AbstractIn the past two decades, meshfree methods have emerged into a new class of computational methods with considerable success. In addition, a significant amount of progress has been made in ...addressing the major shortcomings that were present in these methods at the early stages of their development. For instance, essential boundary conditions are almost trivial to enforce by employing the techniques now available, and the need for high order quadrature has been circumvented with the development of advanced techniques, essentially eliminating the previously existing bottleneck of computational expense in meshfree methods. Given the proper treatment, nodal integration can be made accurate and free of spatial instability, making it possible to eliminate the need for a mesh entirely. Meshfree collocation methods have also undergone significant development, which also offer a truly meshfree solution. This paper gives an overview of many classes of meshfree methods and their applications, and several advances are described in detail.
GPOPS-II Patterson, Michael A.; Rao, Anil V.
ACM transactions on mathematical software,
10/2014, Volume:
41, Issue:
1
Journal Article
Peer reviewed
Open access
A general-purpose MATLAB software program called
GPOPS--II
is described for solving multiple-phase optimal control problems using variable-order Gaussian quadrature collocation methods. The software ...employs a Legendre-Gauss-Radau quadrature orthogonal collocation method where the continuous-time optimal control problem is transcribed to a large sparse nonlinear programming problem (NLP). An adaptive mesh refinement method is implemented that determines the number of mesh intervals and the degree of the approximating polynomial within each mesh interval to achieve a specified accuracy. The software can be interfaced with either quasi-Newton (first derivative) or Newton (second derivative) NLP solvers, and all derivatives required by the NLP solver are approximated using sparse finite-differencing of the optimal control problem functions. The key components of the software are described in detail and the utility of the software is demonstrated on five optimal control problems of varying complexity. The software described in this article provides researchers a useful platform upon which to solve a wide variety of complex constrained optimal control problems.
A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations (TFSEs) subject to ...initial-boundary and non-local conditions. The first step depends mainly on shifted Legendre Gauss–Lobatto collocation (SL-GL-C) method for spatial discretization; an expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the TFSE is investigated. In addition, the Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions. Thereby, the expansion coefficients are then determined by reducing the TFSE with its nonlocal conditions to a system of fractional differential equations (SFDEs) for these coefficients. The second step is to propose a shifted Legendre Gauss–Radau collocation (SL-GR-C) scheme, for temporal discretization, to reduce such system into a system of algebraic equations which is far easier to be solved. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to solve the two-dimensional TFSE. Numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms. By selecting relatively limited Legendre Gauss–Lobatto and Gauss–Radau collocation nodes, we are able to get very accurate approximations, demonstrating the utility and high accuracy of the new approach over other numerical methods.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK
We propose a new reduced integration rule for isogeometric analysis (IGA) based on the concept of variational collocation. It has been recently shown that, if a discrete space is constructed by ...smooth and pointwise non-negative basis functions, there exists a set of points - named Cauchy-Galerkin (CG) points - such that collocation performed at these points can reproduce the Galerkin solution of various boundary value problems exactly. Since CG points are not known a-priori, estimates are necessary in practice and can be found based on superconvergence theory. In this contribution, we explore the use of estimated CG points (i.e. superconvergent points) as numerical quadrature points to obtain an efficient and stable reduced quadrature rule in IGA. We use the weighted residual formulation as basis for our new quadrature rule, so that the proposed approach can be considered intermediate between the standard (accurately integrated) Galerkin variational formulation and the direct evaluation of the strong form in collocation approaches. The performance of the method is demonstrated by several examples. For odd degrees of discretization, we obtain spatial convergence rates and accuracy very close to those of accurately integrated standard Galerkin with a quadrature rule of two points per parametric direction independently of the degree.
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For control problems with control constraints, a local convergence rate is established for an
hp
-method based on collocation at the Radau quadrature points in each mesh interval of the ...discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the
hp
-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OBVAL, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
Multi-Index Stochastic Collocation for random PDEs Haji-Ali, Abdul-Lateef; Nobile, Fabio; Tamellini, Lorenzo ...
Computer methods in applied mechanics and engineering,
07/2016, Volume:
306
Journal Article
Peer reviewed
Open access
In this work we introduce the Multi-Index Stochastic Collocation method (MISC) for computing statistics of the solution of a PDE with random data. MISC is a combination technique based on mixed ...differences of spatial approximations and quadratures over the space of random data. We propose an optimization procedure to select the most effective mixed differences to include in the MISC estimator: such optimization is a crucial step and allows us to build a method that, provided with sufficient solution regularity, is potentially more effective than other multi-level collocation methods already available in literature. We then provide a complexity analysis that assumes decay rates of product type for such mixed differences, showing that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one dimensional problem. We show the effectiveness of MISC with some computational tests, comparing it with other related methods available in the literature, such as the Multi-Index and Multilevel Monte Carlo, Multilevel Stochastic Collocation, Quasi Optimal Stochastic Collocation and Sparse Composite Collocation methods.
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A local convergence rate is established for an orthogonal collocation method based on Gauss quadrature applied to an unconstrained optimal control problem. If the continuous problem has a smooth ...solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as the number of collocation points increases, the discrete solution converges to the continuous solution at the collocation points, exponentially fast in the sup-norm. Numerical examples illustrating the convergence theory are provided.
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CEKLJ, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
The physical model of convective–radiative porous fin with temperature dependent properties and heat generation. Display omitted
•SCM has high accuracy and exponential convergence rate for porous ...fin.•Radiation and convection effects on porous fin are considered.•Temperature dependent thermal properties of porous fin are considered.•Effects of porous parameters on temperature and fin efficiency are analyzed.
In this work, spectral collocation method is presented to predict the thermal performance of convective–radiative porous fin with temperature dependent convective heat transfer coefficient, fin surface emissivity and internal heat generation. In this approach, the dimensionless fin temperature distribution is approximated by Lagrange interpolation polynomials at spectral collocation points. The differential form of the governing equation is formulated by the Darcy model, and is transformed to a matrix form of algebraic equation. The accuracy of the SCM is verified by compared with numerical results by the homotopy perturbation method and the finite volume method. The node convergence rate of the SCM approximately follows an exponential law, and the computational time of the SCM do not significantly increase with the increasing of collocation points. The effects of various geometric and thermo-physical parameters on the dimensionless fin temperature, fin efficiency and heat transfer rate are comprehensively analyzed. In addition, optimum design analysis is also carried out.
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The aim of this work is to identify numerically, for the first time, the time‐dependent potential coefficient in a fourth‐order pseudo‐parabolic equation with nonlocal initial data, nonlocal boundary ...conditions, and the boundary data as overdetermination condition. This problem emerges significantly in the modeling of various phenomena in physics and engineering. From literature we already know that this inverse problem has a unique solution. However, the problem is still ill‐posed by being unstable to noise in the input data. For the numerical realization, we apply the quintic B‐spline (QB‐spline) collocation method for discretizing the pseudo‐parabolic problem and the Tikhonov regularization for finding a stable and accurate solution. The resulting nonlinear minimization problem is solved using the MATLAB subroutine lsqnonlin. Moreover, the von Neumann stability analysis is also discussed.
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters ...in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss–Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.
•A rigorous and unified framework for solving and learning nonlinear PDEs.•The proposed framework is based on techniques derived from Gaussian process regression and kernel methods.•It is provably convergent and inherits complexity vs accuracy guarantees of state of the art dense kernel matrix solvers.•It is interpretable and amenable to numerical analysis.
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