Smolyak Sparse Grid Research Articles

A two-stage adaptive stochastic collocation method on nested sparse grids for multiphase flow in randomly heterogeneous porous media

A new computational method is proposed for efficient uncertainty quantification of multiphase flow in porous media with stochastic permeability. For pressure estimation, it combines the dimension-adaptive stochastic collocation method on Smolyak sparse grids and the Kronrod–Patterson–Hermite nested quadrature formulas. For saturation estimation, an additional stage is developed, in which the pressure and velocity samples are first generated by the sparse grid interpolation and then substituted into the transport equation to solve for the saturation samples, to address the low regularity problem of the saturation. Numerical examples are presented for multiphase flow with stochastic permeability fields to demonstrate accuracy and efficiency of the proposed two-stage adaptive stochastic collocation method on nested sparse grids.

Non‐intrusive reduced order modelling with least squares fitting on a sparse grid

SummaryThis paper presents a non‐intrusive reduced order model for general, dynamic partial differential equations. Based upon proper orthogonal decomposition (POD) and Smolyak sparse grid collocation, the method first projects the unknowns with full space and time coordinates onto a reduced POD basis. Then we introduce a new least squares fitting procedure to approximate the dynamical transition of the POD coefficients between subsequent time steps, taking only a set of full model solution snapshots as the training data during the construction. Thus, neither the physical details nor further numerical simulations of the original PDE model are required by this methodology, and the level of non‐intrusiveness is improved compared with existing reduced order models. Furthermore, we take adaptive measures to address the instability issue arising from reduced order iterations of the POD coefficients. This model can be applied to a wide range of physical and engineering scenarios, and we test it on a couple of problems in fluid dynamics. It is demonstrated that this reduced order approach captures the dominant features of the high‐fidelity models with reasonable accuracy while the computation complexity is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.

Uncertainty Quantification of Modelling of Equiaxed Solidification

Numerical simulations of metal alloy solidification are used to gain insight into physical phenomena that cannot be observed experimentally. Often validation of such models has been done through comparison to sparse experimental data, to which agreement can be misinterpreted due to both model and experimental uncertainty. Uncertainty quantification (UQ) and sensitivity analysis are performed on a transient model of solidification of Al-4.5 wt.% Cu in a rectangular cavity, with equiaxed (grain refined) solidification morphology. This model solves equations for momentum, temperature, and species conservation; UQ and sensitivity analysis are performed for the degree of macrosegregation. A Smolyak sparse grid algorithm is used to select input values to construct a response surface fit to model outputs. The response surface is then used as a surrogate for the solidification model to determine the sensitivities and probability density functions of the model outputs. Uncertain model inputs of interest include the secondary dendrite arm spacing, equiaxed particle size, and fraction solid at which the rigid mushy zone forms. Similar analysis was also performed on a transient model of direct chill casting of the same alloy.

Non‐intrusive reduced‐order modeling for multiphase porous media flows using Smolyak sparse grids

SummaryIn this article, we describe a non‐intrusive reduction method for porous media multiphase flows using Smolyak sparse grids. This is the first attempt at applying such an non‐intrusive reduced‐order modelling (NIROM) based on Smolyak sparse grids to porous media multiphase flows. The advantage of this NIROM for porous media multiphase flows resides in that its non‐intrusiveness, which means it does not require modifications to the source code of full model. Another novelty is that it uses Smolyak sparse grids to construct a set of hypersurfaces representing the reduced‐porous media multiphase problem. This NIROM is implemented under the framework of an unstructured mesh control volume finite element multiphase model. Numerical examples show that the NIROM accuracy relative to the high‐fidelity model is maintained, whilst the computational cost is reduced by several orders of magnitude. Copyright © 2016 John Wiley & Sons, Ltd.

Stochastic analysis of the impact of freestream conditions on the aerodynamics of a rectangular 5:1 cylinder

Uncertainty plays a significant role in the Benchmark on the Aerodynamics of a Rectangular Cylinder (BARC) with a chord-to-depth ratio of 5. In particular, besides modeling and numerical errors, in numerical simulations it is difficult to exactly reproduce the experimental conditions due to uncertainties in the set-up parameters, which sometimes cannot be exactly controlled or characterized. In this study, the impact of the uncertainties in the inflow conditions of the BARC configuration is investigated by using probabilistic methods and two-dimensional URANS simulations. The following uncertain set-up parameters are investigated: the angle of incidence, the freestream longitudinal turbulence intensity and the freestream turbulence length scale. The stochastic collocation method is employed to perform the probabilistic propagation of the uncertainty in the three set-up parameters. This results in 25 URANS simulations based on the Smolyak sparse grid extension of the level-2 Clenshaw-Curtis quadrature points. The discretization error is estimated by repeating the same analysis on different grid sizes. Similarly, the effect of turbulence modeling is appraised by carrying out the uncertainty quantification for the Reynolds stress and the SST k−ω models. Finally, the results obtained for different assumed probability density functions of the set-up parameters are compared. The propagation of the considered uncertainties does not explain alone the dispersion of the BARC experimental data. For certain quantities of interest, the effect of turbulence modeling is more important than the impact of the uncertainties in inflow conditions. The sensitivity to the considered uncertainties also varies with the turbulence model, with a larger variability of the results obtained with the Reynolds stress model. The inflow turbulence length scale is in all cases the least important parameter.

Collocation methods for uncertain heat convection-diffusion problem with interval input parameters

This paper proposes a full grid interval collocation method (FGICM) and a sparse grid interval collocation method (SGICM) to solve the uncertain heat convection-diffusion problem with interval input parameters in material properties, applied loads and boundary conditions. The Legendre polynomial series is adopted to approximate the functional dependency of temperature response with respect to the interval parameters. In the process of calculating the expansion coefficients, FGICM evaluates the deterministic solutions directly on the full tensor product grids, while the Smolyak sparse grids are reconstructed in SGICM to avoid the curse of dimensionality. The eventual lower and upper bounds of temperature responses are easily predicted based on the continuously-differentiable property of the approximate function. Comparing results with traditional Monte Carlo simulations and perturbation method, the numerical example evidences the remarkable accuracy and effectiveness of the proposed methods for interval temperature field prediction in engineering.

Uncertainty propagation of heat conduction problem with multiple random inputs

Uncertainty propagation analysis in engineering systems constitutes a significant challenge. To effectively solve the uncertain heat conduction problem with multiple random inputs, a random collocation method (RCM) and a modified random collocation method (MRCM) are established based on the spectral analysis theory. In both methods, the truncated high-order polynomial series is adopted to approximate the temperature responses with respect to random parameters, and the eventual probabilistic moments are derived by using the orthogonal relationship of polynomial bases. In the pivotal process of calculating the expansion coefficients, RCM evaluates the deterministic solutions directly on full tensor product grids, whereas the Smolyak sparse grids are reconstructed in MRCM to avoid the huge computational cost caused by high dimensions. Comparing the results with traditional Monte Carlo simulation, two numerical examples verify the remarkable accuracy and effectiveness of the proposed methods for random temperature field prediction in engineering.

A Pseudospectral Collocation Approach for Flood Inundation Modelling with Random Input Fields

In this study, an efficient framework of pseudospectral collocation approach combined with the generalized polynomial chaos (gPC) and Karhunen-Loevè expansion (gPC/KLE) was introduced to examine the flood flow fields within a two-dimensional flood modelling system. In the proposed framework, the heterogeneous random input field (logarithmic Manning’s roughness) was approximated by the normalized KLE and the output field of flood flow depth was represented by the gPC expansion, whose coefficients were obtained with a nodal set construction via Smolyak sparse grid quadrature. In total, 3 scenarios (with different levels of input spatial variability) were designed for gPC/KLE application and the results from Monte Carlo simulations were provided as the benchmark for comparison. This study demonstrated that the gPC/KLE approach could predict the statistics of flood flow depth (i.e., means and standard deviations) with significantly less computational requirement than MC; it also outperformed the probabilistic collocation method (PCM) with KLE (PCM/KLE) in terms of fitting accuracy. This study made the first attempt to apply gPC/KLE to flood inundation field and evaluated the effects of key parameters (like the number of eigenpairs and the order of gPC expansion) on model performances.

Insights into the Spin-State Transitions in [Fe(tpy)2]2+: Importance of the Terpyridine Rocking Motion.

Iron(II) polypyridine complexes have the potential for numerous applications on a global scale, such as sensitizers, sensors, and molecular memory. The excited-state properties of these systems, particularly the intersystem crossing (ISC) rates, are sensitive to the choice of ligands and can be significantly altered depending on the coordination environment. We employ density functional theory and Smolyak's sparse grid interpolation algorithm to construct potential energy surfaces (PESs) for the photophysically relevant states ((1)A, (3,5)MC, and (1,3)MLCT) of the [Fe(tpy)2](2+) (tpy = 2,2':6',2"-terpyridine) complex, with the goal of obtaining a deeper understanding of the ground- and excited-state electronic structure of this system. The three dimensions that define our adiabatic PESs consist of equatorial and axial metal-ligand bond length distortions and a terpyridine ligand "rocking angle", which has not previously been investigated. The intersection crossing seams and minimum energy crossing points (MECPs) between surfaces are also determined. Overall, we find that the PESs of all electronic excited states investigated are characterized by low-energy valleys along the tpy rocking-angle coordinate. This results in the presence of large low-energy areas around the MECPs on the intersection seams of different electronic states and indicates that inclusion of this third coordinate is crucial for an adequate description of the PESs and surface crossing seams of the [Fe(tpy)2](2+) complex. Finally, we suggest that tuning the energetics of the tpy ligand rocking motion could provide a way to control the ISC process in this complex.

Non-intrusive reduced order modelling of the Navier–Stokes equations

Abstract This article presents two new non-intrusive reduced order models based upon proper orthogonal decomposition (POD) for solving the Navier–Stokes equations. The novelty of these methods resides in how the reduced order models are formed, that is, how the coefficients of the POD expansions are calculated. Rather than taking a standard approach of projecting the underlying equations onto the reduced space through a Galerkin projection, here two different techniques are employed. The first method applies a second order Taylor series to calculate the POD coefficients at each time step from the POD coefficients at earlier time steps. The second method uses a Smolyak sparse grid collocation method to calculate the POD coefficients, where again the coefficients at earlier time steps are used as the inputs. The advantage of both approaches are that they are non-intrusive and so do not require modifications to a system code; they are therefore very easy to implement. They also provide accurate solutions for modelling flow problems, and this has been demonstrated by the simulation of flows past a cylinder and within a gyre. It is demonstrated that accuracy relative to the high fidelity model is maintained whilst CPU times are reduced by several orders of magnitude in comparison to high fidelity models.

Uncertainty Quantification in Solidification Modelling

Numerical models have been used to simulate solidification processes, to gain insight into physical phenomena that cannot be observed experimentally. Often validation of such models has been done through comparison to a few or single experiments, in which agreement is dependent on both model and experimental uncertainty. As a first step to quantifying the uncertainty in the models, sensitivity and uncertainty analysis were performed on a simple steady state 1D solidification model of continuous casting of weld filler rod. This model includes conduction, advection, and release of latent heat was developed for use in uncertainty quantification in the calculation of the position of the liquidus and solidus and the solidification time. Using this model, a Smolyak sparse grid algorithm constructed a response surface that fit model outputs based on the range of uncertainty in the inputs to the model. The response surface was then used to determine the probability density functions (PDF's) of the model outputs and sensitivities of the inputs. This process was done for a linear fraction solid and temperature relationship, for which there is an analytical solution, and a Scheil relationship. Similar analysis was also performed on a transient 2D model of solidification in a rectangular domain.

PUQ: A code for non-intrusive uncertainty propagation in computer simulations

We present a software package for the non-intrusive propagation of uncertainties in input parameters through computer simulation codes or mathematical models and associated analysis; we demonstrate its use to drive micromechanical simulations using a phase field approach to dislocation dynamics. The PRISM uncertainty quantification framework (PUQ) offers several methods to sample the distribution of input variables and to obtain surrogate models (or response functions) that relate the uncertain inputs with the quantities of interest (QoIs); the surrogate models are ultimately used to propagate uncertainties. PUQ requires minimal changes in the simulation code, just those required to annotate the QoI(s) for its analysis. Collocation methods include Monte Carlo, Latin Hypercube and Smolyak sparse grids and surrogate models can be obtained in terms of radial basis functions and via generalized polynomial chaos. PUQ uses the method of elementary effects for sensitivity analysis in Smolyak runs. The code is available for download and also available for cloud computing in nanoHUB. PUQ orchestrates runs of the nanoPLASTICITY tool at nanoHUB where users can propagate uncertainties in dislocation dynamics simulations using simply a web browser, without downloading or installing any software. Program summaryProgram title: PUQCatalogue identifier: AEWP_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEWP_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: MIT licenseNo. of lines in distributed program, including test data, etc.: 45075No. of bytes in distributed program, including test data, etc.: 3318862Distribution format: tar.gzProgramming language: Python, C.Computer: Workstations.Operating system: Linux, Mac OSX.Classification: 4.11, 4.12, 4.13.External routines: SciPy, Matplotlib, h5pyNature of problem: Uncertainty propagation and creation of response surfaces.Solution method: Generalized Polynomial Chaos (gPC) using Smolyak sparse grids.Running time: PUQ performs uncertainty quantification and sensitivity analysis by running a simulation multiple times using different values for input parameters. Its run time will be the product of the run time of the chosen simulation code and the number of runs required to achieve the desired accuracy.

A Sparse Interpolation Algorithm for Dynamical Simulations in Computational Chemistry

In this paper we present a new implementation of Smolyak's sparse grid interpolation algorithm designed for dynamical simulations. The implementation is motivated by an application to quantum chemistry where the goal is to simulate photo-induced molecular transformations. A molecule conforms to a geometry that minimizes its potential energy, and many molecules have multiple potential energy minima. These geometries correspond to local minima of the molecule's potential energy surface, and one can simulate how a molecule transitions from one geometry to another by following the steepest descent path, or reaction path, on potential energy surfaces. Molecular vibrations and thermal fluctuations cause randomness in dynamics, so one must follow several paths simultaneously to more accurately simulate possible reaction paths. Current algorithms for reaction path following are too computationally burdensome for molecules of moderate size, but Smolyak's interpolation algorithm offers a cheap surrogate for potential energy surfaces. While current implementations of Smolyak's algorithm are not designed to simultaneously follow multiple reaction paths efficiently, our implementation of Smolyak's algorithm achieves this efficiency by recursively defining Lagrange basis polynomials and making use of an efficient reformulation of Smolyak's algorithm. In this paper we describe our new implementation of Smolyak's algorithm and compare performance times to MATLAB's Sparse Grid Interpolation Toolbox to demonstrate its computational savings. We also present dynamical simulations for the photoisomerization of 2-butene as an example of our reaction path following method.

Robust Topology Optimization Based on Stochastic Collocation Methods under Loading Uncertainties

A robust topology optimization (RTO) approach with consideration of loading uncertainties is developed in this paper. The stochastic collocation method combined with full tensor product grid and Smolyak sparse grid transforms the robust formulation into a weighted multiple loading deterministic problem at the collocation points. The proposed approach is amenable to implementation in existing commercial topology optimization software package and thus feasible to practical engineering problems. Numerical examples of two- and three-dimensional topology optimization problems are provided to demonstrate the proposed RTO approach and its applications. The optimal topologies obtained from deterministic and robust topology optimization designs under tensor product grid and sparse grid with different levels are compared with one another to investigate the pros and cons of optimization algorithm on final topologies, and an extensive Monte Carlo simulation is also performed to verify the proposed approach.

A new sampling scheme for developing metamodels with the zeros of Chebyshev polynomials

The accuracy of metamodelling is determined by both the sampling and approximation. This article proposes a new sampling method based on the zeros of Chebyshev polynomials to capture the sampling information effectively. First, the zeros of one-dimensional Chebyshev polynomials are applied to construct Chebyshev tensor product (CTP) sampling, and the CTP is then used to construct high-order multi-dimensional metamodels using the ‘hypercube’ polynomials. Secondly, the CTP sampling is further enhanced to develop Chebyshev collocation method (CCM) sampling, to construct the ‘simplex’ polynomials. The samples of CCM are randomly and directly chosen from the CTP samples. Two widely studied sampling methods, namely the Smolyak sparse grid and Hammersley, are used to demonstrate the effectiveness of the proposed sampling method. Several numerical examples are utilized to validate the approximation accuracy of the proposed metamodel under different dimensions.

Reaction Path Following with Sparse Interpolation.

Computing the potential energy of an N-atom molecule is an expensive optimization process of 3N - 6 molecular coordinates, so following reaction pathways as a function of all 3N - 6 coordinates is unfeasible for large molecules. In this paper, we present a method that isolates d < 3N - 6 molecular coordinates and continuously follows reaction paths on d-dimensional potential energy surfaces approximated by a Smolyak's sparse grid interpolation algorithm.1 Compared to dense grids, sparse grids efficiently improve the ratio of invested storage and computing time to approximation accuracy and thus allow one to increase the number of coordinates d in molecular reaction path following simulations. Furthermore, evaluation of the interpolant is much less expensive than the evaluation of the actual energy function, so our technique offers a computationally efficient way to simulate reaction paths on ground and excited state potential energy surfaces. To demonstrate the capabilities of our method, we present simulation results for the isomerization of 2-butene with two, three, and six degrees of freedom.

A Recursive Sparse Grid Collocation Method for Differential Equations with White Noise

We consider a sparse grid collocation method in conjunction with a time discretization of the differential equations for computing expectations of functionals of solutions to differential equations perturbed by time-dependent white noise. We first analyze the error of Smolyak's sparse grid collocation used to evaluate expectations of functionals of solutions to stochastic differential equations discretized by the Euler scheme. We show theoretically and numerically that this algorithm can have satisfactory accuracy for small magnitude of noise or small integration time, however it does not converge neither with decrease of the Euler scheme's time step size nor with increase of Smolyak's sparse grid level. Subsequently, we use this method as a building block for proposing a new algorithm by combining sparse grid collocation with a recursive procedure. This approach allows us to numerically integrate linear stochastic partial differential equations over longer times, which is illustrated in numerical tests on a stochastic advection-diffusion equation.

Smolyak-Grid-Based Flutter Analysis with the Stochastic Aerodynamic Uncertainty

How to estimate the stochastic aerodynamic parametric uncertainty on aeroelastic stability is studied in this current work. The aerodynamic uncertainty is more complicated than the structural one, and it takes more significant effect on the flutter boundary. First, the nominal unsteady aerodynamic influence coefficients were calculated with the doublet lattice method. Based on this nominal model, the stochastic uncertainty model for unsteady aerodynamic pressure coefficients was constructed with physical meaning. Afterwards, the methodology for flutter uncertainty quantification due to aerodynamic perturbation was developed, based on the nonintrusive polynomial chaos expansion theory. In order to enhance the computational efficiency, the integration algorithm, namely, Smolyak sparse grids, was employed to calculate the coefficients of the stochastic polynomial basis. Finally, the flutter uncertainty analysis methodology was applied to an aircraft's wing model. The influence of uncertainty with uniform distribution for aerodynamic pressure coefficients on flutter boundary was quantified. The numerical results indicate that, the influence of unsteady aerodynamic pressure due to the motion of coupling modes takes significant effect on flutter boundary. It is validated that the flutter uncertainty analysis based on Smolyak sparse grids integration is efficient and accurate for quantifying input uncertainty with high dimensions.

Proper orthogonal decomposition‐based model order reduction via radial basis functions for molecular dynamics systems

SUMMARYModel order reduction for molecular dynamics (MD) systems exhibits intrinsic complexities because of the highly nonlinear and nonlocal multi‐atomic interactions in high dimensions. In the present work, we introduce a proper orthogonal decomposition‐based method in conjunction with the radial basis function (RBF) approximation of the nonlinear and nonlocal potential energies and inter‐atomic forces for MD systems. This approach avoids coordinate transformation between the physical and reduced‐order coordinates, and allows the potentials and inter‐atomic forces to be calculated directly in the reduced‐order space. The RBF‐approximated potential energies and inter‐atomic forces in the reduced‐order space are discretized on the basis of the Smolyak sparse grid algorithm to further enhance the effectiveness of the proposed method. The good approximation properties of RBFs in interpolating scattered data make them ideal candidates for the reduced‐order approximation of MD inter‐atomic force calculations. The proposed approach is validated by performing the reduced‐order simulations of DNA molecules under various external loadings. Copyright © 2013 John Wiley &amp; Sons, Ltd.

A priori testing of sparse adaptive polynomial chaos expansions using an ocean general circulation model database

This work explores the implementation of an adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed pseudo-spectral algorithm that is based on a direct application of the Smolyak sparse grid formula and that allows the use of arbitrary admissible sparse grids. The adaptive algorithm is tested using an existing simulation database of the oceanic response to Hurricane Ivan in the Gulf of Mexico. The a priori tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling in the present setting.