Waad Subber

University of Notre Dame, Aerospace and Mechanical Engineering, Post-Doc

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Papers by Waad Subber

Data-based Discovery of Governing Equations

Most common mechanistic models are traditionally presented in mathematical forms to explain a giv... more Most common mechanistic models are traditionally presented in mathematical forms to explain a given physical phenomenon. Machine learning algorithms, on the other hand, provide a mechanism to map the input data to output without explicitly describing the underlying physical process that generated the data. We propose a Data-based Physics Discovery (DPD) framework for automatic discovery of governing equations from observed data. Without a prior definition of the model structure, first a free-form of the equation is discovered, and then calibrated and validated against the available data. In addition to the observed data, the DPD framework can utilize available prior physical models, and domain expert feedback. When prior models are available, the DPD framework can discover an additive or multiplicative correction term represented symbolically. The correction term can be a function of the existing input variable to the prior model, or a newly introduced variable. In case a prior mode...

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Data-Informed Decomposition for Localized Uncertainty Quantification of Dynamical Systems

Industrial dynamical systems often exhibit multi-scale response due to material heterogeneities, ... more Industrial dynamical systems often exhibit multi-scale response due to material heterogeneities, operation conditions and complex environmental loadings. In such problems, it is the case that the smallest length-scale of the systems dynamics controls the numerical resolution required to effectively resolve the embedded physics. In practice however, high numerical resolutions is only required in a confined region of the system where fast dynamics or localized material variability are exhibited, whereas a coarser discretization can be sufficient in the rest majority of the system. To this end, a unified computational scheme with uniform spatio-temporal resolutions for uncertainty quantification can be very computationally demanding. Partitioning the complex dynamical system into smaller easier-to-solve problems based of the localized dynamics and material variability can reduce the overall computational cost. However, identifying the region of interest for high-resolution and intensiv...

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Domain decomposition methods for uncertainty quantification

A thesis subm itted to the Faculty of Graduate and Postdoctoral Affairs in partial fulfillment o ... more

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Industrial Applications of Intelligent Adaptive Sampling Methods for Multi-Objective Optimization

Design Engineering and Manufacturing [Working Title], 2019

Multi-objective optimization is an essential component of nearly all engineering design. However,... more Multi-objective optimization is an essential component of nearly all engineering design. However, for industrial applications, the design process typically demands running expensive computer code and/or real-world experiments putting the design process at risk of finding suboptimal solutions and/or not meeting budget constraints. As a first step toward a remedy, meta-models are built to mimic the response surface at a much lower query cost. We cover a time-tested technology specifically tailored to limited-data scenarios called Bayesian hybrid modeling (GEBHM) developed and maintained at General Electric (GE) research. GEBHM offers Bayesian mean and principled uncertainty predictions allowing a second technology called intelligent design and analysis of experiments (GE-IDACE/ IDACE) to perform the optimization task using an adaptive sampling strategy. This chapter first covers the theoretical framework of both GEBHM and GE-IDACE. Then, the impact of GEBHM/GE-IDACE is demonstrated on multiple real-world engineering applications including additive manufacturing, combustion testing, and computational fluid dynamic design modeling. GEBHM and GE-IDACE are used daily and extensively within GE with huge impact in the form of 30-90% cost reduction and superior engineering designs of competitive products.

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Uncertainty quantification of the reverse Taylor impact test and localized asynchronous space-time algorithm

The reverse Taylor impact is a common experiment to investigate the dynamical response of materia... more The reverse Taylor impact is a common experiment to investigate the dynamical response of materials at high strain rates. To better understand the physical phenomena and to provide a platform for code validation and Uncertainty Quantification (UQ), a co-designed simulation and experimental paradigm is investigated. For validation under uncertainty, quantities of interest (QOIs) within subregions of the computational domain are introduced. When regions of interest can be identified, the computational cost for UQ can be reduced by confining the random variability within these regions. This observation inspired us to develop an asynchronous space and time computational algorithm with localized UQ. In every region of interest, high resolution space and time discretization schemes are used for a stochastic model. Apart from the regions of interest, low spatial and temporal resolutions are allowed for a stochastic model with low dimensional representation of uncertainty. The approach is exercised on a linear elastodynamics problem and shows a potential in reducing the UQ computational cost.

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A scalable parallel uncertainty analysis and data assimilation framework applied to some hydraulic problems

A signiﬁcant uncertainty exists in numerical predictions of hydraulic and hydrologic phenomena su... more A signiﬁcant uncertainty exists in numerical predictions of hydraulic and hydrologic phenomena such as solute transport in river and subsurface ﬂow due to (a) insufﬁcient knowledge on the spatio-temporal variability of the hydraulic and geological parameters across multiple length scales and (b) lack of complete knowledge of the physics and mechanics of the associated phenomena. The conventional approach of treating the system parameters as deterministic quantities may therefore be unacceptable for such models. Due to the uncertainty and complexity of such spatially-distributed interconnected systems, it is necessary to quantify conﬁdence in computer simulation models for their acceptability as reliable alternatives to ﬁeld experiments. This paperdescribes a scalable parallel uncertainty quantiﬁcation and data assimilation framework for some hydraulicand hydrologic systems that can exploit modern supercomputers and blend sensor data to reduce uncertainty in numerical predictions.

Schwarz Preconditioner for the Stochastic Finite Element Method

Lecture Notes in Computational Science and Engineering, 2016

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Data-driven discovery of free-form governing differential equations

ArXiv, 2019

We present a method of discovering governing differential equations from data without the need to... more We present a method of discovering governing differential equations from data without the need to specify a priori the terms to appear in the equation. The input to our method is a dataset (or ensemble of datasets) corresponding to a particular solution (or ensemble of particular solutions) of a differential equation. The output is a human-readable differential equation with parameters calibrated to the individual particular solutions provided. The key to our method is to learn differentiable models of the data that subsequently serve as inputs to a genetic programming algorithm in which graphs specify computation over arbitrary compositions of functions, parameters, and (potentially differential) operators on functions. Differential operators are composed and evaluated using recursive application of automatic differentiation, allowing our algorithm to explore arbitrary compositions of operators without the need for human intervention. We also demonstrate an active learning process ...

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Remarks for Scaling Up a General Gaussian Process to Model Large Dataset with Sub-models

Efficient Bayesian Inverse Method Using Robust Gaussian Processes For Design Under Uncertainty

A Domain Decomposition Method of Stochastic PDEs: An Iterative Solution Technique Using A Two-level

Recent advances in high performance computing systems and sensing technologies motivate computati... more Recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions. A two-level domain decomposition method is reported in this investigation to devise a linear solver for the large-scale system in the Galerkin spectral stochastic finite element method (SSFEM). In particular, a two-level scalable preconditioner is introduced in order to iteratively solve the largescale linear system in the intrusive SSFEM using an iterative substructuring based domain decomposition solver. The implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains. The numerical and parallel scalabilities of

Data-based Discovery of Governing Equations

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Advances in Bayesian Probabilistic Modeling for Industrial Applications

ASCE-ASME J Risk and Uncert in Engrg Sys Part B Mech Engrg, Mar 26, 2020

Industrial applications frequently pose a notorious challenge for state-of-the-art methods in the... more Industrial applications frequently pose a notorious challenge for state-of-the-art methods in the contexts of optimization, designing experiments and modeling unknown physical response. This problem is aggravated by limited availability of clean data, uncertainty in available physics-based models and additional logistic and computational expense associated with experiments. In such a scenario, Bayesian methods have played an impactful role in alleviating the aforementioned obstacles by quantifying uncertainty of different types under limited resources. These methods, usually deployed as a framework, allows decision makers to make informed choices under uncertainty while being able to incorporate information on the the fly, usually in the form of data, from multiple sources while being consistent with the physical intuition about the problem. This is a major advantage that Bayesian methods bring to fruition especially in the industrial context. This paper is a compendium of the Bayesian modeling methodology that is being consistently developed at GE Research. The methodology, called GE's Bayesian Hybrid Modeling (GEBHM), is a probabilistic modeling method, based on the Kennedy and O'Hagan framework, that has been continuously scaled-up and industrialized over several years. In this work, we explain the various advancements in GEBHM's methods and demonstrate their impact on several challenging industrial problems.

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Bayesian-entropy gaussian process for constrained metamodeling

Reliability Engineering & System Safety

A Strategy for Adaptive Sampling of Multi-Fidelity Gaussian Processes to Reduce Predictive Uncertainty

Volume 2B: 45th Design Automation Conference

Multi-fidelity Gaussian process (GP) modeling is a common approach to employ in resource-expensiv... more Multi-fidelity Gaussian process (GP) modeling is a common approach to employ in resource-expensive computationally demanding algorithms such as optimization, calibration and uncertainty quantification where multiple datasets of varying fidelities are encountered. Briefly, in its simplest form, a multi-fidelity GP is trained on two separate sources of datasets each with its own fidelity level, e.g., a software code/simulator for the low-fidelity source and real-world experiments for the high-fidelity source. Adaptive sampling for multi-fidelity Gaussian processes is a challenging task since we not only seek to estimate the next sampling location of the design variable, but also account for the data fidelities. This issue is often addressed by including the cost of the data sources as an another element in the search criterion in conjunction with an uncertainty reduction metric. In this work, we extent the traditional design of experiment framework for multi-fidelity GPs by partitioni...

Chemo-thermal model and Gaussian process emulator for combustion synthesis of Ni/Al composites

Combustion and Flame

A parallel time integrator for noisy nonlinear oscillatory systems

Journal of Computational Physics

Asynchronous space–time domain decomposition method with localized uncertainty quantification

Computer Methods in Applied Mechanics and Engineering

Domain Decomposition Methods of Stochastic PDEs

Lecture Notes in Computational Science and Engineering, 2013

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Schwarz preconditioners for stochastic elliptic PDEs

Computer Methods in Applied Mechanics and Engineering, 2014

Increasingly the spectral stochastic finite element method (SSFEM) has become a popular computati... more Increasingly the spectral stochastic finite element method (SSFEM) has become a popular computational tool for uncertainty quantification in numerous practical engineering problems. For large-scale problems however, the computational cost associated with solving the arising linear system in the SSFEM still poses a significant challenge. The development of efficient and robust preconditioned iterative solvers for the SSFEM linear system is thus of paramount importance for uncertainty quantification of large-scale industrially relevant problems. In the context of high performance computing, the preconditioner must scale to a large number of processors. Therefore in this paper, a two-level additive Schwarz preconditioner is described for the iterative solution of the SSFEM linear system. The proposed preconditioner can be viewed as a generalization of the mean based block-diagonal preconditioner commonly used in the literature. For the numerical illustrations, two-dimensional steady-state diffusion and elasticity problems with spatially varying random coefficients are considered. The performance of the algorithm is investigated with respect to the geometric parameters, strength of randomness, dimension and order of the stochastic expansion.

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