| [1] |
Peherstorfer, B., Willcox, K. & Gunzburger, M. Survey of multifidelity methods in uncertainty propagation, inference, and optimization.
SIAM Review, 60(3):550-591, 2018. [Abstract]Abstract
In many situations across computational science and engineering, multiple computational models are available that describe a system of interest. These different models have varying evaluation costs and varying fidelities. Typically, a computationally expensive high-fidelity model describes the system with the accuracy required by the current application at hand, while lower-fidelity models are less accurate but computationally cheaper than the high-fidelity model. Outer-loop applications, such as optimization, inference, and uncertainty quantification, require multiple model evaluations at many different inputs, which often leads to computational demands that exceed available resources if only the high-fidelity model is used. This work surveys multifidelity methods that accelerate the solution of outer-loop applications by combining high-fidelity and low-fidelity model evaluations, where the low-fidelity evaluations arise from an explicit low-fidelity model (e.g., a simplified physics approximation, a reduced model, a data-fit surrogate, etc.) that approximates the same output quantity as the high-fidelity model. The overall premise of these multifidelity methods is that low-fidelity models are leveraged for speedup while the high-fidelity model is kept in the loop to establish accuracy and/or convergence guarantees. We categorize multifidelity methods according to three classes of strategies: adaptation, fusion, and filtering. The paper reviews multifidelity methods in the outer-loop contexts of uncertainty propagation, inference, and optimization. [BibTeX]@article{PWG17MultiSurvey,
title = {Survey of multifidelity methods in uncertainty propagation, inference, and optimization},
author = {Peherstorfer, B. and Willcox, K. and Gunzburger, M.},
journal = {SIAM Review},
volume = {60},
number = {3},
pages = {550-591},
year = {2018},
} |
| [2] |
Peherstorfer, B. & Marzouk, Y. A transport-based multifidelity preconditioner for Markov chain Monte Carlo.
Advances in Computational Mathematics, 45:2321-2348, 2019. [Abstract]Abstract
Markov chain Monte Carlo (MCMC) sampling of posterior distributions arising in Bayesian inverse problems is challenging when evaluations of the forward model are computationally expensive. Replacing the forward model with a low-cost, low-fidelity model often significantly reduces computational cost; however, employing a low-fidelity model alone means that the stationary distribution of the MCMC chain is the posterior distribution corresponding to the low-fidelity model, rather than the original posterior distribution corresponding to the high-fidelity model. We propose a multifidelity approach that combines, rather than replaces, the high-fidelity model with a low-fidelity model. First, the low-fidelity model is used to construct a transport map that deterministically couples a reference Gaussian distribution with an approximation of the low-fidelity posterior. Then, the high-fidelity posterior distribution is explored using a non-Gaussian proposal distribution derived from the transport map. This multifidelity preconditioned MCMC approach seeks efficient sampling via a proposal that is explicitly tailored to the posterior at hand and that is constructed efficiently with the low-fidelity model. By relying on the low-fidelity model only to construct the proposal distribution, our approach guarantees that the stationary distribution of the MCMC chain is the high-fidelity posterior. In our numerical examples, our multifidelity approach achieves significant speedups compared to single-fidelity MCMC sampling methods. [BibTeX]@article{PM18MultiTM,
title = {A transport-based multifidelity preconditioner for Markov chain Monte Carlo},
author = {Peherstorfer, B. and Marzouk, Y.},
journal = {Advances in Computational Mathematics},
volume = {45},
pages = {2321-2348},
year = {2019},
} |
| [3] |
Peherstorfer, B. Multifidelity Monte Carlo estimation with adaptive low-fidelity models.
SIAM/ASA Journal on Uncertainty Quantification, 7:579-603, 2019. [Abstract]Abstract
Multifidelity Monte Carlo (MFMC) estimation combines low- and high-fidelity models to speedup the estimation of statistics of the high-fidelity model outputs. MFMC optimally samples the low- and high-fidelity models such that the MFMC estimator has minimal mean-squared error for a given computational budget. In the setup of MFMC, the low-fidelity models are static, i.e., they are given and fixed and cannot be changed and adapted. We introduce the adaptive MFMC (AMFMC) method that splits the computational budget between adapting the low-fidelity models to improve their approximation quality and sampling the low- and high-fidelity models to reduce the mean-squared error of the estimator. Our AMFMC approach derives the quasi-optimal balance between adaptation and sampling in the sense that our approach minimizes an upper bound of the mean-squared error, instead of the error directly. We show that the quasi-optimal number of adaptations of the low-fidelity models is bounded even in the limit case that an infinite budget is available. This shows that adapting low-fidelity models in MFMC beyond a certain approximation accuracy is unnecessary and can even be wasteful. Our AMFMC approach trades-off adaptation and sampling and so avoids over-adaptation of the low-fidelity models. Besides the costs of adapting low-fidelity models, our AMFMC approach can also take into account the costs of the initial construction of the low-fidelity models (``offline costs''), which is critical if low-fidelity models are computationally expensive to build such as reduced models and data-fit surrogate models. Numerical results demonstrate that our adaptive approach can achieve orders of magnitude speedups compared to MFMC estimators with static low-fidelity models and compared to Monte Carlo estimators that use the high-fidelity model alone. [BibTeX]@article{P19AMFMC,
title = {Multifidelity Monte Carlo estimation with adaptive low-fidelity models},
author = {Peherstorfer, B.},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
volume = {7},
pages = {579-603},
year = {2019},
} |
