In this paper, we consider an orthogonal spline collocation (OSC) method to solve the fourth-order multi-term subdiffusion equation. The L1 method on graded meshes is employed in temporal direction. ...In the spatial direction, we develop the orthogonal spline collocation method. The stability and convergence of the above mentioned method are proved theoretically, and it's still valid if α1→1−. A new robust error bound is obtained, which is a more natural and desirable result and does not blow up as α1→1−. Some numerical results are also provided to verify our theoretical analysis and demonstrate the effectiveness of the OSC method.
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This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present ...an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.
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Satellite SSS products are usually validated by comparing them with collocated in situ observations from Argo, mooring buoys, and ship measurements. However, the remote sensed data are the ...measurements averaged spatially within the satellite footprint and in situ observations are pointwise measurements. The different spatial resolution of satellite and in situ data causes the problem of representativeness error (RE) that affects the validation results of satellite SSS. Many efforts have been devoted to properly estimating the REs. REs can be resolved naturally by a quadruple collocated dataset and linear algebra method if the source of REs is known beforehand. Therefore, identifying the source of REs is more important than estimating them. In this study, we find that, for a given quadruple dataset, the results in the error estimation from triple collocation subsets show different characteristics. These differences can be used to identify the source of RE. Then, we present a novel methodology of RE identification based on the characteristics of the error estimation. Using the match-up SSS dataset provided by Pi-MEP and numerical experiments, the feasibility of this method is verified. The results show that for the quadruple collocation (QC) of SMOS, SMAP, Argo, and WOA, the RE occurs in SMOS and SMAP which have similar resolutions with a value of 0.10 psu.
Weighted least squares collocation methods Brugnano, Luigi; Iavernaro, Felice; Weinmüller, Ewa B.
Applied numerical mathematics,
September 2024, 2024-09-00, Volume:
203
Journal Article
Peer reviewed
Open access
We consider overdetermined collocation methods and propose a weighted least squares approach to derive a numerical solution. The discrete problem requires the evaluation of the Jacobian of the vector ...field which, however, appears in a O(h) term, h being the stepsize. We show that, by neglecting this infinitesimal term, the resulting scheme becomes a low-rank Runge–Kutta method. Among the possible choices of the weights distribution, we analyze the one based on the quadrature formula underlying the collocation conditions. A few numerical illustrations are included to better elucidate the potential of the method.
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Collocation is a ubiquitous phenomenon in languages and accurate collocation recognition and extraction is of great significance to many natural language processing tasks. Collocations can be ...differentiated from simple bigram collocations to collocation frames (referring to distant multi-gram collocations). So far little focus is put on collocation frames. Oriented to translation and parsing, this study aims to recognize and extract the longest possible collocation frames from given sentences. We first extract bigram collocations with distributional semantics based method by introducing collocation patterns and integrating some state-of-the-art association measures. Based on bigram collocations extracted by the proposed method, we get the longest collocation frames according to recursive nature and linguistic rules of collocations. Compared with the baseline systems, the proposed method performs significantly better in bigram collocation extraction both in precision and recall. And in extracting collocation frames, the proposed method performs even better with the precision similar to its bigram collocation extraction results.
This paper is an introductory tutorial for numerical trajectory optimization with a focus on direct collocation methods. These methods are relatively simple to understand and effectively solve a wide ...variety of trajectory optimization problems. Throughout the paper we illustrate each new set of concepts by working through a sequence of four example problems. We start by using trapezoidal collocation to solve a simple one-dimensional toy problem and work up to using Hermite-Simpson collocation to compute the optimal gait for a bipedal walking robot. Along the way, we cover basic debugging strategies and guidelines for posing well-behaved optimization problems. The paper concludes with a short overview of other methods for trajectory optimization. We also provide an electronic supplement that contains well-documented MATLAB code for all examples and methods presented. Our primary goal is to provide the reader with the resources necessary to understand and successfully implement their own direct collocation methods.
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In the present paper, we construct the numerical solution for time fractional (1 + 1)- and (1 + 2)-dimensional Schrödinger equations (TFSEs) subject to initial boundary. The solution is expanded in a ...series of shifted Jacobi polynomials in time and space. A collocation method in two steps is developed and applied. First step depends mainly on application of shifted Jacobi Gauss-Lobatto-collocation method for spatial discretization on the approximate solution and its spatial derivatives occurring in the TFSE and substitution in the boundary conditions or treatment of the non-local conservation conditions by the Jacobi Gauss-Lobatto quadrature rule. As a result, a system of fractional differential equation for the expansion coefficients is obtained. The second step is to use a shifted Jacobi Gauss-Radau- collocation scheme, for temporal discretization, to reduce such system into a system of nonlinear Newton iterative method. Numerical results carried out to confirm the spectral accuracy and efficiency of the proposed algorithms demonstrating superiority over other methods.
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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
Periodic dynamical systems ubiquitously exist in science and engineering. The harmonic balance (HB) method and its variants have been the most widely‐used approaches for such systems, but are either ...confined to low‐order approximations or impaired by aliasing and improper‐sampling problems. Here we propose a collocation‐based harmonic balance framework to successfully unify and reconstruct the HB‐like methods. Under this framework a new conditional identity, which exactly bridges the gap between frequency‐domain and time‐domain harmonic analyses, is discovered by introducing a novel aliasing matrix. Upon enforcing the aliasing matrix to vanish, we propose a powerful reconstruction harmonic balance (RHB) method that obtains extremely high‐order (>100) nonaliasing solutions, previously deemed out‐of‐reach, for a range of complex nonlinear systems including the cavitation bubble dynamics, the three‐body problem and the two dimensional airfoil dynamics. We show that the present method is 2–3 orders of magnitude more accurate and simultaneously much faster than the state‐of‐the‐art method. Hence, it has immediate applications in multidisciplinary problems where highly accurate periodic solutions are sought.
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39.
Highly stable multivalue collocation methods Conte, Dajana; D'Ambrosio, Raffaele; D'Arienzo, Maria Pia ...
Journal of physics. Conference series,
06/2020, Volume:
1564, Issue:
1
Journal Article
Peer reviewed
Open access
The paper is focused on the development of A-stable collocation based multivalue methods for stiff problems. This methods are dense output extensions of discrete multivalue methods, since the ...solution is approximated by a piecewise collocation polynomial with high global regularity. The underlying multivalue method is assumed to be diagonally implicit and with uniform order of convergence, thus it does not suffer from order reduction, as it happens for classical one-step collocation methods. The effectiveness of the approach is also confirmed by the numerical evidence.
This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions ...(TDF) to overcome the C1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points. Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO.
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