Tensor numerical methods in scientific computing

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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