Positive Approximations of the Inverse of Fractional Powers of SPD M-Matrices Stanislav Harizanov and Svetozar Margenov Abstract This study is motivated by the recent development in the fractional calculus and its applications. During last few years, several different techniques are proposed to localize the nonlocal fractional diffusion operator. They are based on transformation of the original problem to a local elliptic or pseudoparabolic problem, or to an integral representation of the solution, thus increasing the dimen- sion of the computational domain. More recently, an alternative approach aimed at reducing the computational complexity was developed. The linear algebraic system αu f, 0<α< 1 is considered, where is a properly normalized (scalded)A symmetric= and positive definite matrixA obtained fromfinite element or finite difference approximation of second order elliptic problems inΩ R d , d 1,2, 3. The method is based on best uniform rational approximations (BURA)⊂ = β α of the functiont − for 0<t 1 and naturalβ. The maximum principles≤ are among the major qualitative properties of linear elliptic operators/PDEs. In many studies and applications, it is important that such properties are preserved by the selected numerical solution method. In this α paper we present and analyze the properties of positive approximations of − obtained by the BURA technique. Sufficient conditions for positiveness are proven,A complemented by sharp error estimates. The theoretical results are supported by representative numerical tests. 1 Introduction This work is inspired by the recent development in the fractional calculus and its various applications, i.e., to Hamiltonian chaos, Zaslavsky (2002), anomalous diffusion in complex systems, Bakunin (2008), long-range interaction in elastic S. Harizanov · S. Margenov (�) Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria e-mail: [email protected] © Springer International Publishing AG, part of Springer Nature 2018 147 G. Feichtinger et al. (eds.), Control Systems and Mathematical Methods in Economics, Lecture Notes in Economics and Mathematical Systems 687, https://doi.org/10.1007/978-3-319-75169-6_8 [email protected] 148 S. Harizanov and S. Margenov deformations, Silling (2000), nonlocal electromagneticfluidflows, McCay and Narasimhan (1981), image processing, Gilboa and Osher (2008). A more recent impressive examples of anomalous diffusion models in chemical engineering are provided in Metzler et al. (2014). Such kind of applications lead to fractional order partial differential equations that involve in general non-symmetric elliptic operators see, e.g. Kilbas et al. (2006). An important subclass of this topic are the fractional powers of self-adjoint elliptic operators, which are nonlocal but self-adjoint. In particular, the fractional Laplacian (Pozrikidis 2016) describes an unusual diffusion process associated with random excursions. In general, the parabolic equations with fractional derivatives in time are associated with sub- diffusion, while the fractional elliptic operators are related to super-diffusion. Let us consider the elliptic boundary value problem in a weak form:findu V such that ∈ a(u, v) (a(x) u(x) v(x) q(x) ) dx f (x)v(x)dx, v V, := Ω ∇ ·∇ + = Ω ∀ ∈ (1) where 1 V v H (Ω) v(x) 0 onΓ D , := { ∈ : = } Γ ∂Ω, andΓ Γ Γ . We assume thatΓ has positive measure, q(x) 0 ¯D ¯N D in=Ω, anda(x) =is an SPDd∪ d matrix, uniformly bounded inΩ, i.e., ≥ × 2 T 2 d c z z a(x)z C z z R , x Ω, (2) ≤ ≤ ∀ ∈ ∀ ∈ for some positive constantsc andC. Also,Ω is a polygonal domain inR d ,d 1,2,3 , and f (x) is a given Lebesgue integrable function onΩ that belongs∈ to { } the spaceL 2(Ω). Further, the case whena(x) does not depend onx is referred to as problem in homogeneous media, while the general case models processes in non homogeneous media. The bilinear forma( , ) defines a linear operator V V ∗ withV being the dual ofV . Namely, for· all· u, v V a(u, v) u,L: v →, where ∗ ∈ :=L , is the pairing betweenV andV ∗. · ·One possible way to introduce α, 0<α< 1, is through its spectral decomposition, i.e. L α ∞ α ∞ u(x) λ ci ψ (x), where u(x) ciψ (x). (3) L = i i = i i 1 i 1 = = Here ψ i(x) i∞1 are the eigenfunctions of , orthonormal inL 2-inner product and { } = L λi i∞1 are the corresponding positive real eigenvalues. This definition general- izes{ } = the concept of equally weighted left and right Riemann-Liouville fractional derivative, defined in one space dimension, to the multidimensional case. There is still ongoing research about the relations of the different definitions and their applications, see, e.g. Bates (2006). [email protected] Fractional Power of Monotone SPD Matrices 149 The numerical solution of nonlocal problems is rather expensive. The following three approaches (A1–A3) are based on transformation of the original problem αu f (4) L = to a local elliptic or pseudo-parabolic problem, or on integral representation of the solution, thus increasing the dimension of the original computational domain. The Poisson problem is considered in the related papers refereed bellow, i.e. a(u, v) u(x) v(x)dx. := Ω ∇ ·∇ A1 Extension to a mixed boundary value problem in the semi-infinite cylinder d 1 C Ω R R + =A Neumann× + ⊂ to Dirichlet map is used in Chen et al. (2016). Then, the solution of fractional Laplacian problem is obtained by u(x) v(x,0) wherev Ω = : × R R is a solution of the equation + → 1 2α div y − v(x, y) 0, (x, y) Ω R , − ∇ = ∈ × + wherev( , y) satisfies the boundary conditions of ( 1) y R , · ∀ ∈ + lim v(x, y) 0, x Ω, y →∞ = ∈ as well as 1 2α lim y − vy (x, y) f (x), x Ω. y 0 − = ∈ → + It is shown that the variational formulation of this equation is well posed in the related weighted Sobolev space. Thefinite element approximation uses the rapid decay of the solution v(x, y) in they direction, thus enabling truncation of the semi-infinite cylinder to a bounded domain of modest size. The proposed multilevel method is based on the Xu-Zikatanov identity (Xu and Zikatanov 2002). The numerical tests forΩ (0,1) andΩ (0,1) 2 confirm the theoretical estimates of almost optimal= computational= complexity. A2 Transformation to a pseudo-parabolic problem The problem (1) is considered in Vabishchevich (2014, 2015) assuming the boundary condition ∂u a(x) μ(x)u 0, x ∂Ω, ∂n + = ∈ [email protected] 150 S. Harizanov and S. Margenov which ensures ∗ δ ,δ> 0. Then the solution of fractional power diffusion problemL=uL canbefound≥ I as α u(x) w(x,1), w(x,0) δ − f, = = where w(x, t),0<t< 1, is the solution of pseudo-parabolic equation dw (t δ ) α w 0, D+ I dt + D = and δ 0. Stability conditions are obtained for the fully discreteD=L schemes− I≥ under consideration. A further development of this approach is presented in Lazarov and Vabishchevich (2017) where the case of fractional order boundary conditions is studied. A3 Integral representation of the solution The following representation of the solution of (1) is used in Bonito and Pasciak (2015): 1 α 2 sin(πα) ∞ 2α 1 2 − − t − t dt L = π 0 I+ L Among others, the authors introduce an exponentially convergent quadrature scheme. Then, the approximate solution ofu only involves evaluations of 1 ( t i )− f , wheret i (0, ) is related to the current quadrature node, andI+ whereA and stand for∈ the∞ identity and thefinite element stiffness matrix correspondingI toA the Laplacian. The computational complexity of the method depends on the number of quadrature nodes. For instance, the presented analysis shows that approximately 50 auxiliary linear systems have to be solved to get 5 accuracy of the quadrature scheme of orderO(10 − ) forα 0.25,0.5,0.75 . A further development of this approach is available in∈ Bonito{ and Pasciak} (2016), where the theoretical analysis is extended to the class of regularly accretive operators. An alternative approach is applied in Harizanov et al. (2016) where a class of optimal solvers for linear systems with fractional power of symmetric and positive N N definite (SPD) matrices is proposed. Let R × be a normalized SPD matrix generated by afinite element orfinite differenceA∈ approximation of some self- adjoint elliptic problem. An efficient method for solving algebraic systems of linear equations involving fractional powers of the matrix is considered, namely for solving the system A αu f, where 0<α<1. (5) A = The fractional power of SPD matrix , similarly to the infinite dimensional counterpart , is expressed through theA spectral representation ofu through the L N eigenvalues and eigenvectors (Λ i, i ) i 1 of , assuming that the eigenvectors { } = A [email protected] Fractional Power of Monotone SPD Matrices 151 T arel 2-orthonormal, i.e. i j δ ij andΛ 1 Λ 2 ...Λ N 1. Then T , α α T , where= theN N matrices≤ ≤ and are≤ defined asA= WDW TA T=WD TW × W αD α T ( 1 , 2 ,..., N ) and diag(Λ 1,...,Λ N ), − − , and theW= solution of αu f can be expressedD= as A =WD W A = α α T u − f − f. (6) =A =WD W α β β Instead of the system (5), one can solve the equivalent system − u − f F withβ 1 an integer. Then the idea is to approximately evaluateA β α=FA using:= a set ≥ A − of equations involving inversion of and d j I, forj 1,...,k. The integer parameterk 1 is the number of partialA A fractions− of the= best uniform rational ≥ β β α approximation (BURA)r α (t) oft − on the interval(0,1 .