Fast solver for fractional differential equations based on Hierarchical Matrices

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2 International Conference on Fractional Differentiation and its Applications, Novi Sad, Seria, July 18-20, 2016 Fast solver for fractional differential equations ased on Hierarchical Matrices Xuan Zhao 2,1, Xiaozhe Hu 3, Wei Cai 1,4, George Em Karniadakis 5 1 Division of Algorithms, Beijing Computational Science Research Center, Beijing, P. R. China 2 Department of Mathematics, Southeast University, Nanjing , P. R. China xuanzhao11@seu.edu.cn; xuanzhao11@gmail.com 3 Department of Mathematics, Tufts University, Medford, MA, USA xiaozhe.hu@tufts.edu 4 Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC, USA wcai@csrc.ac.cn 5 Division of Applied Mathematics, Brown University, Providence, RI, USA george karniadakis@rown.edu Astract A roust and fast solver for the fractional differential equation (FDEs) involving the Riesz fractional derivative is developed using an adaptive finite element method on nonuniform meshes. It is ased on the utilization of hierarchical matrices (H-Matrices) for the representation of the stiffness matrix resulting from the finite element discretization of the FDEs. We employ a geometric multigrid method for the solution of the algeraic system of equations. We comine it with an adaptive algorithm ased on a posteriori error estimation to deal with general-type singularities arising in the solution of the FDEs. Through various test examples we demonstrate the efficiency of the method and the highaccuracy of the numerical solution even in the presence of singularities. To the est of our knowledge, there are currently no other methods for FDEs that resolve singularities accurately at linear complexity as the one we propose here. Key words: Hierarchical Matrices, Riesz derivative, geometric multigrid method, adaptivity, non-smooth solutions, finite element method 1 Introduction Numerical methods for differential equations involving fractional derivatives either in time or space have een studied widely since they have various physical and iological applications [3, 15, 17]. However, as the size of the prolem increases, the time required to solve the final system of equations increases consideraly due to the nonlocality of the

3 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices fractional differential operators. Generally, there are two main difficulties in solving space fractional prolems. First, the matrix otained y the numerical discretization is fully populated. This leads to increased storage memory requirements as well as increased solution time. Since the matrix is dense, the memory required to store its coefficients is of order O(N 2 ), where N denotes the numer of unknowns, and the solution of the system requires O(N 3 ) operations if direct solvers are used. Second, the solution of a space fractional prolem has singularities around the oundaries even with smooth input data since the definition involves the integration of weak singular kernels. So far good progress has een made on reducing the computational complexity of the space FDEs for uniform mesh discretizations ased on the oservation that the coefficient matrices are Toeplitz-like if a uniform mesh is employed. This results in efficient matrixvector multiplications [11, 14, 18], which in conjunction with effective preconditioners lead to enhanced efficiency [8, 10, 12]. The multigrid method [4, 9, 14] has also een employed to reduce the computational cost to O(N log(n)). However, to the est of our knowledge, most of the existing fast solvers depend on the Toeplitz-like structure of the matrices, which implies that the underlying meshes have to e uniform. Therefore, the existing efficient solvers cannot e readily applied when a nonuniform mesh is used, including the important case of non-uniform meshes generated y adaptive discretizations employed to deal with singularities. Another effective approach to deal with such singularities is y tuning the appropriate asis, e.g.. in Galerkin and collocation spectral methods. For example, the weighted Jacoi polynomials can e used to accommodate the weak singularity if one has some information aout the solution [5, 19, 20, 21]; the proper choice of the asis results in significant improvement in the accuracy of numerical solutions. However, in the current work we assume that we do not have any information on the non-smoothness of the solution. H-Matrices [6] have een developed over the last twenty years as a powerful datasparse approximation of dense matrices. This representation has een used for solving integral equations and elliptic partial differential equations [1, 2, 7]. The main advantage of H-Matrices are the reduction of storage requirement and efficiency in solving linear system, e.g. when storing a dense matrix, which requires O(N 2 ) units of storage, while H- Matrices provide an approximation requiring only O(N k log(n)) units of storage, where k is a parameter controlling the accuracy of the approximation. In this work, instead of using the Topelitz-like structure of the matrices to reduce the computational complexity, we adapt the H-Matrices representation to approximate the dense matrices arising from the discretization of the FDEs. Our H-Matrices approach does not restrict to the uniform meshes and can e easily generalized to the non-uniform meshes. Therefore, it is suitale for the adaptive finite element method (AFEM) for FDEs. We will show theoretically that the error of such H-Matrices representation decays like O(3 k ) while the storage complexity is O(Nk log(n)). Thanks to the newly designed algorithm for solving the linear system of equations, the new AFEM for FDEs achieves the optimal computational complexity, while it maintains the optimal convergence order. The key to such AFEM algorithm for FDEs is the optimal linear solver we developed ased on H-Matrices representation and the geometric multigrid (GMG) method. The remainder of the paper is structured as follows. The finite element discretization and H-Matrices representation for FDE are discussed in Section 2. The overall AFEM algorithm is discussed in detail in Section 3, and numerical experiments are presented in Section 4 to demonstrate the high efficiency and accuracy of the proposed new method.

4 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices 2 Discretization of the prolem ased on H-Matrices representation We consider the following fractional differential equation D α x u(x) = f(x), x (, c), 1 < α < 2, (1) where f L 2 ([, c]) and Dx α denotes the Riesz fractional derivative, sujected to the oundary conditions u() = 0, u(c) = 0. Following the Galerkin approach, we solve equation (1) projected onto the finite dimensional space V := span{ϕ 1,, ϕ N } and V H0 1 ([, c]), where H0 1 ([, c]) is the standard Soolev space on [, c] and {ϕ i } are standard piecewise linear asis functions defined on a mesh = x 0 < x 1 < < x N < x N+1 = c with meshsize h i = x i+1 x i, i = 1, 2,, N. We multiply v V y (1) and integrate over [, c], D α x u(x)ϕ i (x) dx = f(x)ϕ i (x) dx. (2) By integration y parts, we otain the following weak formulation of (1): find u(x) V, such that [ 1 d c ] x ξ 1 α u(ξ) dξ v (x) dx = f(x)v(x) dx, v V 2c(α) dx where c(α) = cos(απ/2)γ(2 α). We rewrite the discrete solution u n = N j=1 u jϕ j V and then the coefficient vector u = (u 1, u 2,, u N ) is the solution of the linear system where and A ij := [ 1 2c(α) d dx f i := Au = f, ] x ξ 1 α ϕ j (ξ) dξ ϕ i(x) dx, (3) ϕ i (x)f(x) dx. (4) The matrix A is dense as all entries are nonzero. Our aim is to approximate A y a matrix Ã which can e stored in a data-sparse (not necessarily sparse) format. The idea is to replace the kernel S(x, ξ) = x ξ 1 α y a degenerate kernel 2.1 Taylor Expansion of the Kernel k 1 S(x, ξ) = p ν (x)q ν (ξ). (5) ν=0 Let τ := [a, ], σ := [c, d ], τ σ [c, d] [c, d] e a sudomain with the property < c such that the intervals are disjoint: τ σ =. Then the kernel function is nonsingular in τ σ.

5 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices Lemma 1 Taylor expansions of the left/right kernel ν x ν x [ (x ξ) 1 α ] ν = ( 1) ν (α + l 2)(x ξ) 1 α ν, [ (ξ x) 1 α ] = l=1 ν (α + l 2)(ξ x) 1 α ν. l=1 Then we can use the truncated Talyor series at x 0 := (a + )/2 to approximate the kernel and eventually otain an approximation of the stiffness matrix, i. e., where S(x, ξ) := k 1 ν=0 [ ν ] 1 (α + l 2)(ξ x 0 ) 1 α ν (x x 0 ) ν := ν! l=1 k 1 p ν (x)q ν (ξ), (6) ν=0 p ν (x) = (x x 0 ) ν, (7) q ν (ξ) = 1 ν (α + l 2)(ξ x 0 ) 1 α ν. ν! (8) l=1 2.2 Low rank approximation of Matrix Blocks If τ σ is admissile, then we can approximate the kernel S in this sudomain y the truncated Taylor series S from (6) and replace the matrix entries A ij y the use of the degenerate kernel S(x, ξ) for the indices (i, j) t s : Ã ij = [ 1 2c(α) d c S(x, ξ)ϕ dx j (ξ) dξ ] ϕ i(x) dx, (9) in which the doule integral is separated into two integrals: [ c ] 1 d c k 1 Ã ij = p ν (x)q ν (ξ)ϕ 2c(α) dx j (ξ) dξ ϕ i(x) dx (10) = 1 2c(α) ν=0 k 1 [ ] [ p ν(x)ϕ i(x) dx ν=0 ] q ν (ξ)ϕ j (ξ) dξ Thus, the sumatrix A t s can e represented in a factorized form A t s = 1 2c(α) CRT, C R t {0,,k 1} s {0,,k 1}, R R where the entries of the matrix factors C and R are C iν := p ν(x)ϕ i(x) dx, R jν := (11) q ν (ξ)ϕ j (ξ) dξ. (12) Now we have the following error estimation for the H-Matrix representation.

6 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices Theorem 2 (Element-wise Approximation Error) Let τ := [a, ], σ := [c, d ], < c, x 0 = (a + )/2, and k 2, we have A ij Ãij 1 [ ] 3 [ ] k k(k 1)(h i + h i+1 )(h j + h j+1 ) diam(τ) + 2dist(τ, σ) dist(τ, σ) 8c(α) ( x 0 a + c ) α , x 0 a 2 2dist(τ, σ) diam(τ) (13) where diam(τ) is the diameter of τ and dist(τ, σ) is the distance etween the intervals τ and σ. If dist(τ,σ) diam(τ) 1, we have A ij Ãij 27 1 k(k 1)(h i + h i+1 )(h j + h j+1 ) 64 c(α) ( x 0 a + c ) α 1 x 0 a 2 (3) k. (14) Figure 1: Error in the Froenius norm ( A Ã F, where M F := i j M ij ) mainly depends on the ratio rather than on α. Left: ratio := dist(τ,σ) diam(τ) with diam(τ) the diameter of the interval τ and dist(τ, σ) the distance etween the intervals τ and σ; Right: α (1, 2) is the order of the fractional derivative. [ ] k, In the error estimate (13), we can see that the dominating term is dist(τ,σ) diam(τ) which determines the decaying rate and, thus, the quality of the approximation. If dist(τ, σ) 0, the approximation will degenerate. However, if we require diam(τ) dist(τ, σ) as in the Theorem 2, we can have a nearly uniform ound A ij Ãij = O(3 k ). Moreover, the element-wise error mainly depends on the ratio dist(τ,σ) diam(τ) if we assume diam(τ) dist(τ, σ), and the igger the ratio the etter the approximation. The error decays exponentially with respect to the order k. Figure 1 shows how the error decays with respect to k when we vary the ratio or α; we can see clearly that the error depends strongly on the ratio rather than α, which supports our error estimates.

7 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices 3 Adaptive Finite Element Method for Fractional PDEs In this section, we discuss the adaptive finite element method (AFEM) for solving FDEs. We follow the idea of standard AFEM, which is characterized y the following iteration SOLVE ESTIMATE MARK REFINE. Such iteration generates a sequence of discrete solutions converging to the exact one. We want to emphasize that one of the main difficulties of applying the AFEM to the FDEs is the SOLVE step. As usual, the SOLVE module should take the current grid as the input and output the corresponding finite element approximation. Note here that the current grid, in general, is otained y adaptive refinement and, therefore, it is an unstructured grid. To the est of our knowledge, all the existing fast solvers can not e used which makes AFEM infeasile. Our SOLVE module use the hierarchical matrix representation mentioned in Section 2 to assemle the linear system of equations stored in H-Matrix format and we also design efficient GMG method to solve it ased on the H-matrix format. More precisely, adaptive refinement gives us a natural nested unstructured grids ased on which we uild the GMG hierarchy. As usual, the main components of our GMG method are the smoother and prolongation. The latter is defined y the standard inclusion operator etween the grids ecause the grids are nested. For the smoother, we adapt the Gauss-Seidel smoother ut modify it ased on the H-Matrix representation in order to keep the computational complexity optimal. Finally, V-cycle GMG is defined y recursion on coarser grids and it is easy to check the overall computational complexity of our GMG method is O(kN l log N l ) where N l is the numer of the degrees of freedom on level l. Given a grid T l and finite element approximation ũ l V l, the ESTIMATE module computes a posteriori error estimators {η l (ũ l, τ)} τ Tl, which should e computale on each element τ T l and indicate the true error. In this work, we adopt the gradient recovery ased error estimators due to its simplicity and prolem-independence. It is easy to check that the computational complexity of ESTIMATE module is O(N l ) ecause all the computations are done locally on the elements τ and on each element τ, the operations are finite and independent of N l. The MARK module selects elements τ T l whose local error η l (U l, τ) is relatively large and needs to e refined in the refinement. This module is independent of the model prolems and we can directly use the strategies developed for second order elliptic PDEs for FDES here. In this work, we use the so-called Döflers marking strategy. The REFINE module is also prolem independent. It takes the marked elements and current grid T k as inputs and outputs a refined grid T l+1, which will e used as a new grid. In this work, ecause we are only considering the 1D case, the refinement procedure is just isection. Namely, if an element [x l i 1, xl i ] T l is marked, it will e divided into two suintervals [x l i 1, xl i ] and [ xl i, xl i ] y the midpoint xl i = (xl i 1 + xl i )/2. The overall computational cost of each iteration of the AFEM method is O(kN l log N l ) or O(N l log N l ) when k N l. 4 Numerical example Example 1: Solving model prolem (1) with right hand side f(x) = 1 + sin(x).

8 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices The aove example we consider here does not have exact solution. However, due to the property of the FDEs, we expect the solution to have singularities near the oundaries, which leads to degenerated convergence rate in the errors of finite element approximations on uniform grids. This is confirmed y the numerical results as shown in Figure 2 for α = 1.5. The convergence rates of the L 2 errors for oth full matrix and H-Matrix approaches are aout 1.24, which reflects the singularity of the solution and the necessity for the AFEM method. These comparisons show that the AFEM algorithm can achieve etter accuracy with less computational cost, which demonstrates the effectiveness of the AFEM algorithm for FDEs. Figure 2: Convergence rates for full matrix and H-Matrix representation on uniform grids. Next we apply the AFEM algorithm to solve Example 1. The results are shown in Figure 3. We can see that, using the AFEM method, the optimal convergence rates of oth L 2 error and L have een recovered for α = 1.5. This demonstrates the effectiveness and roustness of the our AFEM methods. In Figure 4, we plot the numerical solutions on adaptive meshes for α = 1.5. The adaptive refinement near the oundary points demonstrates that our error estimate captures the singularities well and overall roustness of our AFEM algorithm. As mentioned in Section 3, one distinct feature of our proposed AFEM method is that in the SOLVE module, the H-Matrix representation and the multigrid method are used, hence providing nearly optimal computational complexity O(N log N). In Figure 5, we show the CPU time of the GMG method for the H-Matrix used in the SOLVE module for different fractional orders. We can see that, for all cases, the computational complexity is optimal, which confirms our expectation. Next we compare the computational costs of FEM on uniform grids and AFEM. The results are shown in Tale 1. Here DoFs denotes the degrees of freedom. For AFEM, we start the adaptive refinement from a coarse grid of size 32. For AFEM, T-DoFs denotes the sum of the DoFs of all the adaptive grids starting from the coarse grid to current adaptive grid and T-Time denotes the total CPU time of the whole AFEM algorithm while Time denotes the CPU time of solving the FDES on the current adaptive grid. For FEM on a uniform grid of size 16, 383, the L 2 error is and the CPU time is aout seconds. However, if we use AFEM, solving the prolem on an adaptive

9 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices Figure 3: Convergence rates of FEM on uniform grid and AFEM for α = 1.5. Figure 4: Numerical solution on adaptive meshes. Figure 5: CPU time of GMG method on nonuniform grid grid of size 1, 059 leads to accuracy in the L 2 norm. This means that the AFEM achieves aout 2.7 times etter results using a 15.5 times smaller grid. Even the total DoFs, which is 6, 382, is aout 2.6 times smaller than the size of the uniform grid. The speed up is aout 18.3 if we only consider the final adaptive grid and is aout 3.2 if we consider the whole AFEM procedure. In Figure 6, we report the reakdown of computational cost of the AFEM on the finest adaptive grid. As we can see, the SOLVE module (Assemling and H-Matrix & MG Solve) dominates the whole AFEM algorithm. The computational cost of the other three modules, ESTIMATE module (Estimate Error), MARK module (Mark), and REFINE module (Adaptive Refine), are roughly the same and could e ignored compared with the SOLVE module. This is mainly ecause our experiments are in 1D in the current paper. We can expect those three modules to ecome more and more time consuming in 2D and 3D cases. However, the SOLVE module should still e the dominant module in terms of computational cost and that is why we introduce the H-Matrix and the GMG method together to make sure that we achieve optimal computational complexity.

X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices Tale 1: (α = 1.5): Computational cost comparison etween FEM on uniform grids and AFEM.

10 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices Tale 1: (α = 1.5): Computational cost comparison etween FEM on uniform grids and AFEM. (The time unit is second) Uniform Adaptive L 2 error DoFs Time L 2 error DoFs Time T-DoFs T-Time , , , , , , Figure 6: (α = 1.5): Breakdown of CPU time of AFEM 5 Conclusions In this paper, we presented an adaptive FEM (AFEM) method for a fractional differential equation with Riesz derivative, targeting in particular non-smooth solutions that they may arise even in the presence of smooth right-hand-sides in the equation. To this end, uniform grids result in suoptimal and in fact su-linear convergence rate, while the AFEM yields optimal second-order accuracy. The demonstrated efficiency of the method is ased on comining two effective ideas, which act synergistically. First, we approximated the singular kernel in the fractional derivative using an H-Matrix representation, and second, we employed a geometric multigrid method with nearly linear overall computational complexity. In the current paper, we developed these ideas for the one-dimensional case ut the greater challenge is to consider higher dimensions, where adaptive refinement has to resolve oth solution singularities and geometric singularities around the oundaries. Acknowledgments This work was supported y the OSD/ARO/MURI on Fractional PDEs for Conservation Laws and Beyond: Theory, Numerics and Applications (W911NF ) and Fundamental Research Funds for the Central Universities.

11 X. Zhao, X. Hu, W. Cai, G. E. Karniadakis: Fast solver for FDEs ased on H-Matrices References [1] Beendorf M., Approximation of oundary element matrices, Numer. Math., 86, , [2] Brm, S., Grasedyck, L., Hyrid cross approximation of integral operators, Numer. Math., 101, , [3] Carpinteri, A., Mainardi, F., Fractals and fractional calculus in continuum mechanics, Springer, 378, [4] Chen, M., wang, Y., Cheng, X., Deng, W., Second-order LOD multigrid method for multidimensional Riesz fractional diffusion equation, BIT Numer. Math., 54, , [5] Chen, S., Shen, J., Wang, L.L., Generalized Jacoi functions and their applications to fractional differential equations, Math. Comp., 85, , [6] Hackusch, W., A sparse matrix arithmetic ased on H-matrices. Part I: Introduction to H-matrices, Computing, 62, , [7] Ho, K.L., Ying, L., Hierarchical interpolative factorization for elliptic operators: differential equations, Comm. Pure. Appl. Math., [8] Jia, J., Wang, H., A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299, , [9] Jiang, Y., Xu, X., Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302, , [10] Lei, S.L., Sun, H.W., A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242, , [11] Lin, F.R., Yang, S.W., Jin, X.Q., Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256, , [12] Moroney, T., Yang, Q., A anded preconditioner for the two-sided, nonlinear spacefractional diffusion equation, Comput. Math. Appl., 66, , [14] Pang, H.K., Sun, H.W., Multigrid method for fractional diffusion equations, J. Comput. Phys., 231, , [15] Reenshtok, A., Denisov, S., Hnggi, P., Barkai, E., Non-normalizale densities in strong anomalous diffusion: eyond the central limit theorem, Phys. Rev. Lett., 112, , [17] Sun, H., Chen, W., Chen, Y., Variale-order fractional differential operators in anomalous diffusion modeling, Phys. A, 388, , [18] Wang, H., Du, N., A fast finite difference method for three-dimensional timedependent space-fractional diffusion equations and its efficient implementation, J. Comput. Phys., 253, 50 63, [19] Zayernouri, M., Karniadakis, G.E., Fractional spectral collocation method, SIAM J. Sci. Comput., 36, A40 A62, [20] Zeng, F., Zhang, Z., Karniadakis, G.E., A generalized spectral collocation method with tunale accuracy for variale-order fractional differential equations, SIAM J. Sci. Comput., 37, A2710 A2732, [21] Zhao, X., Zhang, Z., Superconvergence points of fractional spectral interpolation, SIAM J. Sci. Comput., 38, A598 A613, 2016.

arxiv: v1 [math.na] 4 Mar 2016

arxiv: v1 [math.na] 4 Mar 2016 ADAPTIVE FINITE ELEMENT METHOD FOR FRACTIONAL DIFFERENTIAL EQUATIONS USING HIERARCHICAL MATRICES XUAN ZHAO, XIAOZHE HU, WEI CAI, AND GEORGE EM KARNIADAKIS arxiv:1603.01358v1 [math.na] 4 Mar 2016 Astract.