Fast Tensor Solutions For Several Classes Of Fractional Partial Differential Equations

The study of fractional calculus operators began almost simultaneously with that of integer calculus operators,but the development and application of fractional calculus are still far from enough due to the lack of a specific physical model background.In recent years,the multi-scale nature of fractional operators has stimulated the interest in fractional calculus,and has led to a large number of practical applications,especially for the simulation of physical problems,including the constitutive equations of viscoelastic materials,transport processes in complex media,mechanics,nonlocal elasticity,biomedical engineering,and so on.Although fractional equations can simulate some complex physical models well,it is often difficult to obtain the analytical solutions of the models.Therefore,the research of efficient numerical solution methods for fractional equations is of great interest.The non-local nature of the fractional operators leads to the dense nature of the discrete linear system,which makes the numerical computation very difficult.Therefore,the efficient solution of related linear systems has become a hot topic of research.With the increasing complexity of the solution problem,it has become an important research topic to compute very large linear systems in a shorter time.As a powerful tool to solve the “curse of dimensionality” problem,the low-rank tensor has become a hot issue in numerical algebra.This paper mainly studies the following four types of fractional partial differential equations: high-dimensional space Riesz fractional equation,high-dimensional timespace Caputo-Riesz fractional diffusion equation,spatial fractional Sturm-Liouville differential equation and variable-order time-space fractional diffusion equations for fast solutions.There are seven chapters in the whole paper.Chapter 1 introduces the research background and current status of the topic,and briefly introduces the main research content and innovations of this paper.The solution of high-dimensional fractional partial differential equations is a challenge in numerical computation.In Chapter 2,a low-rank preconditioned conjugate gradient(PCG)method is proposed for the fast solution of the high-dimensional fractional Riesz equation.In the Krylov subspace iteration,the low-rank tensor technique combined with the matrix-free algorithm is used to accelerate the multiplication of multilevel Toeplitz matrices with vectors.In addition,based on the spectral information of the coefficient matrix,we construct a low-rank circulant preconditioner.Numerical results show that the new method is competitive in terms of computational efficiency and memory requirements compared to the state-of-the-art matrix-free methods.In Chapter 3,we study a fast tensor solution method for the time-space CaputoRiesz fractional equations.The equations are discretized to obtain a nonsymmetric multilevel Toeplitz linear system,and the right-hand side of the system is assumed to have a low-rank form.We symmetrize the linear system and propose a low-rank preconditioned minimum residual method(LRPMINRES)for solving this system.Tensor techniques are used to accelerate the multiplication of multilevel Toeplitz matrices and vectors in the iterative process.Combining the circulant matrix and the-matrix,we propose a fast diagonalizable preconditioner and implement matrix-free computation.The eigenvalues of the preconditioned matrix are mostly distributed in the interval [-3/2,3/2]/{0},which ensures fast convergence of the iterations of the Krylov subspace.Further,based on the structure of the proposed preconditioner,we construct an approximate low-rank preconditioner using the Sinc integral method.Numerical results show that the low-rank method has a great advantage in terms of computational time and degrees of freedom in solving the problem compared to the state-of-the-art matrix-free methods.In Chapter 4,we propose a robust preconditioning technique for the time-space Caputo-Riesz fractional equation.The robust preconditioner consists of -circulant matrix and -matrix,which can be easily implemented for matrix-free and low-rank calculations.Theoretical analysis shows that the singular values of the preconditioned matrix are mainly clustered in and around the interval [1/2,3/2].Further,to solve larger-scale problems,we design a flexible GMRES method for solving linear systems with low-rank preconditioners.The results of numerical experiments show that the proposed preconditioner is efficient and robust,and the number of iteration steps is almost independent of the grid parameters.In Chapter 5,we consider a fast solution of the linear system arising from the fractional Sturm-Liouville problem.The coefficient matrix of the linear system contains a Toeplitz or Toeplitz-like structure.Based on a proper circulant approximation of the coefficient matrix,we construct a matching preconditioner in matrix-free form.Theoretically,the spectrum of the preconditioned matrix is shown to be clustered in and around [1/2,1),which ensures the accelerated convergence of the proposed preconditioner in the Krylov subspace iterative method.In addition,we use the tensor algorithm to solve the linear system efficiently.The performance of the proposed preconditioner in accelerating the GMRES method is tested numerically.The computational results show that the proposed preconditioner is very efficient compared to some other existing preconditioning methods.Compared with the constant-order fractional equation,the variable-order fractional equation can model the order change in a more detailed and accurate way.Therefore,the variable-order fractional model has attracted much attention in recent years.Due to the nonlocal property of the variable-order fractional operators,the linear system after discretization is dense and has no special structure.Therefore,it is very difficult to solve the variable-order fractional equations.In Chapter 6,we consider a fast algorithm for the high-dimensional variable-order space-time fractional diffusion equation.First,an implicit discretization scheme based on Grunwald's formula is proposed,and its stability and convergence are discussed.Then,to reduce the computational cost,two preconditioning methods are designed based on the structure of the coefficient matrix.In addition,we design two low-rank tensor minimal energy reduction algorithms for high-dimensional problems to solve the associated huge linear systems.The results of numerical experiments show the effectiveness of the proposed method.Finally,a summary of the article and future research directions are given in Chapter 7.