arXiv subjects

Alireza Doostan

Publications and source records attributed to Alireza Doostan.

At least 55 records · Page 3Linked to original sources

arXiv2021

Bi-fidelity Reduced Polynomial Chaos Expansion for Uncertainty Quantification

A ubiquitous challenge in design space exploration or uncertainty quantification of complex engineering problems is the minimization of computational cost. A useful tool to ease the burden of solving such systems is model reduction. This work considers a stochastic model reduction method (SMR), in the context of polynomial chaos (PC) expansions, where low-fidelity (LF) samples are leveraged to form a stochastic reduced basis. The reduced basis enables the construction of a bi-fidelity (BF) estimate of a quantity of interest from a small number of high-fidelity (HF) samples. A successful BF estimate approximates the quantity of interest with accuracy comparable to the HF model and computational expense close to the LF model. We develop new error bounds for the SMR approach and present a procedure to practically utilize these bounds in order to assess the appropriateness of a given pair of LF and HF models for BF estimation. The effectiveness of the SMR approach, and the utility of the error bound are presented in three numerical examples.

math.NA↗

arXiv2021

Prediction of Ultrasonic Guided Wave Propagation in Solid-fluid and their Interface under Uncertainty using Machine Learning

Structural health monitoring (SHM) systems use the non-destructive testing principle for damage identification. As part of SHM, the propagation of ultrasonic guided waves (UGWs) is tracked and analyzed for the changes in the associated wave pattern. These changes help identify the location of a structural damage, if any. We advance existing research by accounting for uncertainty in the material and geometric properties of a structure. The physics model used in this study comprises of a monolithically coupled system of acoustic and elastic wave equations, known as the wave propagation in fluid-solid and their interface (WpFSI) problem. As the UGWs propagate in the solid, fluid, and their interface, the wave signal displacement measurements are contrasted against the benchmark pattern. For the numerical solution, we develop an efficient algorithm that successfully addresses the inherent complexity of solving the multiphysics problem under uncertainty. We present a procedure that uses Gaussian process regression and convolutional neural network for predicting the UGW propagation in a solid-fluid and their interface under uncertainty. First, a set of training images for different realizations of the uncertain parameters of the inclusion inside the structure is generated using a monolithically-coupled system of acoustic and elastic wave equations. Next, Gaussian processes trained with these images are used for predicting the propagated wave with convolutional neural networks for further enhancement to produce high-quality images of the wave patterns for new realizations of the uncertainty. The results indicate that the proposed approach provides an accurate prediction for the WpFSI problem in the presence of uncertainty.

eess.SP↗

arXiv2021

Task-parallel in-situ temporal compression of large-scale computational fluid dynamics data

Present day computational fluid dynamics simulations generate extremely large amounts of data, sometimes on the order of TB/s. Often, a significant fraction of this data is discarded because current storage systems are unable to keep pace. To address this, data compression algorithms can be applied to data arrays containing flow quantities of interest to reduce the overall amount of storage. Compression methods either exactly reconstruct the original dataset (lossless compression) or provide an approximate representation of the original dataset (lossy compression). The matrix column interpolative decomposition (ID) can be implemented as a type of lossy compression for data matrices that factors the original data matrix into a product of two smaller factor matrices. One of these matrices consists of a subset of the columns of the original data matrix, while the other is a coefficient matrix which approximates the columns of the original data matrix as linear combinations of the selected columns. Motivating this work is the observation that the structure of ID algorithms makes them a natural fit for the asynchronous nature of task-based parallelism; they are able to operate independently on sub-domains of the system of interest and, as a result, provide varied levels of compression. Using the task-based Legion programming model, a single-pass ID algorithm (SPID) for CFD applications is implemented. Performance studies, scalability, and the accuracy of the compression algorithms are presented for an analytical Taylor-Green vortex problem, followed by a large-scale implementation of a compressible Taylor-Green vortex using a high-order Navier-Stokes solver. In both cases, compression factors exceeding 100 are achieved with relative errors at or below 10e-3. Moreover, strong and weak scaling results demonstrate that introducing SPID to solvers leads to negligible increases in runtime.

cs.CE↗

arXiv2020

Sparse Identification of Nonlinear Dynamical Systems via Reweighted $\ell_1$-regularized Least Squares

This work proposes an iterative sparse-regularized regression method to recover governing equations of nonlinear dynamical systems from noisy state measurements. The method is inspired by the Sparse Identification of Nonlinear Dynamics (SINDy) approach of {\it [Brunton et al., PNAS, 113 (15) (2016) 3932-3937]}, which relies on two main assumptions: the state variables are known {\it a priori} and the governing equations lend themselves to sparse, linear expansions in a (nonlinear) basis of the state variables. The aim of this work is to improve the accuracy and robustness of SINDy in the presence of state measurement noise. To this end, a reweighted $\ell_1$-regularized least squares solver is developed, wherein the regularization parameter is selected from the corner point of a Pareto curve. The idea behind using weighted $\ell_1$-norm for regularization -- instead of the standard $\ell_1$-norm -- is to better promote sparsity in the recovery of the governing equations and, in turn, mitigate the effect of noise in the state variables. We also present a method to recover single physical constraints from state measurements. Through several examples of well-known nonlinear dynamical systems, we demonstrate empirically the accuracy and robustness of the reweighted $\ell_1$-regularized least squares strategy with respect to state measurement noise, thus illustrating its viability for a wide range of potential applications.

stat.ML↗

arXiv2020

S-Frame Discrepancy Correction Models for Data-Informed Reynolds Stress Closure

Despite their well-known limitations, RANS models remain the most commonly employed tool for modeling turbulent flows in engineering practice. RANS models are predicated on the solution of the RANS equations, but these equations involve an unclosed term, the Reynolds stress tensor, which must be modeled. The Reynolds stress tensor is often modeled as an algebraic function of mean flow field variables and turbulence variables. This introduces a discrepancy between the Reynolds stress tensor predicted by the model and the exact Reynolds stress tensor. This discrepancy can result in inaccurate mean flow field predictions. In this paper, we introduce a data-informed approach for arriving at Reynolds stress models with improved predictive performance. Our approach relies on learning the components of the Reynolds stress discrepancy tensor associated with a given Reynolds stress model in the mean strain-rate tensor eigenframe. These components are typically smooth and hence simple to learn using state-of-the-art machine learning strategies and regression techniques. Our approach automatically yields Reynolds stress models that are symmetric, and it yields Reynolds stress models that are both Galilean and frame invariant provided the inputs are themselves Galilean and frame invariant. To arrive at computable models of the discrepancy tensor, we employ feed-forward neural networks and an input space spanning the integrity basis of the mean strain-rate tensor, the mean rotation-rate tensor, the mean pressure gradient, and the turbulent kinetic energy gradient, and we introduce a framework for dimensional reduction of the input space to further reduce computational cost. Numerical results illustrate the effectiveness of the proposed approach for data-informed Reynolds stress closure for a suite of turbulent flow problems of increasing complexity.

physics.flu-dyn↗

arXiv2020

On transfer learning of neural networks using bi-fidelity data for uncertainty propagation

Due to their high degree of expressiveness, neural networks have recently been used as surrogate models for mapping inputs of an engineering system to outputs of interest. Once trained, neural networks are computationally inexpensive to evaluate and remove the need for repeated evaluations of computationally expensive models in uncertainty quantification applications. However, given the highly parameterized construction of neural networks, especially deep neural networks, accurate training often requires large amounts of simulation data that may not be available in the case of computationally expensive systems. In this paper, to alleviate this issue for uncertainty propagation, we explore the application of transfer learning techniques using training data generated from both high- and low-fidelity models. We explore two strategies for coupling these two datasets during the training procedure, namely, the standard transfer learning and the bi-fidelity weighted learning. In the former approach, a neural network model mapping the inputs to the outputs of interest is trained based on the low-fidelity data. The high-fidelity data is then used to adapt the parameters of the upper layer(s) of the low-fidelity network, or train a simpler neural network to map the output of the low-fidelity network to that of the high-fidelity model. In the latter approach, the entire low-fidelity network parameters are updated using data generated via a Gaussian process model trained with a small high-fidelity dataset. The parameter updates are performed via a variant of stochastic gradient descent with learning rates given by the Gaussian process model. Using three numerical examples, we illustrate the utility of these bi-fidelity transfer learning methods where we focus on accuracy improvement achieved by transfer learning over standard training approaches.

stat.ML↗

arXiv2020

Pass-efficient methods for compression of high-dimensional turbulent flow data

The future of high-performance computing, specifically on future Exascale computers, will presumably see memory capacity and bandwidth fail to keep pace with data generated, for instance, from massively parallel partial differential equation (PDE) systems. Current strategies proposed to address this bottleneck entail the omission of large fractions of data, as well as the incorporation of $\textit{in situ}$ compression algorithms to avoid overuse of memory. To ensure that post-processing operations are successful, this must be done in a way that a sufficiently accurate representation of the solution is stored. Moreover, in situations where the input/output system becomes a bottleneck in analysis, visualization, etc., or the execution of the PDE solver is expensive, the the number of passes made over the data must be minimized. In the interest of addressing this problem, this work focuses on the utility of pass-efficient, parallelizable, low-rank, matrix decomposition methods in compressing high-dimensional simulation data from turbulent flows. A particular emphasis is placed on using coarse representation of the data -- compatible with the PDE discretization grid -- to accelerate the construction of the low-rank factorization. This includes the presentation of a novel single-pass matrix decomposition algorithm for computing the so-called interpolative decomposition. The methods are described extensively and numerical experiments on two turbulent channel flow data are performed. In the first (unladen) channel flow case, compression factors exceeding $400$ are achieved while maintaining accuracy with respect to first- and second-order flow statistics. In the particle-laden case, compression factors of 100 are achieved and the compressed data is used to recover particle velocities.

physics.comp-ph↗

arXiv2019

Bi-fidelity Stochastic Gradient Descent for Structural Optimization under Uncertainty

The presence of uncertainty in material properties and geometry of a structure is ubiquitous. The design of robust engineering structures, therefore, needs to incorporate uncertainty in the optimization process. Stochastic gradient descent (SGD) method can alleviate the cost of optimization under uncertainty, which includes statistical moments of quantities of interest in the objective and constraints. However, the design may change considerably during the initial iterations of the optimization process which impedes the convergence of the traditional SGD method and its variants. In this paper, we present two SGD based algorithms, where the computational cost is reduced by employing a low-fidelity model in the optimization process. In the first algorithm, most of the stochastic gradient calculations are performed on the low-fidelity model and only a handful of gradients from the high-fidelity model are used per iteration, resulting in an improved convergence. In the second algorithm, we use gradients from low-fidelity models to be used as control variate, a variance reduction technique, to reduce the variance in the search direction. These two bi-fidelity algorithms are illustrated first with a conceptual example. Then, the convergence of the proposed bi-fidelity algorithms is studied with two numerical examples of shape and topology optimization and compared to popular variants of the SGD method that do not use low-fidelity models. The results show that the proposed use of a bi-fidelity approach for the SGD method can improve the convergence. Two analytical proofs are also provided that show the linear convergence of these two algorithms under appropriate assumptions.

math.OC↗

arXiv2019

Topology Optimization under Uncertainty using a Stochastic Gradient-based Approach

Topology optimization under uncertainty (TOuU) often defines objectives and constraints by statistical moments of geometric and physical quantities of interest. Most traditional TOuU methods use gradient-based optimization algorithms and rely on accurate estimates of the statistical moments and their gradients, e.g., via adjoint calculations. When the number of uncertain inputs is large or the quantities of interest exhibit large variability, a large number of adjoint (and/or forward) solves may be required to ensure the accuracy of these gradients. The optimization procedure itself often requires a large number of iterations, which may render TOuU computationally expensive, if not infeasible. To tackle this difficulty, we here propose an optimization approach that generates a stochastic approximation of the objective, constraints, and their gradients via a small number of adjoint (and/or forward) solves, per iteration. A statistically independent (stochastic) approximation of these quantities is generated at each optimization iteration. The total cost of this approach is only a small factor larger than that of the corresponding deterministic TO problem. We incorporate the stochastic approximation of objective, constraints and their design sensitivities into two classes of optimization algorithms. First, we investigate the stochastic gradient descent (SGD) method and a number of its variants, which have been successfully applied to large-scale optimization problems for machine learning. Second, we study the use of the proposed stochastic approximation approach within conventional nonlinear programming methods, focusing on the Globally Convergent Method of Moving Asymptotes (GCMMA). The performance of these algorithms is investigated with structural design optimization problems utilizing a Solid Isotropic Material with Penalization (SIMP), as well as an explicit level set method.

math.OC↗

arXiv2019

Stochastic Subspace Descent

We present two stochastic descent algorithms that apply to unconstrained optimization and are particularly efficient when the objective function is slow to evaluate and gradients are not easily obtained, as in some PDE-constrained optimization and machine learning problems. The basic algorithm projects the gradient onto a random subspace at each iteration, similar to coordinate descent but without restricting directional derivatives to be along the axes. This algorithm is previously known but we provide new analysis. We also extend the popular SVRG method to this framework but without requiring that the objective function be written as a finite sum. We provide proofs of convergence for our methods under various convexity assumptions and show favorable results when compared to gradient descent and BFGS on non-convex problems from the machine learning and shape optimization literature. We also note that our analysis gives a proof that the iterates of SVRG and several other popular first-order stochastic methods, in their original formulation, converge almost surely to the optimum; to our knowledge, prior to this work the iterates of SVRG had only been known to converge in expectation.

math.OC↗

arXiv2019

Physics-Informed Tolerance Allocation: A Surrogate-Based Framework for the Control of Geometric Variation on System Performance

In this paper, we present a novel tolerance allocation algorithm for the assessment and control of geometric variation on system performance that is applicable to any system of partial differential equations. In particular, we parameterize the geometric domain of the system in terms of design parameters and subsequently measure the effect of design parameter variation on system performance. A surrogate model via a tensor representation is constructed to map the design parameter variation to the system performance. A set of optimization problems over this surrogate model restricted to nested hyperrectangles represents the effect of prescribing design tolerances, where the maximizer of this restricted function depicts the worst-case member, i.e. the worst-case design. Moreover, the loci of these tolerance hyperrectangles with maximizers attaining, but not surpassing, the performance constraint represents the boundary to the feasible region of allocatable tolerances. Every tolerance in this domain is measured through a user-specified, weighted norm which is informed by design considerations such as cost and manufacturability. The boundary of the feasible set is elucidated as an immersed manifold of codimension one, over which a suite of optimization routines exist and are employed to efficiently determine an optimal feasible tolerance with respect to the specified measure. Examples of this algorithm are presented with applications to a plate with a hole described by two design parameters, a plate with a hole described by six design parameters, and an L-Bracket described by seventeen design parameters.

math.NA↗

arXiv2019

Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

Stochastic physical problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zero of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods.

math.NA↗

arXiv2018

Bi-fidelity approximation for uncertainty quantification and sensitivity analysis of irradiated particle-laden turbulence

Efficiently performing predictive studies of irradiated particle-laden turbulent flows has the potential of providing significant contributions towards better understanding and optimizing, for example, concentrated solar power systems. As there are many uncertainties inherent in such flows, uncertainty quantification is fundamental to improve the predictive capabilities of the numerical simulations. For large-scale, multi-physics problems exhibiting high-dimensional uncertainty, characterizing the stochastic solution presents a significant computational challenge as many methods require a large number of high-fidelity solves. This requirement results in the need for a possibly infeasible number of simulations when a typical converged high-fidelity simulation requires intensive computational resources. To reduce the cost of quantifying high-dimensional uncertainties, we investigate the application of a non-intrusive, bi-fidelity approximation to estimate statistics of quantities of interest associated with an irradiated particle-laden turbulent flow. This method relies on exploiting the low-rank structure of the solution to accelerate the stochastic sampling and approximation processes by means of cheaper-to-run, lower fidelity representations. The application of this bi-fidelity approximation results in accurate estimates of the QoI statistics while requiring a small number of high-fidelity model evaluations.

physics.comp-ph↗

arXiv2018

Multi-fidelity uncertainty quantification of irradiated particle-laden turbulence

The study of complex systems is often based on computationally intensive, high-fidelity, simulations. To build confidence in the prediction accuracy of such simulations, the impact of uncertainties in model inputs on the quantities of interest must be measured. This, however, requires a computational budget that is a possibly large multiple of the cost of a single simulation, and thus may become infeasible for expensive simulation models featuring a large number of uncertain inputs and highly nonlinear behavior. Therefore, this work explores multi-fidelity strategies to accelerate the estimation of the effect of uncertainties. The main idea behind multi-fidelity models is to utilize cheaper, lower-fidelity models - than the intended high-fidelity, expensive model of the problem - to generate a baseline solution that together, with relatively small number of high-fidelity simulations can lead to accurate predictions. The methods are briefly presented, and their performance assessed on an irradiated particle-laden turbulent flow case related to Stanford's PSAAP II particle-based solar energy receiver.

physics.comp-ph↗

arXiv2017

Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Design

In the field of uncertainty quantification, sparse polynomial chaos (PC) expansions are commonly used by researchers for a variety of purposes, such as surrogate modeling. Ideas from compressed sensing may be employed to exploit this sparsity in order to reduce computational costs. A class of greedy compressed sensing algorithms use least squares minimization to approximate PC coefficients. This least squares problem lends itself to the theory of optimal design of experiments (ODE). Our work focuses on selecting an experimental design that improves the accuracy of sparse PC approximations for a fixed computational budget. We propose DSP, a novel sequential design, greedy algorithm for sparse PC approximation. The algorithm sequentially augments an experimental design according to a set of the basis polynomials deemed important by the magnitude of their coefficients, at each iteration. Our algorithm incorporates topics from ODE to estimate the PC coefficients. A variety of numerical simulations are performed on three physical models and manufactured sparse PC expansions to provide a comparative study between our proposed algorithm and other non-adaptive methods. Further, we examine the importance of sampling by comparing different strategies in terms of their ability to generate a candidate pool from which an optimal experimental design is chosen. It is demonstrated that the most accurate PC coefficient approximations, with the least variability, are produced with our design-adaptive greedy algorithm and the use of a studied importance sampling strategy. We provide theoretical and numerical results which show that using an optimal sampling strategy for the candidate pool is key, both in terms of accuracy in the approximation, but also in terms of constructing an optimal design.

stat.ME↗

arXiv2017

Parametric/Stochastic Model Reduction: Low-Rank Representation, Non-Intrusive Bi-Fidelity Approximation, and Convergence Analysis

For practical model-based demands, such as design space exploration and uncertainty quantification (UQ), a high-fidelity model that produces accurate outputs often has high computational cost, while a low-fidelity model with less accurate outputs has low computational cost. It is often possible to construct a bi-fidelity model having accuracy comparable with the high-fidelity model and computational cost comparable with the low-fidelity model. This work presents the construction and analysis of a non-intrusive (i.e., sample-based) bi-fidelity model that relies on the low-rank structure of the map between model parameters/uncertain inputs and the solution of interest, if exists. Specifically, we derive a novel, pragmatic estimate for the error committed by this bi-fidelity model. We show that this error bound can be used to determine if a given pair of low- and high-fidelity models will lead to an accurate bi-fidelity approximation. The cost of this error bound is relatively small and depends on the solution rank. The value of this error estimate is demonstrated using two example problems in the context of UQ, involving linear and non-linear partial differential equations.

math.NA↗

arXiv2017

Least Squares Polynomial Chaos Expansion: A Review of Sampling Strategies

As non-institutive polynomial chaos expansion (PCE) techniques have gained growing popularity among researchers, we here provide a comprehensive review of major sampling strategies for the least squares based PCE. Traditional sampling methods, such as Monte Carlo, Latin hypercube, quasi-Monte Carlo, optimal design of experiments (ODE), Gaussian quadratures, as well as more recent techniques, such as coherence-optimal and randomized quadratures are discussed. We also propose a hybrid sampling method, dubbed alphabetic-coherence-optimal, that employs the so-called alphabetic optimality criteria used in the context of ODE in conjunction with coherence-optimal samples. A comparison between the empirical performance of the selected sampling methods applied to three numerical examples, including high-order PCE's, high-dimensional problems, and low oversampling ratios, is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand. We observed that the alphabetic-coherence-optimal technique outperforms other sampling methods, specially when high-order ODE are employed and/or the oversampling ratio is low.

stat.CO↗

arXiv2017

Time-dependent global sensitivity analysis with active subspaces for a lithium ion battery model

Renewable energy researchers use computer simulation to aid the design of lithium ion storage devices. The underlying models contain several physical input parameters that affect model predictions. Effective design and analysis must understand the sensitivity of model predictions to changes in model parameters, but global sensitivity analyses become increasingly challenging as the number of input parameters increases. Active subspaces are part of an emerging set of tools for discovering and exploiting low-dimensional structures in the map from high-dimensional inputs to model outputs. We extend linear and quadratic model-based heuristic for active sub- space discovery to time-dependent processes and apply the resulting technique to a lithium ion battery model. The results reveal low-dimensional structure and sensitivity metrics that a designer may exploit to study the relationship between parameters and predictions.

physics.comp-ph↗