1. Introduction. We address the problem of solving a set of linear equations

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1 SIAM J. SCI. COMPUT. Vol. 0, No. 0, pp c 20XX Society for Industrial and Applied Mathematics THE ITERATIVE SOLVER RISOLV WITH APPLICATION TO THE EXTERIOR HELMHOLTZ PROBLEM HILLEL TAL-EZER AND E. TURKEL Abstract. The innermost computational kernel of many large-scale scientific applications is often a large set of linear equations of the form Ax = b which typically consumes a significant portion of the overall computational time required by the simulation. The traditional approach for solving this problem is to use direct methods. This approach is often preferred in industry because direct solvers are robust and effective for moderate size problems. However, direct methods can consume a huge amount of memory, and CPU time, in large-scale cases. In these cases, iterative techniques are the only viable alternative. Unfortunately, iterative methods lack the robustness of direct methods. The situation is especially difficult when the matrix is nonsymmetric. A lot of research has been devoted to trying to develop a robust iterative algorithm for nonsymmetric systems. The present paper describes a new robust and efficient algorithm aimed at solving iteratively nonsymmetric linear systems. It is based on looking for an approximation to the optimal polynomial P m(z) which satisfies P m(z) =min Q Πm Q(z), z D, whereπ m is the set of all polynomials of degree m which satisfies Q m(0) = 1 and D is a domain in the complex plane which includes all the eigenvalues of A. The resulting algorithm is an efficient one, especially in the case where we have a set of linear systems which share the same matrix A. We present several applications, including the exterior Helmholtz problem, which leads to a large indefinite, nonsymmetric, and complex system. Key words. Risolv, Krylov, Helmholtz exterior, preconditioning AMS subject classifications. 15A15, 15A09, 15A23 DOI / X 1. Introduction. We address the problem of solving a set of linear equations (1.1) Ax = b, where A is a large, general N N matrix and b is an N 1 vector. In real-life problems, preconditioning is mandatory. In this paper we do not address this issue. When A is a symmetric, positive definite matrix, conjugate gradient [13] is shown to be the optimal algorithm. Similarly, there are optimal algorithms for the indefinite, symmetric case (e.g., MINRES, SYMMLQ [23]). The situation is significantly more complicated in the general, nonsymmetric case. A popular algorithm for this type of problem is the well-known GMRES algorithm [25], which is optimal in the following sense. Let x m 1 be the solution vector after applying m 1 matrix-vector multiplications: (1.2) x m 1 = x 0 + Q m 1 (A)r 0, where x 0, r 0 are the initial guess and residual, respectively. Hence, (1.3) r m 1 = b Ax m 1 = b A(x 0 + Q m 1 r 0 )=(I AQ m 1 (A)) r 0 = P m (A)r 0, Received by the editors May 19, 2008; accepted for publication (in revised form) August 5, 2009; published electronically DATE. School of Computer Science, Academic College of Tel-Aviv Yaffo, Tel Aviv 64044, Israel (hillel@ mta.ac.il). School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (turkel@post.tau. ac.il). 1

2 2 HILLEL TAL-EZER AND E. TURKEL and it satisfies (1.4) r m 1 2 = min T Π m T (A)r 0 2. Π m is the set of polynomials, T,ofdegreem with T (0) = 1. The major disadvantage of the GMRES algorithm is that, in order to get the optimal solution, one has to store m vectors. For real-life problems, the algorithm poses stringent demands on computer memory and CPU time. In order to overcome this obstacle, the standard approach is to implement the so-called restarted GMRES. In this approach, one gives up optimality and restarts the algorithm after k (k m) matrix-vector multiplications. This tactic can result in a highly efficient algorithm, but, as frequently has been observed, the rate of convergence can deteriorate or even complete stagnation can occur. In the last 20 years, many studies have aimed to improve upon this drawback, for example, [22, 34, 6, 35, 20, 21]. Another family of methods aimed at solving nonsymmetric linear systems is based on the Lanczos biconjugate algorithm. These methods include QMR [9], TFQMR [10], and BICGSTAB [34]. The main advantage of these algorithms is the elimination of the need to store many vectors. On the other hand, since these methods do not incorporate minimization of the residual, erratic behavior of the error can be observed. Recently, methods which combine the biconjugate and GMRES approaches have been developed [27, 29]. A third approach is based on a polynomial approximation in the complex plane. Since, in a general polynomial approach, the residual vector satisfies (1.3), we look for P m (z) such that (1.5) P m (z) = min Q Π m Q(z), z D. D is a domain in the complex plane which includes all the eigenvalues of A. When D is a real interval, then P m is the scaled Chebyshev polynomial. Manteuffel [15, 16] had developed an algorithm which considers the situation when D is an ellipse in the complex plane. This algorithm had been generalized in [17]. In [28], the authors treat general domains in the complex plane where P m is expanded in Faber polynomials. The Risolv algorithm, presented in this paper, belongs to the third family of algorithms described above. It is based on efficiently computing the roots of P m (z) such that P m (z) is close to optimal. 2. Risolv. Although Risolv belongs to the family of methods based on polynomial approximation in the complex plane, it can also be considered as a slight modification of restarted GMRES. Applying GMRES(s), at the j restart, we have (2.1) x j+1 = x j + V j y j, where V j is an N s matrix whose columns constitute an orthonormal basis of the Krylov space (2.2) K s = Span{r j,ar j,...,a s 1 r j }, and y j is chosen such that r j+1 2 is minimized. It can be shown that r j+1 satisfies s ( (2.3) r j+1 = I 1 ) A r j, z ij i=1

3 RISOLV HELMHOLTZ EQUATION 3 where z 1j,...,z sj are harmonic Ritz values (the eigenvalues of the modified Hessenberg matrix) [11]. Hence, after n restarts we have n s ( (2.4) r m = I 1 ) A r 0 (m = ns). z ij j=1 i=1 The difference between Risolv and restarted GMRES is in the choice of the vector y j. For Risolv, y j is chosen so that z 1j,...,z sj are uniformly (not equally) distributed in D. This can be achieved by employing the Leja algorithm [24] as follows. Let H be the Hessenberg matrix that satisfies (2.5) AV = V H = VH+ h s+1,s v s+1 e H s, and Ĥ is the modified Hessenberg defined as (2.6) Ĥ = H + h 2 s+1,sfe H s, where f = H H e s and e s =[0,...,1] H. At each restart we have in our possession a set of 2s matrices whose eigenvalues can be considered as approximated points in the domain D. These matrices are H i, Ĥi, 1 i s. Since each matrix is of dimension i i, all together we have a set of s 2 points. From this set we choose, using the Leja algorithm, a set of s points uniformly distributed in D (taking into account the already chosen sj points). Hence, after n restarts, the residual is the same as (2.4), but now the set of points (2.7) {z ij } 1 i s,1 j n is almost uniformly distributed in D. By (2.3), we have (2.8) r j+1 = Vy j, (2.9) x j+1 = x j + Vμ j, where y j and μ j are vectors of length s +1ands, respectively. These vectors can be computed by the following algorithm. NewVectors. μ =[0,...,0] H,y=[1, 0,...,0] H, for i =1:s w = 1 z i ȳ (ȳ i = y i, 1 i s) μ = μ + w y = y Hw end Based on the above algorithm the Risolv algorithm is described as follows. Risolv algorithm. r 0 = b Ax 0,z=emptyset, tol for j = 0 : until satisfied [ V, H] = Arnoldi(r j ) z new =Leja( H,z) z = z z new [μ j,y j ] = NewVectors( H,z new ) x j+1 = x j + Vμ j r j+1 = Vy j if r j+1 2 tol b 2, break end

4 4 HILLEL TAL-EZER AND E. TURKEL Fig. 1. Eigenvalue distribution of matrix A, (3.1). Risolv is especially efficient for solving sets of linear systems of the form (2.10) Ax i = b i, 1 i k. After solving first system, we have a set of points which is almost uniformly distributed, and Risolv becomes essentially a simple Richardson algorithm: (2.11) x k+1 = x k + 1 w k (b Ax k ), where w k is a point in this set. This stage is free of inner products. If needed, a few additional matrix-vector multiplications will reduce the error to the desired accuracy. 3. Numerical examples. In this section we consider a variety of numerical problems and compare the results shown by Risolv and GMRES. All the computations described below were done in MATLAB. Example 1. We consider a modified Grcar matrix (3.1) A = and the right-hand-side vector b =[1,...,1] T. The exact eigenvalues of A, ascomputed by MATLAB, are shown in Figure 1. We solved this system both with restarted GMRES and Risolv. The size of the Krylov space is 20 and the desired accuracy is ItrequiredGMRESandRISOLV 3226 and 1195 matrix-vector multiplications, respectively, to reach an approximate solution of the desired accuracy, The set z ij (2.3) for the two algorithms is given in Figures 2(a) and 2(b). The modified Grcar matrix is a highly nonnormal matrix. Its nonnormality is in-

5 RISOLV HELMHOLTZ EQUATION 5 (a) GMRES (b) RISOLV Fig. 2. z ij for matrix (3.1). Fig. 3. Eigenvalue distribution of matrix B, (3.2). creased by adding upper diagonals. Adding upper diagonals to (3.1) yields (3.2) B = The exact eigenvalues of (3.2) are shown in Figure 3. The set {z ij } for (3.2) for GMRES and RISOLV is given in Figures 4(a) and 4(b). Due to the fact that now the pseudospectra of the matrix almost surround the origin, solving the linear system iteratively becomes a much more difficult task. While

6 6 HILLEL TAL-EZER AND E. TURKEL (a) GMRES (b) RISOLV Fig. 4. z ij for matrix (3.2). Table 1 Matrices from Matrix-Market. Matrix Risolv GMRES mat-vecs mat-vecs ADD MEMPLUS ORSIRR ORSIRR ORSREG RAEFSKY SAYLR GMRES stagnated completely, it took 1400 matrix-vector multiplications for RISOLV to reach an accuracy of Example 2. We next present several linear systems with matrices taken from Matrix-Market [18]. In Table 1 we compare the number of matrix-vector multiplies required for RISOLV and GMRES to compute solutions with a relative accuracy of 10 4 with right-hand-side vector [1,...,1] T. Example 3. We now demonstrate the high efficiency of RISOLV when several right-hand sides are given. Hence, we consider the following: (3.3) Ax i = b i, 1 i s. In this experiment we solve 10 linear systems. A is the ORSIRR1 matrix and b i are random vectors. The results are shown in Table 2. We see that the number of matrix multiplies is almost the same for each right-hand side. However, the number of inner products reduces substantially for the second right-hand side and becomes negligible for further solutions of the system with different right-hand sides. Example 4. Finally, this example is for a convection-diffusion problem taken from SPARSKIT2 of Y. Saad. The grid is shown in Figure 5. In this case N(size of the matrix) = and NNZ (number of nonzeros) = We used ILUT as the preconditioner. A drop tolerance of 0.1 was good enough for RISOLV yielding

7 RISOLV HELMHOLTZ EQUATION 7 Table 2 Operation count for problem with multiple right-hand sides. No. system Mat-vecs Inn-prod Fig. 5. Grid for convection-diffusion problem. nonzeros in the ILU matrices. A drop tolerance of was needed for GMRES resulting in nonzeros in the ILU matrices Exterior Helmholtz problem. We now consider the Helmholtz equation for scattering exterior to a body. This leads to a difficult indefinite, nonsymmetric and complex equation (3.4) Δu + k 2 u =0. Since we consider the exterior of a body it is necessary to truncate the infinite domain and introduce an artificial outer surface. On this surface we introduce boundary conditions that reduce reflections back into the physical domain. With this boundary condition the solution is complex. The resultant discretized system is not positive definite and is not self-adjoint. Several recent surveys discuss the difficulties in numerically solving the Helmholtz equation exterior to a body; see, for example, [12, 30, 31, 33]. We consider two dimensional scattering about a general boundary. The total field is given by u tot = u scat + u inc,withu inc given by a plane wave. Then [5] (3.5) Δu + k 2 u = 0 exterior to D on D, u = u scat + e ik x d (soft body) u or n = n (uscat + e ik x d ) (hard body), ( u scat ) (x) lim r iku scat (x) = 0 (Sommerfeld radiation condition). r r

8 8 HILLEL TAL-EZER AND E. TURKEL We discretize the equation in the interior of D by a standard linear finite element method in polar coordinates. To approximate the Sommerfeld radiation condition we truncate the domain by introducing an artificial elliptical surface. On this artificial surface we impose the absorbing boundary condition of Kriegsmann, Taflove, and Umashankar [14] and Medvinsky, Turkel, and Hetmaniuk [19], which is an extension of BGT [3]. This has been modified to be in divergence-free form [1, 19, 33]: (3.6) u κu = iku n 2 κ2 u 8(ik κ) s ( 1 2(ik κ) We shall consider various Krylov subspace iterations to solve the resultant linear matrix problem. It is well known that Krylov methods converge slowly unless the system is preconditioned. Two types of preconditioners have been developed [33]. In the first kind, the preconditioner utilizes algebraic properties of the matrix. A typical example is ILU. A different class of preconditioners is based on properties of the differential system, i.e., an operator-based preconditioner. An example of this is basing the preconditioner on the solution of the Laplace equation in Bayliss, Goldstein, and Turkel [4]. This works well for low wavenumbers but becomes less efficient as k increases. Erlangga, Vuik, and Oosterlee [7] and Erlangga [8] extended this idea by considering a Helmholtz equation with a complex wavenumber as the preconditioner. One still has to solve a system the same size as the original system. So, the modified Helmholtz equation needs to be also solved by an inner iterative method. We shall also solve the inner preconditioned problem by a Krylov space technique. A complex k introduces dissipation which makes the preconditioned system easier to solve than the original system with a real k. Furthermore, we have a reasonable initial guess for the inner iteration. Hence, we consider a preconditioner which is an approximate solution to (3.7) Δv + k 2 precv =0, k 2 prec =(1+iα)k 2. As a model case we consider scattering about a hard ellipse (natural boundary condition at the scatterer) with an additional tower to resemble a submarine which has an aspect ratio of 10 while the outer surface is an ellipse with a semimajor axis of 1.2 and a semiminor axis of 0.6. A typical grid is shown in Figure 6. The absorbing boundary condition (3.6) is imposed on the outer artificial ellipse. The outer boundary fits snugly about the scatterer. Because of the high aspect ratio of the scatterer and the close position of the outer boundary this is a difficult problem for a solver. For all the iterative solvers we reduce the residual by seven orders of magnitude. We have several layers of iterations. The outer loop is based on solving the original scattering problem with a Krylov method. This is preconditioned by the approximate solution of a modified Helmholtz problem with a complex wavenumber. The initial guess of this outer iteration is based on an ILU(0) decomposition of the modified Helmholtz equation. The modified Helmholtz problem is again solved by a Krylov method. For some of the methods we shall also investigate the use of ILU(0) of the modified Helmholtz equation as the preconditioner. This is not a good preconditioner but has the advantage of not being a variable preconditioner. We will investigate the optimal value of the imaginary portion of k and check the sensitivity to this value. In [32] we presented results using BICGSTABwithanILU(0)preconditioning to solve the preconditioned problem (3.7). For the more difficult cases considered here this did not always converge. Furthermore, the convergence is very sensitive to the mesh. Instead, we first consider the Helmholtz equation solved by BICGSTAB u s ).

9 RISOLV HELMHOLTZ EQUATION 9 Fig. 6. Submarine and grid. but with the modified Helmholtz equation solved by 20 iterations of GMRES with kprec 2 =(1+.3i)k 2. We use ILU(0) as the preconditioning for the inner problem. Again, the number of iterations is very sensitive to the details of the wavenumber and corresponding mesh size. For k =40anda mesh it did not converge. Similarly for the finer meshes this did not converge. This lack of robustness makes this approach less feasible. In summary, when the Helmholtz equation is solved by BICGSTAB we found the iterative method very sensitive whether we used BICGSTAB or GMRES to solve the preconditioned modified Helmholtz equation. Instead we use a restarted GMRES method for the original Helmholtz equation. The inner loop is solved by GMRES with kprec 2 =(1+.5i)k2. GMRES essentially builds a polynomial, P m (z), in the matrix. With each restart we get a different polynomial. Thus, we have a variable preconditioner, and so we use flexible GMRES (FGMRES) to solve the Helmholtz equation rather than GMRES. A maximum of 20 iterations is allowed in the solution of the modified Helmholtz equation with the FGMRES restarted after 10 iterations. In Table 3 we present the results. Interestingly, the number of iterations required to reduce the residual by seven orders of magnitude decreases as the wavenumber increases. For fixed k as we refine the mesh we require more outer iterations than when the modified Helmholtz equation was solved exactly. So solving the inner problem by GMRES reduces the effectiveness of the preconditioner. Also, for a fixed wavenumber the number of iterations required increases with the size of the mesh. Nevertheless, it is more efficient and more robust than the BICGSTAB solver. Finally, we consider Risolv for the Helmholtz equation with a preconditioner based on the modified Helmholtz equation (3.7). Since Risolv, like GMRES, does not allow for a variable preconditioner we instead construct the system AP 1 v = b, Pu = v, wherea corresponds to the original Helmholtz equation (3.5) and P refers to the ILU(0) factorization of the modified Helmholtz equation (3.7). Since we are not solving the full modified Helmholtz equation this is a poorer preconditioner than we used for FGMRES. The convergence rate is fairly insensitive to the imaginary part of k prec. This is despite that the eigenvalue distributions of the various AP 1 differ considerably. One of the advantages of Risolv is that it automatically outputs a set of points distributed in the vicinity of the boundaries of the spectra (or pseudospectra in the nonnormal case) of AP 1. In Figure 7(a) we display this set when k prec = k.

10 10 HILLEL TAL-EZER AND E. TURKEL Table 3 Convergence rate FGMRES; modified problem solved by GMRES. k Mesh # outer iterations # total iterations Time (a) k prec = k (b) k 2 prec =(1+i)k 2 Fig. 7. Effect of k prec on eigenvalues, k =40, grid, ILU (0). In this case the points are almost all real. In Figure 7(b) we plot the set of points when k 2 prec =(1+i)k 2 and see the difference in the distribution. If we choose P 1 as the ILU(0) decomposition of the original equation (3.4), then we save storage and the convergence is affected only mildly. However, we lose robustness since the ILU(0) of the original matrix may not exist. In Table 4 we present the results. We see that the computer times are slightly larger than with FGMRES. Thus, when computing a single case it is preferable to solve the Helmholtz equation and the preconditioner by FGMRES. We again stress that with Risolv we are using a poorer preconditioner than with FGMRES. As described before, a significant advantage of Risolv is its efficiency when subsequent problems are solved with different right-hand sides. This would occur if one is solving a scattering problem with incident waves at various angles. Alternatively, one can consider the Helmholtz equation with various wavenumbers k. Avery, Farhat, and

11 RISOLV HELMHOLTZ EQUATION 11 Table 4 Convergence rate for Risolv solving modified problem by ILU (0). k k prec k Mesh # matrix vector multiplies Time 1 1+i i i i i i i i i i i i i i i i NC NC 40 1+i i Table 5 Comparison of FGMRES with Risolv for multiple right-hand sides. k Scheme k Grid Preconditioner prec k First time Later times FGMRES mod Helm 1+.5i FGMRES ILU(0) 1+.5i Risolv ILU(0) Risolv ILU(0) 1+.2i Risolv ILU(0) 1+.5i Risolv ILU(0) 1+i Risolv ILU(0) Risolv ILU(0) 1+.6i Risolv ILU(0) 1+i Risolv ILU(0) 1i NC NC Risolv ILU(0) 1+.6i Risolv ILU(0) 1+i Reese [2] demonstrate how to reformulate the problem using a Padé expansion. This requires the solution of the Helmholtz equation with a single frequency but multiple right-hand sides. So the CPU time information is misleading since Risolv spends time on computing optimal points in the domain of eigenvalues. These computations do not depend on N (the size of the problem) but on k (the dimension of the Krylov space). Hence, for large N, this part of Risolv is negligible. In Table 5 we compare FGMRES with Risolv when we compute several right-hand sides. In particular we chose incident plane waves at angles of 0, 10, 20, 30. The later times column reflects an average over the later three angles. It contains both differences in convergence rates for various angles of incidence and the effect of the algorithm in computing multiple right-hand sides. For FGMRES we see that using the full modified Helmholtz equation is faster than using than the simplified precon-

12 12 HILLEL TAL-EZER AND E. TURKEL ditioner of the ILU(0) decomposition of the modified Helmholtz equation. For Risolv we cannot use a variable preconditioner, and so we choose only the preconditioner as the ILU(0) decomposition. Nevertheless, depending on k prec Risolv is as efficient or slightly worse than FGMRES for the first angle. For subsequent angles Risolv is almost twice as fast. Note that for this case k prec = k is optimal and we do not need to compute and store the matrix for the modified Helmholtz equation. However, in general the ILU(0) decomposition of the original Helmholtz equation may not exist. For example, for k =40andk prec = k Risolv did not converge because the ILU(0) preconditioner did not exist. This occurred because the mesh was too coarse to resolve this wavenumber. Increasing the imaginary component to k prec =(1+.5i)k the ILU(0) decomposition exists but is a poor preconditioner as seen from the eigenvalue distribution (not shown). Further increasing it to k prec =(1+i)k gives an improved preconditioner and Risolv converges fast. Hence, as seen from the table, for robustness one might wish to choose kprec k =1+i. If the waves are resolved, then k prec = k should be sufficient. 4. Conclusion. We have constructed a new iterative solver, Risolv, based on polynomial approximation in the complex plane. We have shown the advantages of this algorithm for a range of problems. In addition to several matrices from standard libraries we have considered scattering about a submarine-like object using the Helmholtz equation and a suitable preconditioner. This problem involves complex numbers, is nonpositive, and is nonself-adjoint. Risolv is particularly effective when multiple solutions are required with different right-hand sides. A by-product of the algorithm is that it automatically outputs a set of points distributed in the vicinity of the boundaries of the spectra (or pseudospectra in the nonnormal case) of AP 1. REFERENCES [1] X. Antoine, Fast approximate computation of a time-harmonic scattered field using the onsurface radiation condition method, IMA J. Appl. Math., 66 (2001), pp [2] A. Avery, C. Farhat, and G. Reese, Fast frequency sweep computations using a multi-point Padé-based reconstruction method and an efficient iterative solver, Internat. J. Numer. Methods Engrg., 69 (2007), pp [3] A. Bayliss, M. Gunzburger, and E. Turkel, Boundary conditions for the numerical solution of elliptic equations in exterior regions, SIAM J. Appl. Math., 42 (1982) pp [4] A. Bayliss, C. I. Goldstein, and E. Turkel, An iterative method for the Helmholtz equation, J. Comput. Phys., 49 (1983), pp [5] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nded., Springer-Verlag, Berlin, [6] E. De Sturler and D. R. Fokkema, Nested Krylov Methods and Preserving the Orthogonality, Tech. Rep. Preprint 796, Utrecht University, Utrecht, The Netherlands, [7] Y. A. Erlangga, C. Vuik, and C. W. Oosterlee, On a class of preconditioners for the Helmholtz equation, Appl. Numer. Math., 50 (2004), pp [8] Y. A. Erlangga, Advances in iterative methods and preconditioners for the Helmholtz equation, Arch. Comput. Methods Engrg., 15 (2008), pp [9] R. W. Freund and N. Nachtigal, QMR: A quasi-minimal residual method for non-hermitian linear systems, Numer. Math., 60 (1991), pp [10] R. W. Freund, A transpose-free quasi-minimum residual algorithm for non-hermitian linear systems, SIAM J. Sci. Comput., 14 (1993), pp [11] S. Goossens and D. Roose, Ritz and harmonic Ritz values and the convergence of FOM and GMRES, Numer. Linear Algebra Appl., 6 (1999), pp [12] I. Harari, A survey of finite element methods for time-harmonic acoustics, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp [13] M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1954), pp

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