08:30
Multilevel Monte Carlo Sampling on Heterogeneous Computer Architectures
Christiane Adcock | Stanford University | United States
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Authors:
Christiane Adcock | Stanford University | United States
Gianluca Iaccarino | Stanford University | United States
Lluis Jofre | Stanford University | United States
Manolis Papadakis | Stanford University | United States
Yinyu Ye | Stanford University | United States
Multilevel Monte Carlo (MLMC) propagates uncertainty in model parameters using a hierarchy of models, which increase in accuracy and computational cost. The theory developed for MLMC assumes there is a single fixed computational cost to evaluate a model. We extend MLMC to the modern computational setting consisting of multiple computational units and accelerators, in which there are different options to evaluate a model. The costs are determined by the combination of heterogeneous processors, such as CPUs and GPUs, used in the evaluation. We determine for each level the optimal number of samples to evaluate on that level, and for each sample the optimal number of processors of each type to use to evaluate that sample, both to minimize cost for fixed total variance. We show that the common practice in parallel computing of maximizing the number of processors used per problem evaluation leads to unnecessarily high cost for MLMC and propose an alternative method which reduces the cost. We present verification and scaling studies applying the extended MLMC method to stochastic partial differential equations in the portable task-based parallel computing environment, Legion.
08:50
Efficient Automatized Iterative Multilevel and Multifidelity Monte Carlo Simulations for the Compressible Navier Stokes Equations
Jakob Duerrwaechter | Universität Stuttgart | Germany
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Jakob Duerrwaechter | Universität Stuttgart | Germany
Fabian Meyer | Universität Stuttgart | Germany
Thomas Kuhn | Universität Stuttgart | Germany
Andrea Beck | Universität Stuttgart | Germany
Claus-Dieter Munz | Universität Stuttgart | Germany
Christian Rohde | Universität Stuttgart | Germany
In this presentation, we compare different non-intrusive methods for forward uncertainty propagation in computational fluid dynamics. Specifically, we investigate Multilevel Monte Carlo and Multifidelity Monte Carlo methods. As a baseline solver, we use the high order large eddy simulation code FLEXI.
We present our framework POUNCE (Propagation of Uncertainties), which is a management environment for fully automatized non-intrusive UQ simulations on high performance computing systems. Its modular design allows for the implementation of different UQ methods (or even other methods, such as optimization), as well as to run it with different baseline numerical solvers and on different machines.
In Multilevel Monte Carlo and Multifidelity Monte Carlo, the number of samples for every model/level is crucial to the efficiency of the code. Often this number is solution dependent, which necessitates an iterative alogirthm. One focus of POUNCE lies on the efficient scheduling in this iterative multi-resolution problem setting.
We first demonstrate and compare the non-intrusive methods on simple test problems.
As a more complex application, we show non-intrusive UQ simulations of cavity aeroaoustics with uncertain inflow and geometry and compare them to experimental data. We also show preliminary results of wind energy airfoil performance under uncertain icing conditions.
09:10
Generalized Multi-Model Monte Carlo Simulation for Uncertainty Propagation
James Warner | NASA Langley Research Center | United States
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Authors:
James Warner | NASA Langley Research Center | United States
Geoffrey Bomarito | NASA Langley Research Center | United States
Patrick Leser | NASA Langley Research Center | United States
William Leser | NASA Langley Research Center | United States
Anthony Williams | NASA Langley Research Center | United States
This work focuses on the development of a general capability for computing the statistics of outputs from an expensive, high-fidelity model by leveraging faster, low-fidelity models for speedup. For instance, the low-fidelity models could arise from coarsened discretizations in space/time (e.g., Multilevel Monte Carlo - MLMC), or from general data-driven or reduced order models (e.g., Multifidelity Monte Carlo - MFMC). Given a fixed computational budget and a collection of models of varying cost/accuracy, the goal is to determine a sample allocation strategy that results in an estimator with optimal variance reduction. The foundation of the proposed approach is from recent work introducing an approximate control variate (ACV) framework that unified and improved upon traditional MLMC-based/MFMC methods [1]. The numerical optimization problem for optimal sample allocation required by the ACV method is revisited here in an effort to provide more robust schemes that yield improved estimator performance. An open-source Python library for general multi-model uncertainty propagation is introduced that allows a user to easily and efficiently search for an optimal sample allocation and then seamlessly compare its performance to existing methods. The effectiveness of the proposed approach and Python library is demonstrated on a trajectory simulation application where precise, efficient predictions are required.
[1] A. Gorodetsky et al. arXiv:1811.04988v3
09:30
Multi-Fidelity Monte Carlo with Semi-intrusive Algorithm for Multi-scale Simulation
Dongwei Ye | University of Amsterdam | Netherlands
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Dongwei Ye | University of Amsterdam | Netherlands
Anna Nikishova | University of Amsterdam | Netherlands
Lourens Veen | Netherlands eScience Center | Netherlands
Alfons Hoekstra | University of Amsterdam | Netherlands
A standard approach to implement uncertainty quantification is the Monte Carlo method. However, in terms of the multi-scale simulation, the computational model itself is already computational expensive because of the refined grid model at micro-scale or a faster dynamics model which have to run a lot of time before the slow dynamics move to the next time step. The semi-intrusive algorithm [1] has demonstrated its effectiveness for such multi-scale scenarios. The algorithm replaces the most expensive part of the multi-scale model by a surrogate model. The replacement of the submodel inevitably introduces and propagates the error during the computation and therefore leads to a response with certain fidelity. Different levels of the surrogate models result in different fidelity of the final output which can be coupled with multi-fidelity Monte Carlo method (MFMC) [2]. We present our multi-fidelity Monte Carlo with semi-intrusive algorithm and its application to the reaction-diffusion model. The speedup and accuracy of the uncertainty estimation will be shown and compared with a black-box Monte Carlo result and a semi-intrusive algorithm result.
[1] Nikishova, Anna, and Alfons G. Hoekstra. "Semi-intrusive Uncertainty Propagation for multiscale models." Journal of Computational Science (2019).
[2] Peherstorfer, Benjamin, Karen Willcox, and Max Gunzburger. "Survey of multi-fidelity methods in uncertainty propagation, inference, and optimization." Siam Review 60.3 (2018): 550-591.]
09:50
Multifidelity Model-Informed Neural Network in Reduced Order Modeling
Xueyu Zhu | University of Iowa | United States
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Xueyu Zhu | University of Iowa | United States
Chuan Liu | University of Iowa | United States
In this talk, we will discuss a neural network-based reduced basis method with multi-fidelity models to accurately approximate the reduced solutions. The combination of low/high-fidelity solutions helps improve the accuracy of the neural networks as the approximator, and this method demonstrates its ability to produce accurate results with a limited number of high-fidelity simulations with an affordable computational cost. We also provide extensive numerical examples to illustrate the effectiveness of this method.
10:10
Multifidelity Approximate Bayesian Computation
Thomas Prescott | University of Oxford | United Kingdom
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Thomas Prescott | University of Oxford | United Kingdom
Ruth Baker | University of Oxford | United Kingdom
A vital stage in the mathematical modelling of real-world systems is to calibrate a model's parameters to observed data. Likelihood-free parameter inference methods, such as Approximate Bayesian Computation, build Monte Carlo samples of the uncertain parameter distribution by comparing the data with large numbers of model simulations. However, the computational expense of generating these simulations forms a significant bottleneck in the practical application of such methods. We identify how simulations of computationally cheap low-fidelity models can be used to speed the construction of Monte Carlo samples without introducing further bias. We characterise the optimal choice of how often to simulate from cheap, low-fidelity models in place of expensive, high-fidelity models, thereby extending existing multifidelity ABC methods to allow for early acceptance. The resulting early accept/reject multifidelity ABC algorithm is shown to give improved performance over existing multifidelity and high-fidelity approaches to model calibration.
10:30
h- and p- Multilevel Monte Carlo Methods in Geotechnical Engineering
Philippe Blondeel | KU Leuven | Belgium
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Authors:
Philippe Blondeel | KU Leuven | Belgium
Pieterjan Robbe | KU Leuven | Belgium
Stijn François | KU Leuven | Belgium
Geert Lombaert | KU Leuven | Belgium
Stefan Vandewalle | KU Leuven | Belgium
Problems in geotechnical engineering are often characterized by significant uncertainties such as the soil’s Young’s modulus and its cohesion. As such the accurate modeling of those uncertainties is of particular importance. A standard method to account for this uncertainty is the classical Monte Carlo method. While praised for not being intrusive or being affected by the curse of dimensionality, this method suffers from a slow convergence rate. In order to overcome this slow convergence, improved sampling methods known as variance reduction schemes have been devised. A particular successful example of such a scheme is the family of the Multilevel/Multi-Index Monte Carlo methods. This category of methods uses a hierarchical progression of finite element meshes of increasing resolution in order to accelerate the computation. Many computational cheap samples are taken on the coarser meshes and few computational expensive samples are taken on the finer meshes. We apply the Multilevel/Multi-Index (Quasi) Monte Carlo methods to a geotechnical slope stability problem where the goal is to assess the stability of a natural or man made excavation. The uncertainty resides in the soil’s cohesion and is modeled by means of a log-normal random field generated from a Karhunen-Loève expansion. The mesh hierarchy consists of an h- and a p-refined mesh. We compare the computational costs and run times of the Multilevel/Multi-Index methods with those of the classical Monte Carlo method.