ترغب بنشر مسار تعليمي؟ اضغط هنا

Stable High Order Quadrature Rules for Scattered Data and General Weight Functions

120   0   0.0 ( 0 )
 نشر من قبل Jan Glaubitz
 تاريخ النشر 2020
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English
 تأليف Jan Glaubitz




اسأل ChatGPT حول البحث

Numerical integration is encountered in all fields of numerical analysis and the engineering sciences. By now, various efficient and accurate quadrature rules are known; for instance, Gauss-type quadrature rules. In many applications, however, it might be impractical---if not even impossible---to obtain data to fit known quadrature rules. Often, experimental measurements are performed at equidistant or even scattered points in space or time. In this work, we propose stable high order quadrature rules for experimental data, which can accurately handle general weight functions.



قيم البحث

اقرأ أيضاً

123 - Jan Glaubitz 2020
In many applications, it is impractical -- if not even impossible -- to obtain data to fit a known cubature formula (CF). Instead, experimental data is often acquired at equidistant or even scattered locations. In this work, stable (in the sense of n onnegative only cubature weights) high-order CFs are developed for this purpose. These are based on the approach to allow the number of data points N to be larger than the number of basis functions K which are integrated exactly by the CF. This yields an (N-K)-dimensional affine linear subspace from which cubature weights are selected that minimize certain norms corresponding to stability of the CF. In the process, two novel classes of stable high-order CFs are proposed and carefully investigated.
In this work, we propose and investigate stable high-order collocation-type discretisations of the discontinuous Galerkin method on equidistant and scattered collocation points. We do so by incorporating the concept of discrete least squares into the discontinuous Galerkin framework. Discrete least squares approximations allow us to construct stable and high-order accurate approximations on arbitrary collocation points, while discrete least squares quadrature rules allow us their stable and exact numerical integration. Both methods are computed efficiently by using bases of discrete orthogonal polynomials. Thus, the proposed discretisation generalises known classes of discretisations of the discontinuous Galerkin method, such as the discontinuous Galerkin collocation spectral element method. We are able to prove conservation and linear $L^2$-stability of the proposed discretisations. Finally, numerical tests investigate their accuracy and demonstrate their extension to nonlinear conservation laws, systems, longtime simulations, and a variable coefficient problem in two space dimensions.
Data sites selected from modeling high-dimensional problems often appear scattered in non-paternalistic ways. Except for sporadic-clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These features d efy any theoretical treatment that requires local or global quasi-uniformity of distribution of data sites. Incorporating a recently-developed application of integral operator theory in machine learning, we propose and study in the current article a new framework to analyze kernel interpolation of high dimensional data, which features bounding stochastic approximation error by a hybrid (discrete and continuous) $K$-functional tied to the spectrum of the underlying kernel matrix. Both theoretical analysis and numerical simulations show that spectra of kernel matrices are reliable and stable barometers for gauging the performance of kernel-interpolation methods for high dimensional data.
In this paper we design efficient quadrature rules for finite element discretizations of nonlocal diffusion problems with compactly supported kernel functions. Two of the main challenges in nonlocal modeling and simulations are the prohibitive comput ational cost and the nontrivial implementation of discretization schemes, especially in three-dimensional settings. In this work we circumvent both challenges by introducing a parametrized mollifying function that improves the regularity of the integrand, utilizing an adaptive integration technique, and exploiting parallelization. We first show that the mollified solution converges to the exact one as the mollifying parameter vanishes, then we illustrate the consistency and accuracy of the proposed method on several two- and three-dimensional test cases. Furthermore, we demonstrate the good scaling properties of the parallel implementation of the adaptive algorithm and we compare the proposed method with recently developed techniques for efficient finite element assembly.
We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the solution, and we establish their energy stability. This class includes as a special case high order, unconditionally stable schemes obtained via convexity splitting. The new schemes are demonstrated on a variety of gradient flows, including partial differential equations that are gradient flow with respect to the Wasserstein (mass transport) distance.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا