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Tucker Carrington - One of the best experts on this subject based on the ideXlab platform.
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a pruned Collocation based multiconfiguration time dependent hartree approach using a smolyak grid for solving the schrodinger equation with a general potential energy surface
Journal of Chemical Physics, 2019Co-Authors: Robert Wodraszka, Tucker CarringtonAbstract:Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a Collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The Collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with Collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with Collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because Collocation is used, there are no integrals and no need for quadrature. All required matrix-vector products can be evaluated sequentially. We use nested sets of Collocation points and hierarchical basis functions. They permit efficient inversion of the (large) matrix whose elements are basis functions evaluated at points, which is necessary to transform values of functions at points to basis coefficients. The inversion technique could be used outside of chemical physics. We confirm the validity of this new pruned, Collocation-based (PC-)MCTDH approach by calculating the first 50 vibrational eigenenergies of CH2NH.Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a Collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The Collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with Collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with Collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because Collocation is used, there are no integrals and no need for quadrature...
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a new Collocation based multi configuration time dependent hartree mctdh approach for solving the schrodinger equation with a general potential energy surface
Journal of Chemical Physics, 2018Co-Authors: Robert Wodraszka, Tucker CarringtonAbstract:We present a new Collocation-based multi-configuration time-dependent Hartree (MCTDH) approach for solving the Schrodinger equation required to compute (ro-)vibrational spectra, photodissociation cross sections, reaction rate constants, etc., that can be used with general potential energy surfaces. Collocation obviates the need for quadrature and facilitates using complicated kinetic energy operators. When the basis is good, the accuracy of Collocation solutions to the Schrodinger equation is not sensitive to the choice of the Collocation points. We test the Collocation MCTDH equations we derive by showing that they can be used to compute accurate vibrational energy levels of CH3. It is possible to choose (imaginary) time-independent Collocation points with which Collocation-based MCTDH energies are accurate. It is therefore not necessary to calculate potential values many times during the propagation.
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solving the schroedinger equation using smolyak interpolants
Journal of Chemical Physics, 2013Co-Authors: Gustavo Avila, Tucker CarringtonAbstract:In this paper, we present a new Collocation method for solving the Schroedinger equation. Collocation has the advantage that it obviates integrals. All previous Collocation methods have, however, the crucial disadvantage that they require solving a generalized eigenvalue problem. By combining Lagrange-like functions with a Smolyak interpolant, we device a Collocation method that does not require solving a generalized eigenvalue problem. We exploit the structure of the grid to develop an efficient algorithm for evaluating the matrix-vector products required to compute energy levels and wavefunctions. Energies systematically converge as the number of points and basis functions are increased.
Robert Wodraszka - One of the best experts on this subject based on the ideXlab platform.
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a pruned Collocation based multiconfiguration time dependent hartree approach using a smolyak grid for solving the schrodinger equation with a general potential energy surface
Journal of Chemical Physics, 2019Co-Authors: Robert Wodraszka, Tucker CarringtonAbstract:Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a Collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The Collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with Collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with Collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because Collocation is used, there are no integrals and no need for quadrature. All required matrix-vector products can be evaluated sequentially. We use nested sets of Collocation points and hierarchical basis functions. They permit efficient inversion of the (large) matrix whose elements are basis functions evaluated at points, which is necessary to transform values of functions at points to basis coefficients. The inversion technique could be used outside of chemical physics. We confirm the validity of this new pruned, Collocation-based (PC-)MCTDH approach by calculating the first 50 vibrational eigenenergies of CH2NH.Standard multiconfiguration time-dependent Hartree (MCTDH) calculations use a direct product basis and rely on the potential being a sum of products (SOPs). The size of the direct product MCTDH basis scales exponentially with the number of atoms. Accurate potentials may not be SOPs. We introduce an MCTDH approach that uses a pruned basis and a Collocation grid. Pruning the basis significantly reduces its size. Collocation makes it possible to do calculations using a potential that is not a SOP. The Collocation point set is a Smolyak grid. Strategies using pruned MCTDH bases already exist, but they work only if the potential is a SOP. Strategies for using MCTDH with Collocation also exist, but they work only if the MCTDH basis is a direct product. In this paper, we combine a pruned basis with Collocation. This makes it possible to mitigate the direct-product basis size problem and do calculations when the potential is not a SOP. Because Collocation is used, there are no integrals and no need for quadrature...
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a new Collocation based multi configuration time dependent hartree mctdh approach for solving the schrodinger equation with a general potential energy surface
Journal of Chemical Physics, 2018Co-Authors: Robert Wodraszka, Tucker CarringtonAbstract:We present a new Collocation-based multi-configuration time-dependent Hartree (MCTDH) approach for solving the Schrodinger equation required to compute (ro-)vibrational spectra, photodissociation cross sections, reaction rate constants, etc., that can be used with general potential energy surfaces. Collocation obviates the need for quadrature and facilitates using complicated kinetic energy operators. When the basis is good, the accuracy of Collocation solutions to the Schrodinger equation is not sensitive to the choice of the Collocation points. We test the Collocation MCTDH equations we derive by showing that they can be used to compute accurate vibrational energy levels of CH3. It is possible to choose (imaginary) time-independent Collocation points with which Collocation-based MCTDH energies are accurate. It is therefore not necessary to calculate potential values many times during the propagation.
Thomas J R Hughes - One of the best experts on this subject based on the ideXlab platform.
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isogeometric Collocation for large deformation elasticity and frictional contact problems
Computer Methods in Applied Mechanics and Engineering, 2015Co-Authors: Roland Kruse, Nhon Nguyenthanh, L De Lorenzis, Thomas J R HughesAbstract:Abstract Isogeometric Collocation methods have been recently proposed as an alternative to standard Galerkin approaches as they provide a significant reduction in computational cost for higher-order discretizations. In this work, we explore the application of isogeometric Collocation to large deformation elasticity and frictional contact problems. We first derive the non-linear governing equations for the elasticity problem with finite deformation kinematics and provide details on their consistent linearization. Some numerical examples demonstrate the performance of Collocation in its basic and enhanced versions, differing by the enforcement of Neumann boundary conditions. For problems with strong singularities, enhanced Collocation is shown to outperform basic Collocation and to lead to a spatial convergence behavior very similar to Galerkin, whereas for weaker or no singularities enhanced and basic Collocation may give very similar results. A large deformation contact formulation is subsequently developed and tested in the frictional setting, where Collocation confirms the excellent performance already obtained for the frictionless case. Finally, it is shown that the contact formulation in the Collocation framework passes the contact patch test to machine precision in a three-dimensional setting with arbitrarily inclined non-matching discretizations, thus outperforming most of the available contact formulations and all those with pointwise enforcement of the contact constraints.
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isogeometric Collocation cost comparison with galerkin methods and extension to adaptive hierarchical nurbs discretizations
Computer Methods in Applied Mechanics and Engineering, 2013Co-Authors: Dominik Schillinger, John A Evans, Alessandro Reali, Michael A Scott, Thomas J R HughesAbstract:Abstract We compare isogeometric Collocation with isogeometric Galerkin and standard C 0 finite element methods with respect to the cost of forming the matrix and residual vector, the cost of direct and iterative solvers, the accuracy versus degrees of freedom and the accuracy versus computing time. On this basis, we show that isogeometric Collocation has the potential to increase the computational efficiency of isogeometric analysis and to outperform both isogeometric Galerkin and standard C 0 finite element methods, when a specified level of accuracy is to be achieved with minimum computational cost. We then explore an adaptive isogeometric Collocation method that is based on local hierarchical refinement of NURBS basis functions and Collocation points derived from the corresponding multi-level Greville abscissae. We introduce the concept of weighted Collocation that can be consistently developed from the weighted residual form and the two-scale relation of B-splines. Using weighted Collocation in the transition regions between hierarchical levels, we are able to reliably handle coincident Collocation points that naturally occur for multi-level Greville abscissae. The resulting method combines the favorable properties of isogeometric Collocation and hierarchical refinement in terms of computational efficiency, local adaptivity, robustness and straightforward implementation, which we illustrate by numerical examples in one, two and three dimensions.
Narayana R Aluru - One of the best experts on this subject based on the ideXlab platform.
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a point Collocation method based on reproducing kernel approximations
International Journal for Numerical Methods in Engineering, 2000Co-Authors: Narayana R AluruAbstract:A reproducing kernel particle method with built-in multiresolution features in a very attractive meshfree method for numerical solution of partial differential equations. The design and implementation of a Galerkin-based reproducing kernel particle method, however, faces several challenges such as the issue of nodal volumes and accurate and efficient implementation of boundary conditions. In this paper we present a point Collocation method based on reproducing kernel approximations. We show that, in a point Collocation approach, the assignment of nodal volumes and implementation of boundary conditions are not critical issues and points can be sprinkled randomly making the point Collocation method a true meshless approach. The point Collocation method based on reproducing kernel approximations, however, requires the calculation of higher-order derivatives that would typically not be required in a Galerkin method, A correction function and reproducing conditions that enable consistency of the point Collocation method are derived. The point Collocation method is shown to be accurate for several one and two-dimensional problems and the convergence rate of the point Collocation method is addressed. Copyright © 2000 John Wiley & Sons, Ltd.
Judy P Yang - One of the best experts on this subject based on the ideXlab platform.
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a gradient reproducing kernel Collocation method for boundary value problems
International Journal for Numerical Methods in Engineering, 2013Co-Authors: Shengwei Chi, Jiunshyan Chen, Judy P YangAbstract:SUMMARY The earlier work in the development of direct strong form Collocation methods, such as the reproducing kernel Collocation method (RKCM), addressed the domain integration issue in the Galerkin type meshfree method, such as the reproducing kernel particle method, but with increased computational complexity because of taking higher order derivatives of the approximation functions and the need for using a large number of Collocation points for optimal convergence. In this work, we intend to address the computational complexity in RKCM while achieving optimal convergence by introducing a gradient reproduction kernel approximation. The proposed gradient RKCM reduces the order of differentiation to the first order for solving second-order PDEs with strong form Collocation. We also show that, different from the typical strong form Collocation method where a significantly large number of Collocation points than the number of source points is needed for optimal convergence, the same number of Collocation points and source points can be used in gradient RKCM. We also show that the same order of convergence rates in the primary unknown and its first-order derivative is achieved, owing to the imposition of gradient reproducing conditions. The numerical examples are given to verify the analytical prediction. Copyright © 2012 John Wiley & Sons, Ltd.
