerical methods related to data driven stochastic optimization and explore applications of data driven stochastic optimization methods in science and engineering.
16:30
Quasi Monte Carlo Stochastic Gradient Descent Method for Optimal Control Problem
Yanzhao Cao | Auburn University | United States
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Yanzhao Cao | Auburn University | United States
In this talk, we consider the quasi Monte Carlo sample process on the stochastic gradient descent method of optimal controls for systems governed by PDEs with random coefficients. It is shown that QMC will have the effect that reducing the noise in in SGD.
17:00
Adaptive multi-fidelity surrogate modeling for Bayesian inference in inverse problems
Tao Zhou | Chinese Academy of Sciences | China
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Tao Zhou | Chinese Academy of Sciences | China
The polynomial chaos expansion is widely used as a surrogate model in the Bayesian inference to speed up the Markov chain Monte Carlo calculations. However, the use of such a surrogate introduces modeling errors that may severely distort the estimate of the posterior distribution. In this talk, we present an adaptive procedure to construct a multi-fidelity polynomial surrogate. More precisely, the new strategy starts with a low-fidelity surrogate model, and this surrogate will be adaptively corrected using online high-fidelity data. The key idea is to construct and refine the multi-fidelity surrogate over a sequence of samples adaptively determined from data so that the approximation can eventually concentrate to the posterior distribution. We also introduce a multi-fidelity surrogate based on the deep neural networks to deal with problems with high dimensional parameters.
17:30
Low-Rank Functional Representations for Sensitivity Analysis in Earth System Models
Cosmin Safta | Sandia National Laboratory | United States
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Cosmin Safta | Sandia National Laboratory | United States
Uncertainty Quantification studies using state-of-the-art earth system models are challenged by the large number of parameters in these models and the high computational cost required for each model evaluation. Previous sensitivity analysis studies of the land component of the Energy Exascale Earth System Model (E3SM) highlighted non-linear input-output dependencies, which limited the accuracy of surrogate models constructed for subsequent uncertainty quantification and propagation. Here, we present a low-rank functional decomposition approach for constructing sparse surrogate models to simulate the input-output maps in the E3SM land model and uncover interactions between model components and the parameters that control them. We explore tree-based functional representations over the stochastic space that also include space and time dependencies. We discuss challenges in optimizing these low-rank representations including efforts to uncover interactions between model components through a tree-based representation.
18:00
Algorithms for Wave Scattering of Random Media: A FMM for layered media and a phase shift DNN for high frequency learning
Wei Cai | Southern Methodist University | United States
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Wei Cai | Southern Methodist University | United States
In this talk, we will present two algorithms and numerical results for solving electromagnetic wave scattering of random meta-materials. Firstly, a fast multipole method (FMM) for 3-D Helmholtz equation for layered media will be presented based on new multipole expansion (ME) and multipole to local translation (M2L) operators for layered media Green's functions. Secondly, a parallel phase shift deep neural network (PhaseDNN) is proposed for wideband data learning. In order to achieve uniform convergence for low to high frequency content of data, phase shifts are used to convert high frequency learning to low frequency learning. Due to the fast learning of many DNNs in the low frequency range, PhaseDNN is able to learn wideband data uniformly in from low to high frequencies.