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Omar M Knio - One of the best experts on this subject based on the ideXlab platform.
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Sparse Pseudo Spectral Projection Methods with Directional Adaptation for Uncertainty Quantification
Journal of Scientific Computing, 2015Co-Authors: Justin Winokur, Daesang Kim, Fabrizio Bisetti, O. P. Maître, Omar M KnioAbstract:We investigate two methods to build a polynomial approximation of a model output depending on some parameters. The two approaches are based on pseudo-Spectral Projection (PSP) methods on adaptively constructed sparse grids, and aim at providing a finer control of the resolution along two distinct subsets of model parameters. The control of the error along different subsets of parameters may be needed for instance in the case of a model depending on uncertain parameters and deterministic design variables. We first consider a nested approach where an independent adaptive sparse grid PSP is performed along the first set of directions only, and at each point a sparse grid is constructed adaptively in the second set of directions. We then consider the application of aPSP in the space of all parameters, and introduce directional refinement criteria to provide a tighter control of the Projection error along individual dimensions. Specifically, we use a Sobol decomposition of the Projection surpluses to tune the sparse grid adaptation. The behavior and performance of the two approaches are compared for a simple two-dimensional test problem and for a shock-tube ignition model involving 22 uncertain parameters and 3 design parameters. The numerical experiments indicate that whereas both methods provide effective means for tuning the quality of the representation along distinct subsets of parameters, PSP in the global parameter space generally requires fewer model evaluations than the nested approach to achieve similar Projection error. In addition, the global approach is better suited for generalization to more than two subsets of directions.
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global sensitivity analysis in an ocean general circulation model a sparse Spectral Projection approach
Computational Geosciences, 2012Co-Authors: Alen Alexanderian, Omar M Knio, Justin Winokur, Mohamed Iskandarani, Ihab Sraj, Ashwanth Srinivasan, William Carlisle ThackerAbstract:Polynomial chaos (PC) expansions are used to propagate parametric uncertainties in ocean global circulation model. The computations focus on short-time, high-resolution simulations of the Gulf of Mexico, using the hybrid coordinate ocean model, with wind stresses corresponding to hurricane Ivan. A sparse quadrature approach is used to determine the PC coefficients which provides a detailed representation of the stochastic model response. The quality of the PC representation is first examined through a systematic refinement of the number of resolution levels. The PC representation of the stochastic model response is then utilized to compute distributions of quantities of interest (QoIs) and to analyze the local and global sensitivity of these QoIs to uncertain parameters. Conclusions are finally drawn regarding limitations of local perturbations and variance-based assessment and concerning potential application of the present methodology to inverse problems and to uncertainty management.
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Multiscale Stochastic Preconditioners in Non-intrusive Spectral Projection
Journal of Scientific Computing, 2011Co-Authors: Alen Alexanderian, Habib N. Najm, Oliver P. Maître, Mohamed Iskandarani, Omar M KnioAbstract:A preconditioning approach is developed that enables efficient polynomial chaos (PC) representations of uncertain dynamical systems. The approach is based on the definition of an appropriate multiscale stretching of the individual components of the dynamical system which, in particular, enables robust recovery of the unscaled transient dynamics. Efficient PC representations of the stochastic dynamics are then obtained through non-intrusive Spectral Projections of the stretched measures. Implementation of the present approach is illustrated through application to a chemical system with large uncertainties in the reaction rate constants. Computational experiments show that, despite the large stochastic variability of the stochastic solution, the resulting dynamics can be efficiently represented using sparse low-order PC expansions of the stochastic multiscale preconditioner and of stretched variables. The present experiences are finally used to motivate several strategies that promise to yield further advantages in Spectral representations of stochastic dynamics.
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uncertainty quantification in reacting flow simulations through non intrusive Spectral Projection
Combustion and Flame, 2003Co-Authors: Matthew T Reagan, Roger Ghanem, Habib N. Najm, Omar M KnioAbstract:Abstract A Spectral formalism has been developed for the “non-intrusive” analysis of parametric uncertainty in reacting-flow systems. In comparison to conventional Monte Carlo analysis, this method quantifies the extent, dependence, and propagation of uncertainty through the model system and allows the correlation of uncertainties in specific parameters to the resulting uncertainty in detailed flame structure. For the homogeneous ignition chemistry of a hydrogen oxidation mechanism in supercritical water, Spectral Projection enhances existing Monte Carlo methods, adding detailed sensitivity information to uncertainty analysis and relating uncertainty propagation to reaction chemistry. For 1-D premixed flame calculations, the method quantifies the effect of each uncertain parameter on total uncertainty and flame structure, and localizes the effects of specific parameters within the flame itself. In both 0-D and 1-D examples, it is clear that known empirical uncertainties in model parameters may result in large uncertainties in the final output. This has important consequences for the development and evaluation of combustion models. This Spectral formalism may be extended to multidimensional systems and can be used to develop more efficient “intrusive” reformulations of the governing equations to build uncertainty analysis directly into reacting flow simulations.
Jie Shen - One of the best experts on this subject based on the ideXlab platform.
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An Efficient Spectral-Projection Method for the Navier-Stokes Equations in Cylindrical Geometries
Journal of Computational Physics, 2002Co-Authors: Juan Lopez, Francisco Marques, Jie ShenAbstract:An efficient and accurate numerical scheme is presented for the three-dimensional Navier?Stokes equations in primitive variables in a cylinder. The scheme is based on a Spectral-Galerkin approximation for the space variables and a second-order Projection scheme for time. The new Spectral-Projection scheme is implemented to simulate unsteady incompressible flows in a cylinder.
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an efficient Spectral Projection method for the navier stokes equations in cylindrical geometries i axisymmetric cases
Journal of Computational Physics, 1998Co-Authors: Juan Lopez, Jie ShenAbstract:Abstract An efficient and accurate numerical scheme is presented for the axisymmetric Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a new Spectral-Galerkin approximation for the space variables and a second-order Projection scheme for the time variable. The new Spectral-Projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition. A sensible comparison is made with a standard second-order (in time and space) finite difference scheme based on a stream function-vorticity formulation and with available experimental data. The numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition, the Spectral-Projection scheme is still more accurate (in terms of a fixed number of unknowns) and more efficient (in terms of CPU time required for resolving the flow at a fixed Reynolds number to within a prescribed accuracy) than the finite difference scheme. More importantly, the Spectral-Projection scheme can be readily extended to three-dimensional nonaxisymmetric cases.
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An Efficient Spectral-Projection Method for the Navier–Stokes Equations in Cylindrical Geometries: I. Axisymmetric Cases
Journal of Computational Physics, 1998Co-Authors: Juan Lopez, Jie ShenAbstract:Abstract An efficient and accurate numerical scheme is presented for the axisymmetric Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a new Spectral-Galerkin approximation for the space variables and a second-order Projection scheme for the time variable. The new Spectral-Projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition. A sensible comparison is made with a standard second-order (in time and space) finite difference scheme based on a stream function-vorticity formulation and with available experimental data. The numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition, the Spectral-Projection scheme is still more accurate (in terms of a fixed number of unknowns) and more efficient (in terms of CPU time required for resolving the flow at a fixed Reynolds number to within a prescribed accuracy) than the finite difference scheme. More importantly, the Spectral-Projection scheme can be readily extended to three-dimensional nonaxisymmetric cases.
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A numerical study of periodically forced flows using a Spectral-Projection method
Sixteenth International Conference on Numerical Methods in Fluid Dynamics, 1Co-Authors: J. M. Lopez, Jie ShenAbstract:Recent experiments (Weisberg, Kevrekidis & Smits 1997) have demonstrated that the centrifugal instability leading to Taylor vortex flow can be controlled by harmonic oscillations of the inner cylinder in the axial direction. Marques & Lopez (1997) used linear Floquet analysis to study the control of the instability for the flow between two infinite length co-axial cylinders. However, nonlinear end-wall effects have not been investigated previously for the physically realistic case of finite length co-axial cylinders. In this paper, we use an accurate and efficient Spectral-Projection scheme for the nonlinear axisymmetric Navier-Stokes equations to examine the endwalls effects and the breaking of space-time symmetries.
Juan Lopez - One of the best experts on this subject based on the ideXlab platform.
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An Efficient Spectral-Projection Method for the Navier-Stokes Equations in Cylindrical Geometries
Journal of Computational Physics, 2002Co-Authors: Juan Lopez, Francisco Marques, Jie ShenAbstract:An efficient and accurate numerical scheme is presented for the three-dimensional Navier?Stokes equations in primitive variables in a cylinder. The scheme is based on a Spectral-Galerkin approximation for the space variables and a second-order Projection scheme for time. The new Spectral-Projection scheme is implemented to simulate unsteady incompressible flows in a cylinder.
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an efficient Spectral Projection method for the navier stokes equations in cylindrical geometries i axisymmetric cases
Journal of Computational Physics, 1998Co-Authors: Juan Lopez, Jie ShenAbstract:Abstract An efficient and accurate numerical scheme is presented for the axisymmetric Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a new Spectral-Galerkin approximation for the space variables and a second-order Projection scheme for the time variable. The new Spectral-Projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition. A sensible comparison is made with a standard second-order (in time and space) finite difference scheme based on a stream function-vorticity formulation and with available experimental data. The numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition, the Spectral-Projection scheme is still more accurate (in terms of a fixed number of unknowns) and more efficient (in terms of CPU time required for resolving the flow at a fixed Reynolds number to within a prescribed accuracy) than the finite difference scheme. More importantly, the Spectral-Projection scheme can be readily extended to three-dimensional nonaxisymmetric cases.
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An Efficient Spectral-Projection Method for the Navier–Stokes Equations in Cylindrical Geometries: I. Axisymmetric Cases
Journal of Computational Physics, 1998Co-Authors: Juan Lopez, Jie ShenAbstract:Abstract An efficient and accurate numerical scheme is presented for the axisymmetric Navier–Stokes equations in primitive variables in a cylinder. The scheme is based on a new Spectral-Galerkin approximation for the space variables and a second-order Projection scheme for the time variable. The new Spectral-Projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition. A sensible comparison is made with a standard second-order (in time and space) finite difference scheme based on a stream function-vorticity formulation and with available experimental data. The numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition, the Spectral-Projection scheme is still more accurate (in terms of a fixed number of unknowns) and more efficient (in terms of CPU time required for resolving the flow at a fixed Reynolds number to within a prescribed accuracy) than the finite difference scheme. More importantly, the Spectral-Projection scheme can be readily extended to three-dimensional nonaxisymmetric cases.
Gnaneshwar Nelakanti - One of the best experts on this subject based on the ideXlab platform.
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superconvergence results of legendre Spectral Projection methods for weakly singular fredholm hammerstein integral equations
Journal of Computational and Applied Mathematics, 2019Co-Authors: Moumita Mandal, Gnaneshwar NelakantiAbstract:Abstract In this paper, we consider the Galerkin method to approximate the solution of Fredholm–Hammerstein integral equations of second kind with weakly singular kernels, using Legendre polynomial bases. We prove that for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence O ( n − r ) , whereas the iterated Legendre Galerkin method converges with the order O ( n − r − α + 1 2 ) for the algebraic kernel, and order O ( log n n − r − 1 2 ) for logarithmic kernel in both L 2 -norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the solution. We also propose the Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods. We prove that iterated Legendre multi-Galerkin method has order of convergence O ( ( 1 + c l o g n ) n − r − 2 α + 1 2 ) for the algebraic kernel, and order of convergence O ( ( log n ) 2 ( 1 + c l o g n ) n − r − 3 2 ) for logarithmic kernel in both L 2 -norm and infinity norm. Numerical examples are given to illustrate the theoretical results.
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Superconvergence results of Legendre Spectral Projection methods for Volterra integral equations of second kind
Computational and Applied Mathematics, 2018Co-Authors: Moumita Mandal, Gnaneshwar NelakantiAbstract:In this paper, Legendre Spectral Projection methods are applied for the Volterra integral equations of second kind with a smooth kernel. We prove that the approximate solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the order $${\mathcal {O}}(n^{-r})$$ O ( n - r ) in $$L^2$$ L 2 -norm and order $${\mathcal {O}}(n^{-r+\frac{1}{2}})$$ O ( n - r + 1 2 ) in infinity norm, and the iterated Legendre Galerkin solution converges with the order $${\mathcal {O}}(n^{-2r})$$ O ( n - 2 r ) in both $$L^2$$ L 2 -norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order $${\mathcal {O}}(n^{-r })$$ O ( n - r ) in both $$L^2$$ L 2 -norm and infinity norm, n being the highest degree of Legendre polynomials employed in the approximation and r being the smoothness of the kernels. We have also considered multi-Galerkin method and its iterated version, and prove that the iterated multi-Galerkin solution converges with the order $${\mathcal {O}}(n^{-3r})$$ O ( n - 3 r ) in both infinity and $$L^2$$ L 2 norm. Numerical examples are given to illustrate the theoretical results.
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superconvergence of legendre Spectral Projection methods for fredholmhammerstein integral equations
Journal of Computational and Applied Mathematics, 2017Co-Authors: Moumita Mandal, Gnaneshwar NelakantiAbstract:In this paper, we consider the multi-Galerkin and multi-collocation methods for solving the FredholmHammerstein integral equation with a smooth kernel, using Legendre polynomial bases. We show that Legendre multi-Galerkin and Legendre multi-collocation methods have order of convergence O(n3r+34) and O(n2r+12), respectively, in uniform norm, where n is the highest degree of Legendre polynomial employed in the approximation and r is the smoothness of the kernel. Also, one step of iteration method is used to improve the order of convergence and we prove that iterated Legendre multi-Galerkin and iterated Legendre multi-collocation methods have order of convergence O(n4r) and O(n2r), respectively, in uniform norm. Numerical examples are given to illustrate the theoretical results.
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legendre Spectral Projection methods for fredholm hammerstein integral equations
Journal of Scientific Computing, 2016Co-Authors: Payel Das, Mitali Madhumita Sahani, Gnaneshwar Nelakanti, Guangqing LongAbstract:In this paper, we consider the Legendre Spectral Galerkin and Legendre Spectral collocation methods to approximate the solution of Hammerstein integral equation. The convergence of the approximate solutions to the actual solution is discussed and the rates of convergence are obtained. We are able to obtain similar superconvergence rates for the iterated Legendre Galerkin solution for Hammerstein integral equations with smooth kernel as in the case of piecewise polynomial based Galerkin method.
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convergence analysis of legendre Spectral Projection methods for hammerstein integral equations of mixed type
Journal of Applied Mathematics and Computing, 2015Co-Authors: Payel Das, Mitali Madhumita Sahani, Gnaneshwar NelakantiAbstract:In this paper, we consider the Legendre Spectral Galerkin and Legendre Spectral collocation methods to approximate the solution of Hammerstein integral equations of mixed type. We prove that the approximated solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the same orders, $$\mathcal {O}(n^{-r})$$ in $$L^{2}$$ -norm and $$\mathcal {O}(n^{\frac{1}{2}-r})$$ in infinity norm, and the iterated Legendre Galerkin solution converges with the order $$\mathcal {O}(n^{-2r})$$ in both $$L^{2}$$ -norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order $$\mathcal {O}(n^{-r})$$ in both $$L^{2}$$ -norm and infinity norm, n being the highest degree of Legendre polynomial employed in the approximation and r being the smoothness of the kernels.
Howard S. Taylor - One of the best experts on this subject based on the ideXlab platform.
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The calculation of transmission and resonance properties of quantum devices using methods from chemical reactive scattering
Superlattices and Microstructures, 1996Co-Authors: T. R. Ravuri, Vladimir A. Mandelshtam, Howard S. TaylorAbstract:Abstract Two relatively new methods, the Spectral Projection method and the stabilization method, of implementing scattering calculations are described, and are here applied to two devices. Both methods use essentially short range Spectral Projection operators to produce a complete set of solutions of the wave equation that need be valid only inside the interaction region. While the Spectral Projection method is more generic than the stabilization method which is based on using the more difficult to compute Spectral density operator, the latter becomes very efficient when narrow resonances exist. For problems of small size both methods are practical in the sense that they involve only real, symmetric matrices resulting from Hamiltonians represented on L 2 basis sets. For more challenging larger systems the Spectral Projection method lends itself to a very efficient time independent iterative procedure that obtains results simultaneously at all energies. This procedure uses modified Chebyshev recursion relations to essentially expand the operator ( E − H ) −1 . It requires minimal storage and the resulting series converges rapidly in a manner that is uniform in energy.
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Spectral Projection approach to the quantum scattering calculations
The Journal of Chemical Physics, 1995Co-Authors: Vladimir A. Mandelshtam, Howard S. TaylorAbstract:A new method of implementing scattering calculations is presented. For the S‐matrix computation it produces a complete set of solutions of the wave equation that need be valid only inside the interaction region. For problems with small sizes the method is one of several that are practical in the sense that it involves merely a real symmetric Hamiltonian represented in a minimal L2 basis set. For more challenging larger systems it lends itself to a very efficient time independent iterative procedure that obtains results simultaneously at all energies. A modified Chebyshev polynomial expansion of (E−H)−1 is used. This acts on a set of energy independent wave packets located on the edge of the interaction region. The procedure requires minimal storage and is shown to converge rapidly in a manner that is uniform in energy.
