360 research outputs found
A spectral method to the stochastic Stokes equations on the sphere
No full textWe construct numerical solutions to the stochastic Stokes equations on the unit sphere with additive noise. By characterising the noise as a tangential vector field, the weak formulation is derived and a spectral method is used to obtain a numerical solution. The theory is illustrated through a numerical experiment.
References P. Benner and C. Trautwein. Optimal distributed and tangential boundary control for the unsteady stochastic Stokes equations. Technical Report, 2018. URL https://arxiv.org/abs/1809.00911. P. Chen, A. Quarteroni, and G. Rozza. Stochastic optimal Robin boundary control problems of advection-dominated elliptic equations. SIAM J. Numer. Anal., 51(5):2700–2722, 2013. doi:10.1137/120884158. A. Ciraudo, C. D. Negro, A. Herault, and A. Vicari. Advances in modelling methods for lava flow simulation. Commun. SIMAI Cong., 2:1–8, 2007. doi:10.1685/CSC06067. W. Freeden and M. Schreiner. Spherical functions of mathematical geosciences. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer-Verlag, 2009. doi:10.1007/978-3-540-85112-7. M. Ganesh and Q. T. L. Gia. A radial basis Galerkin method for spherical surface Stokes equation. ANZIAM J., 52:C56–C71, 2011. doi:10.21914/anziamj.v52i0.3921. M. Ganesh, Q. T. L. Gia, and I. H. Sloan. A pseudospectral quadrature method for Navier–Stokes equations on rotating spheres. Math. Comput., 80:1397–1430, 2011. doi:10.1090/S0025-5718-2010-02440-8. A. A. Il'in. The Navier–Stokes and Euler equations on two-dimensional manifolds. Math. USSR Sbornik, 69:559–579, 1991. doi:10.1070/sm1991v069n02abeh002116. F. Narcowich, J. Ward, and G. Wright. Divergence-free RBFs on surfaces. J. Fourier Anal. Appl., 13:634–663, 2007. doi:10.1007/s00041-006-6903-2. S. S. Sritharan. Optimal control of viscous flow. SIAM, 1998. doi:10.1137/1.9781611971415. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii. Quantum theory of angular momentum. World Scientific, 2008. doi:10.1142/0270
Continuous and discrete least-squares approximation by radial basis functions on spheres
No full textAbstractIn this paper we discuss Sobolev bounds on functions that vanish at scattered points on the n-sphere Sn in Rn+1. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least-squares surface fits via radial basis functions (RBFs). We also address a stabilization or regularization technique known as spline smoothing
Associated Probabilities in Interactive MADM under Discrimination q-Rung Picture Linguistic Environment
No full textIn some multi-attribute decision-making (MADM) models studying attributes’ interactive phenomena is very important for the minimizing decision risks. Usually, the Choquet integral type aggregations are considered in such problems. However, the Choquet integral aggregations do not consider all attributes’ interactions; therefore, in many cases, when these interactions are revealed in less degree, they do not perceive these interactions and their utility in MADM problems is less useful. For the decision of this problem, we create the Choquet integral-based new aggregation operators’ family which considers all pair interactions between attributes. The problem under the discrimination q-rung picture linguistic and q-rung orthopair fuzzy environments is considered. Construction of a 2-order additive fuzzy measure (TOAFM) involves pair interaction indices and importance values of attributes of a MADM model. Based on the attributes’ pair interactions for the identification of associated probabilities of a 2-order additive fuzzy measure, the Shapley entropy maximum principle is used. The associated probabilities q-rung picture linguistic weighted averaging (APs-q-RPLWA) and the associated probabilities q-rung picture linguistic weighted geometric (APs-q-RPLWG) aggregation operators are constructed with respect to TOAFM. For an uncertainty pole of experts’ evaluations on attributes regarding the possible alternatives, the associated probabilities of a fuzzy measure are used. The second pole of experts’ evaluations as arguments of the aggregation operators by discrimination q-rung picture linguistic values is presented. Discrimination q-rung picture linguistic evaluations specify the attribute’s dominant, neutral and non-dominant impacts on the selection of concrete alternative from all alternatives. Constructed operators consider the all relatedness between attributes in any consonant attribute structure. Main properties on the rightness of extensions are showed: APs-q-RPLWA and APs-q-RPLWG operators match with q-rung picture linguistic Choquet integral averaging and geometric operators for the lower and upper capacities of order two. The conjugation among the constructed operators is also considered. Connections between the new operators and the compositions of dual triangular norms (Tp,Spq) and (Tmin,Smax) are also constructed. Constructed operators are used in evaluation of a selection reliability index (SRI) of candidate service centers in the facility location selection problem, when small degree interactions are observed between attributes. In example MADM, the difference in optimal solutions is observed between the Choquet integral aggregation operators and their new extensions. The difference, however, is due to the need to use indices of all interactions between attributes
Additive Schwarz preconditioners for interpolation of divergence-free vector fields on spheres
No full textThe linear system arising from the interpolation problem of surface divergence-free vector fields using radial basis functions tends to be ill-conditioned when the separation radius of the scattered data is small. When the surface under consideration is the unit sphere, we introduce a preconditioner based on the additive Schwarz method to accelerate the solution process. Theoretical estimates for the condition number of the preconditioned matrix are given. Numerical experiments using scattered data from the MAGSAT satellite show the effectiveness of our preconditioner.
References E. Fuselier, F. Narcowich, J. D. Ward, and G. Wright. Error and stability estimates for surface-divergence free {RBF} interpolants on the sphere. Math. Comp., 78:2157--2186, 2009. doi:10.1090/S0025-5718-09-02214-5. J. R. Holton. An Introduction to Dynamic Meteorology. Academic Press, San Francisco, 3rd ed., 1992. doi:10.1119/1.1987371. Q. T. {Le Gia}, I. H. Sloan, and T. Tran. Overlapping additive {S}chwarz preconditioners for elliptic {PDE}s on the unit sphere. Math. Comp., 78:79--101, 2009. doi:10.1090/S0025-5718-08-02150-9. Q. T. {Le Gia} and T. Tran. An overlapping additive {S}chwarz preconditioner for interpolation on the unit sphere with spherical radial basis functions. J. of Complexity, 26:552--573, 2010. doi:10.1016/j.jco.2010.06.003. F. J. Narcowich, J. D. Ward, and G. B. Wright. Divergence-free {RBFs} on surfaces. J. Fourier Anal. Appl., 13:643--663, 2007. doi:10.1007/s00041-006-6903-2. H. Wendland. Scattered Data Approximation. Cambridge University Press, Cambridge, 2005. doi:10.2277/0521843359
Fast iterative solvers for boundary value problems on a local spherical region
No full textBoundary value problems on local spherical regions arise naturally in geophysics and oceanography when scientists model a physical quantity on large scales. Meshless methods using radial basis functions provide a simple way to construct numerical solutions with high accuracy. However, the linear systems arising from these methods are usually ill-conditioned, which poses a challenge for iterative solvers. We construct preconditioners based on an additive Schwarz method to accelerate the solution process for solving boundary value problems on local spherical regions.
References D. Crowdy. Point vortex motion on the surface of a sphere with impenetrable boundaries. Physics of Fluids, 18:036602 (2006). doi:10.1063/1.2183627. A. E. Gill. Atmosphere-Ocean Dynamics, International Geophysics Series Volume 30. Academic, New York (1982). R. Kidambi and P. K. Newton. Point vortex motion on a sphere with solid boundaries. Physics of Fluids, 12:581 (2000). doi:10.1063/1.870263. Q. T. Le Gia, I. H. Sloan, and T. Tran. Overlapping additive Schwarz preconditioners for elliptic PDEs on the unit sphere. Math. Comp., 78:79--101 (2009). doi:10.1090/S0025-5718-08-02150-9. C. Muller. Spherical Harmonics, Vol. 17 of Lecture Notes in Mathematics. Springer-Verlag, Berlin (1966). M. V. Nezlin. Some remarks on coherent structures out of chaos in planetary atmospheres and oceans. Chaos, 4:109--111 (1994). doi:10.1063/1.165997. T. Tran, Q. T. Le Gia, I. H. Sloan, and E. P. Stephan. Preconditioners for pseudodifferential equations on the sphere with radial basis functions. Numer. Math., 115:141--163 (2009). doi:10.1007/s00211-009-0269-8
Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings
No full textThe class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: Firstly, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms
Decision-making framework with q-rung picture fuzzy linguistic information and its applications to logistics hubs during disaster response establishment
No full textIn the decision-making matrix of modern interactive multi-attribute group decision-making (MAGDM) models, we find such syntactic-semantic forms of presentation of experts' cognitive information that thoroughly ensure the expert's maximum intellectual activity in the decision-making process. Both quantitative and qualitative experts’ activities are considered in q-rung picture fuzzy linguistic sets (q-RPFLSs) environment. The operations based on dual Archimedean t-norms and s-norms on q-rung picture fuzzy linguistic numbers (q-RPFLNs) are defined, which ensure the closure of operation results in q-RPFLNs. The extensions of the Choquet integral averaging (CA) and geometric (CG) operators for q-RPFLNs-arguments are constructed. In order to increase consideration of interaction between attributes in MAGDM models, Archimedean q-RPFLSs associated probability averaging and geometric operators are introduced, which in the q-RPFLNs environment, in a certain sense, represent the extensions of the CA and CG operators. The new operators are averaging type aggregation operators. In order to better illustrate the obtained results, a practical, high-value interactive MAGDM problem - temporary logistics hubs location selection problem is considered. The different results of the extended Choquet aggregations compared to the classical Choquet integral aggregations are explained, which are mainly due to the high degree of attributes interaction in decision-making procedure. The obtained results are also compared to the ranked aggregations of well-known decision-making methods such as TOPSIS, VIKOR and TODIM
Prevalence and visual risk factors for falls in bilateral cataract patients in Ho Chi Minh City, Vietnam
No full textPurpose\ud
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To determine the prevalence of falls in the 12 months prior to cataract surgery and examine the associations between visual and other risk factors and falls among older bilateral cataract patients in Vietnam.\ud
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Methods\ud
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Data collected from 413 patients in the week before scheduled cataract surgery included a questionnaire and three objective visual tests.\ud
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Results\ud
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The outcome of interest was self-reported falls in the previous 12 months. A total of 13% (n = 53) of bilateral cataract patients reported 60 falls within the previous 12 months. After adjusting for age, sex, race, employment status, comorbidities, medication usage, refractive management, living status and the three objective visual tests in the worse eye, women (odds ratio, OR, 4.64, 95% confidence interval, CI, 1.85–11.66), and those who lived alone (OR 4.51, 95% CI 1.44–14.14) were at increased risk of a fall. Those who reported a comorbidity were at decreased risk of a fall (OR 0.43, 95% CI 0.19–0.95). Contrast sensitivity (OR 0.31, 95% CI 0.10–0.95) was the only significant visual test associated with a fall. These results were similar for the better eye, except the presence of a comorbidity was not significant (OR 0.45, 95% CI 0.20–1.02). Again, contrast sensitivity was the only significant visual factor associated with a fall (OR 0.15, 95% CI 0.04–0.53).\ud
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Conclusion\ud
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Bilateral cataract patients in Vietnam are potentially at high risk of falls and in need of falls prevention interventions. It may also be important for ophthalmologists and health professionals to consider contrast sensitivity measures when prioritizing cataract patients for surgery and assessing their risk of falls
Zooming from Global to Local: A Multiscale RBF Approach
No full textBecause physical phenomena on Earth’s surface occur on many different length scales, it makes sense when seeking an efficient approximation to start with a crude global approximation, and then make a sequence of corrections on finer and finer scales. It also makes sense eventually to seek fine scale features locally, rather than globally. In the present work, we start with a global multiscale radial basis function (RBF) approximation, based on a sequence of point sets with decreasing mesh norm, and a sequence of (spherical) radial basis functions with proportionally decreasing scale centered at the points. We then prove that we can “zoom in ” on a region of particular interest, by carrying out further stages of multiscale refinement on a local region. The proof combines multiscale techniques for the sphere from Le Gia, Sloan and Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32 (2012), with those for a bounded region in Rd from Wendland, Numer. Math. 116 (2012). The zooming in process can be continued indefinitely, since the condition numbers of matrices at the different scales remain bounded. A numerical example illustrates the process.
Construction of interlaced polynomial lattice rules with SPOD weights
No full textWe start from a parametrised PDE to define an infinite
dimensional integral which we want to approximate by a QMC rule.
The integral is the expectation over the parameter space of a linear
functional of the solution of the PDE.
It turns out that the regularity of the solution in the parameter domain can
be transferred to the integration problem in the form of SPOD weights.
Then I will show how to construct interlaced polynomial lattice rules for this
setting using a fast CBC algorithm.
This is a report on joint work with J. Dick, F. Y. Kuo, Q. T. Le Gia and
Ch. Schwab.sponsorship: Research Foundation Flanders (FWO)status: Publishe
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