On The TT/QTT Methods For Diffusion Equation With Variable Coefficient And Multivariate Regression Model

This paper mainly discusses QTT methods for solving a class of diffusion equation with variable coefficient and nonparametric multivariate regression model based on TT decomposition.Although considerable research has been devoted to finite difference QTT methods for solving diffusion equation with constant coefficient, rather less atten-tion has been paid to that for solving diffusion equation with variable coefficient. The key to solving variable coefficient diffusion equation lies in constructing the QTT decomposition of finite difference discrete of variable coefficient diffusion op-erator. This paper presents explicit QTT representation of finite difference discrete of variable coefficient diffusion operator using the strong Kronecker representation methods for QTT decomposition. And we propose a fast, accurate method for solving variable coefficient diffusion equation using DMRG algorithm for large-scale linear equation.Nonparametric multivariate regression is one of the important problems of s-tatistic learning. There are many methods such as local model, smoothing splines, additive model and regression trees to handle nonparametric multivariate regres-sion. Our method belongs to smoothing splines method. We assume that the decision function has TT decomposition, and select a set of spline basic function-s, and then fit the smoothing parameters resulting with splines regression model based on TT decomposition. The calculation of learning algorithm for this model is quasi-linear with the number of dimensions and goodness of fit is high.In this paper, We obtain main results as followsWe deduce the strong Kronecker representation methods of TT decomposi-tion and construct the explicit QTT representation of standard finite differ-ence discrete of variable coefficient diffusion operator;We apply DMRG algorithm for solving large-scale linear equation to the ex-plicit QTT representation of standard finite difference discrete of variable coefficient diffusion operator and get accurate numerical solutions of a 3-dimensional steady diffusion equation with variable coefficient and its higher dimensional case and a 3-dimensional unsteady diffusion equation with vari-able coefficient with good results.We propose splines regression model based on TT decomposition for non-parametric multivariate regression and deduce fast learning algorithm based on ALS. And we test our method with three numerical examples and results demonstrate that our method works well.