Generalized Fractional Algebraic Linear System Solvers

Abstract

This paper is devoted to the numerical computation of algebraic linear systems involving several matrix power functions; that is finding x solution to $\sum _{\alpha \in \mathbb {R}}A^{\alpha }x=b$. These systems will be referred to as Generalized Fractional Algebraic Linear Systems (GFALS). In this paper, we derive several gradient methods for solving these very computationally complex problems, which themselves require the solution to intermiediate standard Fractional Algebraic Linear Systems (FALS) $A^{\alpha }x=b$, with $\alpha \in \mathbb {R_+}$. The latter usually require the solution to many classical linear systems $Ax=b$. We also show that in some cases, the solution to a GFALS problem can be obtained as the solution to a first-order hyperbolic system of conservation laws. We also discuss the connections between this PDE-approach and gradient-type methods. The convergence analysis is addressed and some numerical experiments are proposed to illustrate and compare the methods which are proposed in this paper.

A Accelerated FLS Solvers

To accelerate the ODE-based solver for FLS, we propose an original two-step preconditioning technique. The principle is relatively simple and consists in the following steps.

• Construction of a preconditioner P such that $P^{-1}A \approx I$, and such that P and A commute. We refer for instance to [6] where a large number of commuting preconditioners are derived for linear systems solved using a conjugate gradient method. Some of these preconditioners can naturally be used to other iterative Krylov-type solvers.

• Construction on a fine mesh (step $\delta \tau$, and $J_{\tau }$ iterations such that $J_{\tau }\delta \tau =1$) of an approximation to $P^{-\alpha }b$, by efficiently numerically solving on [0, 1] by an ODE-solver

  $$\begin{aligned} y'(\tau ) = -\alpha (P-I)\big (I+\tau (P-I)\big )^{-1}y(\tau ), \qquad y(0)=b \in \mathbb {R}^n\, . \end{aligned}$$

  (37)

  This provides a precise approximation $\{y^{(j)}\}_j$ to $y(\tau )=(I+\tau (P-I))^{-\alpha }b$, such that $y(1)=P^{-\alpha }b$. Practically, the solution to linear systems involving P must be computed much faster than those involving A. Typically, P can be taken as incomplete-LU or Jacobi preconditioners as long as they commute with A.

• Computation on a coarser mesh (step $\varDelta T \gg \varDelta t$ and $J_T\ll J_{\tau }$ iterations) with an ODE-solver of

  $$\begin{aligned} z'(\tau ) = -\alpha (A-P)\big (P+\tau (A-P)\big )^{-1}z(\tau ), \qquad z(0)=P^{-\alpha }b \in \mathbb {R}^n\, . \end{aligned}$$

  (38)

  This provides an accurate approximation $\{z^{(j)}\}_j$, even on a coarse grid, to $z(\tau )=(P+\tau (A-P))^{-\alpha }b$, such that $z(1)=A^{-\alpha }b$. Notice that the commutativity of A and P is required in order to justify that z is indeed solution to (38). Practically, the initial data in (38) will be numerically given by the approximate solution $y^{(J_T)}$ to y(1) in (37).

To analytically justify the approach and for the sake of simplicity, let us assume that solving a linear system involving P (resp. A) has a quadratic (cubic) complexity and that the equations (37) and (38) are solved by using a standard implicit Euler method (although in practice we will use higher order methods), i.e. there exists $c(P)>0$ such that

$$\begin{aligned} \Vert y^{(J_T)}-P^{-\alpha }b\Vert\leqslant & {} c(P)\delta \tau \, . \end{aligned}$$

Hence a direct approach (4) has a complexity $O(J_{\tau }n^3)$ on a fine mesh $\delta \tau$, while the preconditioned approach developed here has a complexity $O(J_{\tau }n^2 + J_T n^3)$, with $J_{T}\ll J_{\tau }$. For instance, if $J_{\tau }\propto n$, we gain an order of complexity. Regarding the accuracy of the preconditioned technique, let us denote by f the following matrix valued function involved in the original direct ODE-solver (4) and g the one appearing in (38)

$$\begin{aligned} f(\tau ) = -\alpha (I-A)(I+\tau (A-I))^{-1}, \, \, \, g(\tau ) = -\alpha (P-A)(P+\tau (A-P))^{-1} \, . \end{aligned}$$

Hence, we have

$$\begin{aligned} df(\tau ) = - \alpha (I-A)^2(I+\tau (A-I))^{-2}, \, \, \, dg(\tau ) = -\alpha (I-P^{-1}A)^2(I+\tau (P^{-1}A-I))^{-2} . \end{aligned}$$

Then denoting $\varLambda _{f}:=\sup _{\tau \in [0,1]}\Vert df(\tau )\Vert$ and $\varLambda _{g}:=\sup _{\tau \in [0,1]}\Vert dg(\tau )\Vert$, for $P^{-1}A \approx I$, we get

$$\begin{aligned} \varLambda _g&\lessapprox&\alpha \Vert I-P^{-1}A\Vert ^2 \, . \end{aligned}$$

For instance, if we assume that A is positive definite, a precise estimate of $\varLambda _f$ is easily obtained as

$$\begin{aligned} \varLambda _f= & {} \alpha \sup _{\tau \in [0,1]}\max _{1\leqslant i\leqslant n}\Big [\frac{1-\lambda _i}{1+\tau (\lambda _i-1)}\Big ]^2 \, . \end{aligned}$$

Beyond the order of convergence, the error of the method is typically dependent of $\exp (\varLambda _{f,g})$ (through the constant in the error estimate) [37]. As a consequence the smaller $\varLambda _{f,g}$, the smaller the error. More precisely, for $\{x^{(j)}\}_j$ (resp. $\{z^{(j)}\}_j$) approximate solution to (4) (resp. (38)), we obtain

$$\begin{aligned} \left. \begin{array}{lcl} \Vert x^{(J_{\tau })}-A^{-\alpha }b\Vert \leqslant \exp (\varLambda _f) \delta \tau \, , \, \, \, \Vert z^{(J_T)}-A^{-\alpha }b\Vert \leqslant \exp (\varLambda _g) (\Vert y^{(J_T)}-P^{-\alpha }b\Vert + \varDelta \tau ) \, . \end{array} \right. \end{aligned}$$

Although $\varDelta \tau \gg \delta \tau$, this effect is counter-balanced by the fact that $\exp (\varLambda _g) \ll \exp (\varLambda _f)$. Whenever we consider non-commutative preconditioners, the construction of efficient ODE-solvers for FLS is a bit more tricky, and in this case we again refer to [4].