Boris N. Choromskij Volgorde van de boeken

The most challenging computational problems today involve higher dimensions. This research monograph introduces tensor numerical methods for solving multidimensional issues in scientific computing, focusing on rank-structured approximations of multivariate functions and operators using various tensor formats. It explores both traditional and new rank-structured tensor formats, emphasizing the innovative quantized tensor approximation method (QTT), which enables function-operator calculus in higher dimensions with logarithmic complexity, facilitating rapid convolution, FFT, and wavelet transforms. The book provides constructive recipes and computational schemes for real-life problems governed by multidimensional partial differential equations. It presents theory and algorithms for sinc-based separable approximations of analytic radial basis functions, including Green’s and Helmholtz kernels. Efficient tensor-based techniques are discussed for electronic structure calculations and grid-based evaluations of long-range interaction potentials in multi-particle systems. Additionally, the QTT numerical approach is examined in many-particle dynamics, tensor techniques for stochastic/parametric PDEs, and the solution and homogenization of elliptic equations with highly oscillating coefficients. Key topics include the theory on separable approximation of multivariate functions, multilinear algebra, nonlinear tensor approximation, and su

During the last decade essential progress has been achieved in the analysis and implementation of multilevel/rnultigrid and domain decomposition methods to explore a variety of real world applications. An important trend in mod ern numerical simulations is the quick improvement of computer technology that leads to the well known paradigm (see, e. g. , [78,179]): high-performance computers make it indispensable to use numerical methods of almost linear complexity in the problem size N, to maintain an adequate scaling between the computing time and improved computer facilities as N increases. In the h-version of the finite element method (FEM), the multigrid iteration real izes an O(N) solver for elliptic differential equations in a domain n c IRd d with N = O(h- ) , where h is the mesh parameter. In the boundary ele ment method (BEM) , the traditional panel clustering, fast multi-pole and wavelet based methods as well as the modern hierarchical matrix techniques are known to provide the data-sparse approximations to the arising fully populated stiffness matrices with almost linear cost O(Nr log? Nr), where 1 d Nr = O(h - ) is the number of degrees of freedom associated with the boundary. The aim of this book is to introduce a wider audience to the use of a new class of efficient numerical methods of almost linear complexity for solving elliptic partial differential equations (PDEs) based on their reduction to the interface.