ترغب بنشر مسار تعليمي؟ اضغط هنا

Fast algorithm for computing nonlocal operators with finite interaction distance

176   0   0.0 ( 0 )
 نشر من قبل Xiaochuan Tian
 تاريخ النشر 2019
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




اسأل ChatGPT حول البحث

Developments of nonlocal operators for modeling processes that traditionally have been described by local differential operators have been increasingly active during the last few years. One example is peridynamics for brittle materials and another is nonstandard diffusion including the use of fractional derivatives. A major obstacle for application of these methods is the high computational cost from the numerical implementation of the nonlocal operators. It is natural to consider fast methods of fast multipole or hierarchical matrix type to overcome this challenge. Unfortunately the relevant kernels do not satisfy the standard necessary conditions. In this work a new class of fast algorithms is developed and analyzed, which is some cases reduces the computational complexity of applying nonlocal operators to essentially the same order of magnitude as the complexity of standard local numerical methods.



قيم البحث

اقرأ أيضاً

We present a new fast algorithm for computing the Boys function using nonlinear approximation of the integrand via exponentials. The resulting algorithms evaluate the Boys function with real and complex valued arguments and are competitive with previously developed algorithms for the same purpose.
We present a fast method for evaluating expressions of the form $$ u_j = sum_{i = 1,i ot = j}^n frac{alpha_i}{x_i - x_j}, quad text{for} quad j = 1,ldots,n, $$ where $alpha_i$ are real numbers, and $x_i$ are points in a compact interval of $mathbb{R }$. This expression can be viewed as representing the electrostatic potential generated by charges on a line in $mathbb{R}^3$. While fast algorithms for computing the electrostatic potential of general distributions of charges in $mathbb{R}^3$ exist, in a number of situations in computational physics it is useful to have a simple and extremely fast method for evaluating the potential of charges on a line; we present such a method in this paper, and report numerical results for several examples.
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.
Kolmogorov famously proved that multivariate continuous functions can be represented as a superposition of a small number of univariate continuous functions, $$ f(x_1,dots,x_n) = sum_{q=0}^{2n+1} chi^q left( sum_{p=1}^n psi^{pq}(x_p) right).$$ Fridma n cite{fridman} posed the best smoothness bound for the functions $psi^{pq}$, that such functions can be constructed to be Lipschitz continuous with constant 1. Previous algorithms to describe these inner functions have only been Holder continuous, such as those proposed by Koppen and Braun and Griebel. This is problematic, as pointed out by Griebel, in that non-smooth functions have very high storage/evaluation complexity, and this makes Kolmogorovs representation (KR) impractical using the standard definition of the inner functions. To date, no one has presented a method to compute a Lipschitz continuous inner function. In this paper, we revisit Kolmogorovs theorem along with Fridmans result. We examine a simple Lipschitz function which appear to satisfy the necessary criteria for Kolmogorovs representation, but fails in the limit. We then present a full solution to the problem, including an algorithm that computes such a Lipschitz function.
The Cadzows algorithm is a signal denoising and recovery method which was designed for signals corresponding to low rank Hankel matrices. In this paper we first introduce a Fast Cadzows algorithm which is developed by incorporating a novel subspace p rojection to reduce the high computational cost of the SVD in the Cadzows algorithm. Then a Gradient method and a Fast Gradient method are proposed to address the non-decreasing MSE issue when applying the Cadzows or Fast Cadzows algorithm for signal denoising. Extensive empirical performance comparisons demonstrate that the proposed algorithms can complete the denoising and recovery tasks more efficiently and effectively.
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