We consider a numerical algorithm for the two-dimensional time-harmonic elastic wave scattering by unbounded rough surfaces with Dirichlet boundary condition.A Nystr¨om method is proposed for the scattering probl...We consider a numerical algorithm for the two-dimensional time-harmonic elastic wave scattering by unbounded rough surfaces with Dirichlet boundary condition.A Nystr¨om method is proposed for the scattering problem based on the integral equation method.Convergence of the Nystr¨om method is established with convergence rate depending on the smoothness of the rough surfaces.In doing so,a crucial role is played by analyzing the singularities of the kernels of the relevant boundary integral operators.Numerical experiments are presented to demonstrate the effectiveness of the method.Mathematics subject classification:35P25,45P05.展开更多
While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approxima...While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper,we analyse the Nystrom solution of one-dimensional nonlinear Volterra integral equation of the second kind and show that approkimate solution admits an asymptotic error expansion in even powers of the step-size h, beginning with a term in h2. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.展开更多
Approximations based on random Fourier features have recently emerged as an efficient and elegant method for designing large-scale machine learning tasks.Unlike approaches using the Nystr?m method,which randomly sampl...Approximations based on random Fourier features have recently emerged as an efficient and elegant method for designing large-scale machine learning tasks.Unlike approaches using the Nystr?m method,which randomly samples the training examples,we make use of random Fourier features,whose basis functions(i.e.,cosine and sine)are sampled from a distribution independent from the training sample set,to cluster preference data which appears extensively in recommender systems.Firstly,we propose a two-stage preference clustering framework.In this framework,we make use of random Fourier features to map the preference matrix into the feature matrix,soon afterwards,utilize the traditional k-means approach to cluster preference data in the transformed feature space.Compared with traditional preference clustering,our method solves the problem of insufficient memory and greatly improves the efficiency of the operation.Experiments on movie data sets containing 100000 ratings,show that the proposed method is more effective in clustering accuracy than the Nystr?m and k-means,while also achieving better performance than these clustering approaches.展开更多
基金supported by the National Key R&D Program of China(Grant 2018YFA0702502)the Beijing Natural Science Foundation(Grant Z210001)+2 种基金the NNSF of China(Grants 12171057,12271515,12201023)the Youth Innovation Promotion Association CAS,by the Education Department of Hunan Province(Grant 21B0299)the Fundamental Research Funds for the Central Universities(Grant YWF-23-Q-1026,YWF-22-T-204)。
文摘We consider a numerical algorithm for the two-dimensional time-harmonic elastic wave scattering by unbounded rough surfaces with Dirichlet boundary condition.A Nystr¨om method is proposed for the scattering problem based on the integral equation method.Convergence of the Nystr¨om method is established with convergence rate depending on the smoothness of the rough surfaces.In doing so,a crucial role is played by analyzing the singularities of the kernels of the relevant boundary integral operators.Numerical experiments are presented to demonstrate the effectiveness of the method.Mathematics subject classification:35P25,45P05.
文摘While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is well understood, there exist no systematic studies of asymptotic error expansion for the approximate solution. In this paper,we analyse the Nystrom solution of one-dimensional nonlinear Volterra integral equation of the second kind and show that approkimate solution admits an asymptotic error expansion in even powers of the step-size h, beginning with a term in h2. So that the Richardson's extrapolation can be done. This will increase the accuracy of numerical solution greatly.
基金supported by the National Natural Science Foundation of China(Nos.61872260 and 61592419)the Natural Science Foundation of Shanxi Province(No.201703D421013).
文摘Approximations based on random Fourier features have recently emerged as an efficient and elegant method for designing large-scale machine learning tasks.Unlike approaches using the Nystr?m method,which randomly samples the training examples,we make use of random Fourier features,whose basis functions(i.e.,cosine and sine)are sampled from a distribution independent from the training sample set,to cluster preference data which appears extensively in recommender systems.Firstly,we propose a two-stage preference clustering framework.In this framework,we make use of random Fourier features to map the preference matrix into the feature matrix,soon afterwards,utilize the traditional k-means approach to cluster preference data in the transformed feature space.Compared with traditional preference clustering,our method solves the problem of insufficient memory and greatly improves the efficiency of the operation.Experiments on movie data sets containing 100000 ratings,show that the proposed method is more effective in clustering accuracy than the Nystr?m and k-means,while also achieving better performance than these clustering approaches.
