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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 541 records · Page 30
All sourcesNASA NTRSDOE OSTI
DOE OSTI2015

Second-order adjoint sensitivity analysis methodology (2ND-ASAM) for computing exactly and efficiently first- and second-order sensitivities in large-scale systems - 14356

This work presents the second-order adjoint sensitivity analysis methodology (2.-ASAM) for computing exactly and efficiently the second-order functional derivatives ('sensitivities') of system responses to the system's model parameters. For a physical system comprising N{sub α} parameters and N{sub r} responses, forward methods require a total of at least (N{sup 2}{sub α}/2+3N{sub α}/2) large-scale computations for obtaining all of the first- and second-order sensitivities, for all N{sub r} system responses. On the other hand, for one functional-type system response, the 2.-ASAM requires one large-scale computation using the first-level adjoint sensitivity system (1.-LASS) for obtaining all of the first-order sensitivities, followed by at most (2N{sub α}+1) large-scale computations using the second-level adjoint sensitivity systems (2.-LASS) for obtaining all of the second-order sensitivities. The second-order sensitivities contribute decisively to causing asymmetries in the response distribution, since they are the leading contributors to the third-order response correlations (skewness). They also cause the 'expected value of the response' to differ from the 'computed nominal value of the response'. The implementation of the 2.-ASAM requires very little additional effort beyond the construction of the adjoint sensitivity system needed for computing the first-order sensitivities. Only the sources on the right-sides of the diffusion (differential) operator needed to be modified; the left-side of the differential equations (and hence the 'solver' in large-scale practical applications) remain unchanged. The application of the 2.-ASAM is illustrated on a benchmark heat conduction/convection problem, which makes transparent the underlying mathematical derivations. For this illustrative problem, 4 'large-scale' adjoint computations suffice for computing exactly all of the 6 first- and 21 distinct second-order derivatives. The 2.-ASAM presented in this work should enable the hitherto very difficult, if not intractable, exact computation of all of the second-order response sensitivities for large-systems involving many parameters, which is expected to affect significantly other fields that need efficiently computed second-order response sensitivities, such as optimization, data assimilation/adjustment, model calibration, and predictive modeling. (author)

42 ENGINEERING↗
DOE OSTI2011

Probability of Initiation in Neutron Transport

We discuss the numerical solution of the nonlinear integro-differential equation for the probability of a divergent neutron chain in a stationary system (i.e., the probability of initiation (POI)). We follow the development described in Bell’s classic paper on the stochastic theory of neutron transport. As noted by Bell, the linearized form of this equation resembles the linear adjoint neutron transport equation. A matrix formalism for the discretized steady state (or forward) neutron equation in slab geometry is first developed, and is then used to derive the discrete adjoint equation. A main advantage of this discrete development is that the resulting discrete adjoint equation does not depend upon how the multigroup cross sections for the forward problem are obtained. That is, we derive the discrete adjoint directly from the discrete forward equations rather than discretizing directly the adjoint equation. This also guarantees that the discrete adjoint operator is consistent with the inner product used to define the adjoint operator. We discuss three approaches for the numerical solution of the POI equations, and present numerical results on several test problems. The three solution methods are a simple fixed point iteration, a second approach that is akin to a nonlinear Power iteration, and a third approach which uses a Newton-Krylov nonlinear solver. We also give sufficient conditions to guarantee the existence and uniqueness of nontrivial solutions to our discrete POI equations when the discrete system is supercritical, and that only the trivial solution exists when the discrete system is subcritical. Our approach is modelled after the analysis presented for the continuous POI equations by Mokhtar-Kharroubi and Jarmouni-Idrissi [12], and by Pazy and Rabinowitz [14].

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS2015

Assessing the Tangent Linear Behaviour of Common Tracer Transport Schemes and Their Use in a Linearised Atmospheric General Circulation Model

The linearity of a selection of common advection schemes is tested and examined with a view to their use in the tangent linear and adjoint versions of an atmospheric general circulation model. The schemes are tested within a simple offline one-dimensional periodic domain as well as using a simplified and complete configuration of the linearised version of NASA's Goddard Earth Observing System version 5 (GEOS-5). All schemes which prevent the development of negative values and preserve the shape of the solution are confirmed to have nonlinear behaviour. The piecewise parabolic method (PPM) with certain flux limiters, including that used by default in GEOS-5, is found to support linear growth near the shocks. This property can cause the rapid development of unrealistically large perturbations within the tangent linear and adjoint models. It is shown that these schemes with flux limiters should not be used within the linearised version of a transport scheme. The results from tests using GEOS-5 show that the current default scheme (a version of PPM) is not suitable for the tangent linear and adjoint model, and that using a linear third-order scheme for the linearised model produces better behaviour. Using the third-order scheme for the linearised model improves the correlations between the linear and non-linear perturbation trajectories for cloud liquid water and cloud liquid ice in GEOS-5.

Tangent Linear Model↗
DOE OSTI2016

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method

Here, we propose two multilevel spectral techniques for constructing coarse discretization spaces for saddle-point problems corresponding to PDEs involving a divergence constraint, with a focus on mixed finite element discretizations of scalar self-adjoint second order elliptic equations on general unstructured grids. We use element agglomeration algebraic multigrid (AMGe), which employs coarse elements that can have nonstandard shape since they are agglomerates of fine-grid elements. The coarse basis associated with each agglomerated coarse element is constructed by solving local eigenvalue problems and local mixed finite element problems. This construction leads to stable upscaled coarse spaces and guarantees the inf-sup compatibility of the upscaled discretization. Also, the approximation properties of these upscaled spaces improve by adding more local eigenfunctions to the coarse spaces. The higher accuracy comes at the cost of additional computational effort, as the sparsity of the resulting upscaled coarse discretization (referred to as operator complexity) deteriorates when we introduce additional functions in the coarse space. We also provide an efficient solver for the coarse (upscaled) saddle-point system by employing hybridization, which leads to a symmetric positive definite (s.p.d.) reduced system for the Lagrange multipliers, and to solve the latter s.p.d. system, we use our previously developed spectral AMGe solver. Numerical experiments, in both two and three dimensions, are provided to illustrate the efficiency of the proposed upscaling technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Fast and Scalable FFT-Based GPU-Accelerated Algorithms for Block-Triangular Toeplitz Matrices with Application to Linear Inverse Problems Governed by Autonomous Dynamical Systems

In this work, we present an efficient and scalable algorithm for performing matrix-vector multiplications (matvecs) for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving inverse problems governed by autonomous systems, and time-invariant systems in particular. In this article, we consider inverse problems that infer unknown parameters from observational data of a linear time-invariant dynamical system given in the form of partial differential equations (PDEs). Matrix-free Newton-conjugate-gradient methods are often the gold standard for solving these inverse problems, but they require numerous actions of the Hessian on a vector. Matrix-free adjoint-based Hessian matvecs require solution of a pair of linearized forward/adjoint PDE solves per Hessian action, which may be prohibitive for large-scale inverse problems. Time invariance of the forward PDE problem leads to a block Toeplitz structure of the discretized parameter-to-observable (p2o) map defining the mapping from inputs (parameters) to outputs (observables) of the PDEs. This block Toeplitz structure enables us to exploit two key properties: (1) compact storage of the p2o map and its adjoint, and (2) efficient fast Fourier transform–based Hessian matvecs. The proposed algorithm is mapped onto large multi-GPU clusters and achieves more than 80% of peak bandwidth on NVIDIA A100 GPUs. Excellent weak scaling is shown for up to 48 A100 GPUs. For the targeted problems, the implementation executes Hessian matvecs within fractions of a second, which is orders of magnitude faster than can be achieved by conventional matrix-free Hessian matvecs via forward/adjoint PDE solves.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1983

Character and stability of axisymmetric thermal convection in spheres and spherical shells

The influence of shell size and mode of heating on the behavior and stability of axisymmetric, infinite Prandtl number convection in a spherical geometry is studied. Heating from within and below features convection onset governed by a self-adjoint system of equations and boundary conditions. For heating only from within or from below, linearized equations and boundary conditions are non-self-adjoint. Identification of the parameter which initiates the departure from self-adjointness, together with the properties of the self-adjoint solution, provide a basis for calculating the heat transfer characteristics of the non-self-adjoint situations. The investigations are an effort to develop a model for heat transfer in planetary interiors. Further development of the technique by modifying the Galerkin method by the introduction of diagonal mode truncation is suggested to permit the consideration of higher values of the Rayleigh numbers, i.e., those more commensurate with terrestrial planet mantles.

Zebib, A.↗
NASA NTRS1974

Topics in structural dynamics: Nonlinear unsteady transonic flows and Monte Carlo methods in acoustics

The results are reported of two unrelated studies. The first was an investigation of the formulation of the equations for non-uniform unsteady flows, by perturbation of an irrotational flow to obtain the linear Green's equation. The resulting integral equation was found to contain a kernel which could be expressed as the solution of the adjoint flow equation, a linear equation for small perturbations, but with non-constant coefficients determined by the steady flow conditions. It is believed that the non-uniform flow effects may prove important in transonic flutter, and that in such cases, the use of doublet type solutions of the wave equation would then prove to be erroneous. The second task covered an initial investigation into the use of the Monte Carlo method for solution of acoustical field problems. Computed results are given for a rectangular room problem, and for a problem involving a circular duct with a source located at the closed end.

Haviland, J. K.↗
DOE OSTI2020

LA17-V-InverseRadTransportProblem-PD2Kb. Final Project Report

In the inverse radiation transport problem, a radiation signal emanating from an unknown object is used to reconstruct geometric and material parameters of the object , e.g., the mass of special nuclear material (SNM), the isotopic composition of the SNM (e.g., the enrichment of 235U), the thickness of shielding, etc. The inverse radiation transport problem is one of the most fundamental and important in the emergency response mission, and has been addressed in previous NA-22 and NA-42/84 investments by using derivative-free optimization methods. Our project was aimed at (i) consolidating previous work by applying polynomial chaos expansion (PCE) for reducing significantly the number of forward model simulations required by derivative-free optimization methods; (ii) developing a general-purpose framework and stand-alone software for solving the inverse radiation transport problem by applying a new “predictive modeling” methodology (predictive modeling of coupled multi-physics systems, PM_CMPS) which enables the fusing of experimental and computational information; and (iii) developing several benchmarks (for unscattered gamma rays) for demonstrating the quantification of non-Gaussian features (e.g., skewness and kurtosis, which characterize asymmetries and long tails) of response distributions by deterministically and efficiently propagating parameter uncertainties (covariances and higher-order moments, where available) using high-order sensitivities computed by the recently developed comprehensive adjoint sensitivity analysis methodology (C-ASAM). With regard to (i), application of PCE, the most significant accomplishment of this project was an implementation of PCE into INVERSE that significantly reduced computation time for the DiffeRential Evolution Adaptive Metropolis (DREAM) and Differential Evolution (DE) methods. For inverse transport problems with measurements of neutron leakage, neutron multiplication, and neutron-induced gamma-rays, the solution times for the DREAM and DE methods were reduced by factors of 30-300, reducing run times from hours to minutes while maintaining the same accuracy as the standard (non-PCE) DREAM and DE. Speedups were further improved by exploring sparse grid quadrature sets using the Toolkit for Adaptive Stochastic Modeling and Non-Intrusive ApproximatioN (Tasmanian) toolset. An initial exploration of sparse grids indicated that further speedups factors of 5-10 are obtainable. The implementation of PCE in INVERSE is not in a form that users can easily access. With regard to (ii), solving the inverse problem using a predictive modeling paradigm, the most significant accomplishment of this project were the derivation and implementation of adjoint-based second derivatives in neutron and photon transport theory. There are two codes that compute these derivatives. SENSPG computes them for uncollided gamma rays and SENSMG computes them for neutron leakage. SENSMG is open-source code available on github. Other significant accomplishments included a stand-alone code, MULTI-PRED, that implements the predictive modeling paradigm, along with a detailed manual. The code was implemented in the existing INVERSE code for inverse problems, but a head-to-head comparison of inverse methods was not achieved. We also derived and implemented the capability to compute sensitivities for multiplicity measurements into the SENSMG code, and this capability was used in a data assimilation application, but it was not used for solving inverse problems. With regard to (iii), developing benchmarks to demonstrate the importance of high-order sensitivity and uncertainty analysis, the most significant accomplishment of this project were two test problems that showed how important second-order sensitivities are.

07 ISOTOPE AND RADIATION SOURCES↗
DOE OSTI2019

Quantum Optimal Control Using High Performance Computing

This feasibility study developed numerical methods for the optimal control problem for realizing logical gates in closed quantum systems, where the evolution of the state vector is governed by the time dependent Schrodinger equation. The number of parameters in the control functions is made independent of the number of time steps by expanding them in terms of B-spline basis functions, with and without carrier waves. We use an interior point gradient-based technique from the IPOPT package to minimize the gate infidelity subject to amplitude constraints on the control functions. The symplectic Stromer-Verlet scheme is used to integrate a real-valued formulation of Schrodinger's equation in time and the gradient of the gate infidelity is obtained by solving the corresponding adjoint equation. This allows all components of the gradient to be calculated at the cost of solving three Schrodinger systems, independently of the number of parameters in the control functions. The correctness of the method is verified and applied to Hamiltonians that model the dynamics of coupled super-conducting qubits.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2004

Sensitivity Equation Derivation for Transient Heat Transfer Problems

The focus of the paper is on the derivation of sensitivity equations for transient heat transfer problems modeled by different discretization processes. Two examples will be used in this study to facilitate the discussion. The first example is a coupled, transient heat transfer problem that simulates the press molding process in fabrication of composite laminates. These state equations are discretized into standard h-version finite elements and solved by a multiple step, predictor-corrector scheme. The sensitivity analysis results based upon the direct and adjoint variable approaches will be presented. The second example is a nonlinear transient heat transfer problem solved by a p-version time-discontinuous Galerkin's Method. The resulting matrix equation of the state equation is simply in the form of Ax = b, representing a single step, time marching scheme. A direct differentiation approach will be used to compute the thermal sensitivities of a sample 2D problem.

Hou, Gene↗
NASA NTRS2016

An Inviscid Decoupled Method for the Roe FDS Scheme in the Reacting Gas Path of FUN3D

An approach is described to decouple the species continuity equations from the mixture continuity, momentum, and total energy equations for the Roe flux difference splitting scheme. This decoupling simplifies the implicit system, so that the flow solver can be made significantly more efficient, with very little penalty on overall scheme robustness. Most importantly, the computational cost of the point implicit relaxation is shown to scale linearly with the number of species for the decoupled system, whereas the fully coupled approach scales quadratically. Also, the decoupled method significantly reduces the cost in wall time and memory in comparison to the fully coupled approach. This work lays the foundation for development of an efficient adjoint solution procedure for high speed reacting flow.

Thompson, Kyle B.↗
DOE OSTI2013

Explicit estimation of higher order modes in fission source distribution of Monte-Carlo calculation

Magnitude of higher order modes in fission source distribution of a multi-group Monte-Carlo calculation is estimated using the orthogonal property of forward and adjoint fission source distributions. Calculation capability of the forward and adjoint fission source distributions for fundamental and higher order modes are implemented in the AEGIS code, which is a two-dimensional transport code based on the method of characteristics. With the calculation results of the AEGIS code, magnitudes of the first to fifth higher order modes in fission source distribution obtained by the multi-group Monte-Carlo code GMVP are estimated. There are two contributions in the present study - (1) establishment of a surrogate model, which represents convergence of fission source distribution taking into account the inherent statistical 'noise' of higher order modes of Monte-Carlo calculations and (2) independent confirmation of the estimated dominance ratio in a Monte-Carlo calculation. The surrogate model would contribute to studies of the inter-cycle correlation and estimation of sufficient number of inactive/active cycles. (authors)

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

A Least-Squares Transport Equation Compatible with Voids

Standard second-order self-adjoint forms of the transport equation, such as the even-parity, odd-parity, and self-adjoint angular flux equation, cannot be used in voids. Perhaps more important, they experience numerical convergence difficulties in near-voids. Here we present a new form of a second-order self-adjoint transport equation that has an advantage relative to standard forms in that it can be used in voids or near-voids. Our equation is closely related to the standard least-squares form of the transport equation with both equations being applicable in a void and having a nonconservative analytic form. However, unlike the standard least-squares form of the transport equation, our least-squares equation is compatible with source iteration. It has been found that the standard least-squares form of the transport equation with a linear-continuous finite-element spatial discretization has difficulty in the thick diffusion limit. Here we extensively test the 1D slab-geometry version of our scheme with respect to void solutions, spatial convergence rate, and the intermediate and thick diffusion limits. We also define an effective diffusion synthetic acceleration scheme for our discretization. Our conclusion is that our least-squares S n formulation represents an excellent alternative to existing second-order S n transport formulations

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2011

Bayesian Radiation Source Localization

Locating illicit radiological sources using gamma ray or neutron detection is a key challenge for both homeland security and nuclear nonproliferation. Localization methods using an array of detectors or a sequence of observations in time and space must provide rapid results while accounting for a dynamic attenuating environment. In the presence of significant attenuation and scatter, more extensive numerical transport calculations in place of the standard analytical approximations may be required to achieve accurate results. Numerical adjoints based on deterministic transport codes provide relatively efficient detector response calculations needed to determine the most likely location of a true source. Probabilistic representations account for uncertainty in the source location resulting from uncertainties in detector responses, the approximations that are used, and the potential for nonunique solutions. A Bayesian approach improves on previous likelihood methods for source localization by allowing the incorporation of all available information to help constrain solutions.

98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL P↗
NASA NTRS1995

Implementation of a multiblock sensitivity analysis method in numerical aerodynamic shape optimization

A multiblock sensitivity analysis method is applied in a numerical aerodynamic shape optimization technique. The Sensitivity Analysis Domain Decomposition (SADD) scheme which is implemented in this study was developed to reduce the computer memory requirements resulting from the aerodynamic sensitivity analysis equations. Discrete sensitivity analysis offers the ability to compute quasi-analytical derivatives in a more efficient manner than traditional finite-difference methods, which tend to be computationally expensive and prone to inaccuracies. The direct optimization procedure couples CFD analysis based on the two-dimensional thin-layer Navier-Stokes equations with a gradient-based numerical optimization technique. The linking mechanism is the sensitivity equation derived from the CFD discretized flow equations, recast in adjoint form, and solved using direct matrix inversion techniques. This investigation is performed to demonstrate an aerodynamic shape optimization technique on a multiblock domain and its applicability to complex geometries. The objectives are accomplished by shape optimizing two aerodynamic configurations. First, the shape optimization of a transonic airfoil is performed to investigate the behavior of the method in highly nonlinear flows and the effect of different grid blocking strategies on the procedure. Secondly, shape optimization of a two-element configuration in subsonic flow is completed. Cases are presented for this configuration to demonstrate the effect of simultaneously reshaping interfering elements. The aerodynamic shape optimization is shown to produce supercritical type airfoils in the transonic flow from an initially symmetric airfoil. Multiblocking effects the path of optimization while providing similar results at the conclusion. Simultaneous reshaping of elements is shown to be more effective than individual element reshaping due to the inclusion of mutual interference effects.

Lacasse, James M.↗
DOE OSTI2018

Calculating Alpha Eigenvalues and Eigenfunctions with a Markov Transition Rate Matrix Monte Carlo Method

For a nuclear system in which the entire α-eigenvalue spectrum is known, eigenfunction expansion yields the time-dependent flux response to any arbitrary source. Applications in which this response is of interest include pulsed-neutron experiments, accelerator-driven subcritical systems, and fast burst reactors, where a steady-state assumption used in neutron transport is invalid for characterizing the time-dependent flux. To obtain the α-eigenvalue spectrum, the transition rate matrix method (TRMM) tallies transition rates describing neutron behavior in a discretized position-direction-energy phase space using Monte Carlo. Interpretation of the resulting Markov process transition rate matrix as the operator in the adjoint α-eigenvalue problem provides an avenue for determining a large finite set of α eigenvalues and eigenfunctions of a nuclear system. Results from the TRMM are verified using analytic solutions, time-dependent Monte Carlo simulations, and modal expansion from diffusion theory. For simplified infinite-medium and one-dimensional geometries, the TRMM accurately calculates eigenvalues, eigenfunctions, and eigenfunction expansion solutions. Applications and comparisons to measurements are made for the small fast burst reactor CALIBAN and the Fort St. Vrain high-temperature gas-cooled reactor. For large three-dimensional geometries, discretization of the large position-energy-direction phase space limits the accuracy of eigenfunction expansion solutions using the TRMM, but it can still generate a fair estimate of the fundamental eigenvalue and eigenfunction. These results show that the TRMM generates an accurate estimate of a large number of α eigenvalues. This is not possible with existing Monte Carlo–based methods.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NASA NTRS2013

ISSM: Ice Sheet System Model

In order to have the capability to use satellite data from its own missions to inform future sea-level rise projections, JPL needed a full-fledged ice-sheet/iceshelf flow model, capable of modeling the mass balance of Antarctica and Greenland into the near future. ISSM was developed with such a goal in mind, as a massively parallelized, multi-purpose finite-element framework dedicated to ice-sheet modeling. ISSM features unstructured meshes (Tria in 2D, and Penta in 3D) along with corresponding finite elements for both types of meshes. Each finite element can carry out diagnostic, prognostic, transient, thermal 3D, surface, and bed slope simulations. Anisotropic meshing enables adaptation of meshes to a certain metric, and the 2D Shelfy-Stream, 3D Blatter/Pattyn, and 3D Full-Stokes formulations capture the bulk of the ice-flow physics. These elements can be coupled together, based on the Arlequin method, so that on a large scale model such as Antarctica, each type of finite element is used in the most efficient manner. For each finite element referenced above, ISSM implements an adjoint. This adjoint can be used to carry out model inversions of unknown model parameters, typically ice rheology and basal drag at the ice/bedrock interface, using a metric such as the observed InSAR surface velocity. This data assimilation capability is crucial to allow spinning up of ice flow models using available satellite data. ISSM relies on the PETSc library for its vectors, matrices, and solvers. This allows ISSM to run efficiently on any parallel platform, whether shared or distrib- ISSM: Ice Sheet System Model NASA's Jet Propulsion Laboratory, Pasadena, California uted. It can run on the largest clusters, and is fully scalable. This allows ISSM to tackle models the size of continents. ISSM is embedded into MATLAB and Python, both open scientific platforms. This improves its outreach within the science community. It is entirely written in C/C++, which gives it flexibility in its design, and the power/speed that C/C++ allows. ISSM is svn (subversion) hosted, on a JPL repository, to facilitate its development and maintenance. ISSM can also model propagation of rifts using contact mechanics and mesh splitting, and can interface to the Dakota software. To carry out sensitivity analysis, mesh partitioning algorithms are available, based on the Scotch, Chaco, and Metis partitioners that ensure equal area mesh partitions can be done, which are then usable for sampling and local reliability methods.

Larour, Eric↗
DOE OSTI2013

Efficient and robust gradient enhanced Kriging emulators.

%E2%80%9CNaive%E2%80%9D or straight-forward Kriging implementations can often perform poorly in practice. The relevant features of the robustly accurate and efficient Kriging and Gradient Enhanced Kriging (GEK) implementations in the DAKOTA software package are detailed herein. The principal contribution is a novel, effective, and efficient approach to handle ill-conditioning of GEK's %E2%80%9Ccorrelation%E2%80%9D matrix, RN%CC%83, based on a pivoted Cholesky factorization of Kriging's (not GEK's) correlation matrix, R, which is a small sub-matrix within GEK's RN%CC%83 matrix. The approach discards sample points/equations that contribute the least %E2%80%9Cnew%E2%80%9D information to RN%CC%83. Since these points contain the least new information, they are the ones which when discarded are both the easiest to predict and provide maximum improvement of RN%CC%83's conditioning. Prior to this work, handling ill-conditioned correlation matrices was a major, perhaps the principal, unsolved challenge necessary for robust and efficient GEK emulators. Numerical results demonstrate that GEK predictions can be significantly more accurate when GEK is allowed to discard points by the presented method. Numerical results also indicate that GEK can be used to break the curse of dimensionality by exploiting inexpensive derivatives (such as those provided by automatic differentiation or adjoint techniques), smoothness in the response being modeled, and adaptive sampling. Development of a suitable adaptive sampling algorithm was beyond the scope of this work; instead adaptive sampling was approximated by omitting the cost of samples discarded by the presented pivoted Cholesky approach.

Dalbey, Keith R.↗