We G Model Reduction Approaches for Solution of Wave Equations for Multiple Frequencies

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1 We G15 5 Moel Reuction Approaches for Solution of Wave Equations for Multiple Frequencies M.Y. Zaslavsky (Schlumberger-Doll Research Center), R.F. Remis* (Delft University) & V.L. Druskin (Schlumberger-Doll Research) SUMMARY We have evelope a novel approach for solving multi-frequency frequency omain wave equation. The approach is base on efficient Krylov subspace approximants an projection-base moel reuction techniques. We have consiere polynomial Krylov an extene Krylov subspaces for approximating the solution given by stability-correcte resolvent. Our numerical examples inicate that polynomial Krylov subspace allows to obtain solution for the whole a priori given frequency range at the cost of solution for minimal frequency (for that frequency range) obtaine using unpreconitione BiCGStab solver. Extene Krylov subspace has been shown to improve the convergence by proviing more uniform rate for the whole frequency range.

2 Introuction Developing fast an robust forwar-moeling methos for frequency-omain wave problems is not only a significant topic by itself, it is also of great importance for full wave inversion. When solutions for multiple frequencies are require, conventional workflow consists of solving iscretize frequencyomain problems for each frequency separately. Typically, however, iscretization gris for 3D problems consist of up to a billion noes an even with state-of-the-art preconitioners such a workflow results in rather computationally intensive tasks. Moel reuction is a well-establishe tool, allowing us to efficiently obtain solutions in the time- or frequency-omains by projecting the large-scale ynamical system on a small Krylov or rational Krylov subspace. For lossy iffusion ominate problems, for example, moel reuction has been shown to provie significant spee ups (see Zaslavsky et al. (211)). For seismic exploration, we note that problems involving lossless meia in unboune omains effectively behave as lossy ones since infinity can be viewe as an absorber of outgoing waves. In fact, two of the authors showe that these problems, polynomial Krylov subspace (PKS) moel reuction outperforms the finite-ifference time-omain metho on large time intervals (see Druskin an Remis (213)). In this paper we exten the PKS approach an use extene Krylov subspaces (EKS) for reuce-orer moel construction (Druskin an Knizhnerman (1998)). An EKS is generate by the system matrix an its inverse an reuce-orer moels taken from such a space usually converge much faster than PKS reuce-orer moels especially if a wie frequency range containing small frequencies is of interest. To compute the moels in an efficient manner, we use a moifie version of the EKS algorithm propose by Jagels an Reichel (29). Specifically, we generate a complex-orthogonal basis of the EKS via short-term recurrences by exploiting the complex-symmetric structure of the system matrix. Problem formulation Consier the multiimensional Helmholtz equation Au + ω 2 u = b. (1) In this equation, A is a self-ajoint nonnegative partial ifferential equation operator on an unboune omain that has an absolutely continuous spectrum. We note that via a proper change of variables, all frequency-omain acoustic an elastic fiel equations can be written in a form as given by Eq. (1). We now iscretize (1) using a secon-orer gri in the interior of the computational omain an use a PML for omain truncation. As a result, we obtain a matrix à N A, where à N C N N is complex symmetric. In practice, the orer of this matrix can be up to billion or even more. When the computational omain is truncate using a conventional time-omain perfectly matche layers (PML) formulation (Berenger (1994)), operator à N becomes frequency-epenent. Inee, that is sufficient for traitional preconitione solvers that treat each frequency one by one. In our approach, however, we inten to reuse the same operator for multi-frequency computations. We therefore follow Druskin an Remis (213) an Druskin et al (213) an apply a fixe-frequency PML with optimal iscrete stretching allowing low cost error control for a prescribe frequency interval. It is tempting to substitute straightforwarly the approximate operator à N in (1) an take the solution u N (ω) =(à N + ω 2 I) 1 b N as an approximation to u. Here, we note that it is out goal to solve Eq. (1) for multiple frequencies employing a fixe-frequency PML such that it is still possible to transform the frequency-omain solution back to the time-omain. However, using the non-hermitian matrix à N

3 instea of its exact Hermitian counterpart A, qualitatively changes the behavior of the solution on the complex plane. In particular, the symmetry relation u(ω)=u( ω) breaks for the approximate solution an the time-omain transform of u N (ω) is unstable. Fortunately, it is shown in Druskin an Remis (213) that we can correct for these efects by using the stabilize approximation to Eq. (1) in the form ũ N (ω)= 1 2 [ B 1 N (B N + iωi) 1 + B 1 ( N BN + iωi ) ] 1 b N, B N = ( ) 1/2 à N (2) Moel reuction an Krylov subspace methos With the stability-correcte fiel approximations available, we can now construct reuce-orer moels base on EKS in the usual way. Specifically, the moels are rawn from the Krylov subspace K m1,m 2 = span{ã m 1+1 N b N,...,à 1 N b N,b N,à N b N...,à m 2 1 N b N }. We note that the PKS K m = span{b N,à N b N,...,à m 1 N b N } correspons to K 1,m2 an PKS reuce-orer moels with application to stability-correcte solutions were investigate in Druskin an Remis (213). The avantage of such a PKS moel-orer reuction approach is that its computational costs are essentially the same as the costs of m 2 steps of the explicit finite-ifference time-omain metho or m 2 steps of the unpreconitione bi-cg metho for the single frequency Helmholtz equation. However, the reuce-orer moels taken from the PKS may not provie us with the fastest convergence if wie frequency ranges with small enough frequencies are of interest. For such problems, we therefore resort to an EKS reuce-orer moeling approach. An EKS can be seen as a special case of a rational Krylov subspace with one expansion point at zero an one at infinity. The action of à 1 N on a vector is require to generate a basis for such a space. This essentially amounts to solving a Poisson-type equation for which efficient solution techniques are available. Computing matrix-vector proucts with the inverse of the system matrix is generally still more expensive than computing matrix-vector proucts with the system matrix itself, however, an from a computational point of view we therefore prefer to eal with EKS K m1,m 2 with m 1 < m 2. In aition, the basis vectors shoul be constructe via short-term recurrence relations, since storage of all basis vectors is generally not practical for large-scale applications. In Jagels an Reichel (29), the authors evelope such an EKS algorithm in which the orthogonal bases V k(i+1) =[v,v 1,...,v i,v 1,...,v k+1,...,v ik ] for the sequence of subspaces K 1,i+1 K 2,2i+1... K k,ki+1 are inee generate via short term recurrences. Here, i is an integer that allows us to optimize the accuracy an computational costs. In this paper, we moify this algorithm an generate complex-orthogonal basis vectors of the EKS by exploiting the complex-symmetric structure of matrix à N. Denote = k(i + 1) an let V C N be matrix with columns being the generate basis vectors. Then all iterations can be summarize into the equation à N V = V H + z e T, where z = h +1, v k + h +2, v ik+1 an h ij is the (i, j) entry of matrix H. Furthermore, matrix H is a pentaiagonal matrix that satisfies D H = V T ÃNV, where D is iagonal matrix with entries δ,δ 1,...,δ i,δ 1,...,δ k+1,...,δ ik. The entries of matrix H can easily be obtaine in explicit form from the pentaiagonal matrix given in Jagels an Reichel (29). The frequency-omain EKS reuceorer moel is given by ũ (ω)= 1 [ 2 δ V B 1 (B + iωi ) 1 +V B 1 ( ) ] 1 B + iωi e 1, (3)

4 45 4 BiCGStab solver 8 7 BiCGStab solver Real part of solution Imaginary part of solution Receiver Receiver Figure 1 SEG/EAGE Salt moel. Real an imaginary parts of the solution for a frequency of 7.5 Hz compute using the BiCGStab an PKS solvers with 1,6 matrix-vector multiplications an 1,8 iterations, respectively. The solutions are almost inistinguishable. ( ) 1/2, where B = D 1/2 H D 1/2 I R is the ientity matrix, an e 1 is its first column. With the help of our moifie EKS algorithm, we now have reuce the computation of functions of a very large finite-ifference matrix à N to the action of functions of a much smaller five-iagonal matrix D 1/2 H D 1/2 times a skinny matrix V. Once both of these matrices have been compute, simulation of the frequency omain curve can be one rather cheaply. Numerical experiments First, we have consiere the 3D SEG/EAGE Salt moel an benchmarke our solver against an inepenently evelope unpreconitione BiCGStab algorithm that is a competitive conventional iterative Helmholtz solver (see Pan et al (212) for etails). Here, we took the case m 1 = 1 which correspons to PKS. Fig. 1 shows excellent agreement between two approaches. Then we note that the PKS approach requires one matrix-vector multiplication per iteration, while BiCGStab nees two. Since this part constitutes the most computationally intensive part of the iteration process, it makes sense to compare the performance of these two methos in terms of matrix-vector multiplications rather than in terms of iterations. We have plotte the convergence rates of BiCGStab against the PKS moel-orer reuction metho on Fig. 2 (left). Clearly, for this single frequency problem, both rates are rather close. We note that both approaches converge faster for higher frequencies an convergence slows own for lower frequencies. Inee, Fig. 2 (right) shows how the reuce-orer moeling metho converges for ifferent frequencies in the range from 2.5 Hz to 7.5 Hz. However, the principal ifference between the PKS metho an BiCGStab is that the former approach obtains solutions for the whole given frequency range at the convergence cost of BiCGStab for the lowest (from that range) frequency. Next, we consier what aing negative powers gives us in terms of performance. In Fig. 3 (left), we have plotte a number of convergence curves (with respect to increasing k) for approximants obtaine using EKS K k,ki+1 for ifferent values of fixe i (i = correspons to PKS) an on a frequency range running from 2.5 Hz to 7.5 Hz. As is clear from this figure, aing negative powers (i < ) visibly improves the convergence of the Krylov subspace metho. Inee, while PKS converges faster for higher frequencies, EKS improves convergence for smaller frequencies an, consequently, convergence is more uniform on the entire frequency range. To confirm that point, in Fig. 3 (right), we have plotte convergence curves for the EKS metho on a frequency range with expane lower part. As one can observe, PKS significantly slows own while EKS performs as efficient as for a narrower frequency range. Conclusions We have evelope a powerful moel-orer reuction tool for solving large-scale multi-frequency wave problems. The PKS metho allows obtaining solutions for a whole range of frequencies at the cost of

5 Convergence for 7.5Hz frequency BiCGStab for fixe frequency Iterations 2 Iterations 3 Iterations Exact solution Error 1 3 Response Number of matrix vector multiplications Frequency, Hz Figure 2 SEG/EAGE Salt moel. Convergence rates of BiCGStab an PKS at a fixe frequency of 7.5 Hz (left) an convergence behavior of PKS for a frequency range running from 2.5 Hz to 7.5 Hz (right) Frequency range 2.5Hz to 7.5Hz i=3 i=5 i=7 i=inf 1 2 Frequency range.1hz to 7.5Hz 1 L2 Error L2 Error Iteration number i=3 1 6 i=5 i=7 i=inf Iteration number Figure 3 SEG/EAGE Salt moel. Convergence of EKS for ifferent values of i an for frequencies ranging from 2.5 Hz to 7.5 Hz (left) an from.1 Hz to 7.5 Hz (right). solving a single-frequency problem with the BiCGStab solver. Moreover, EKS moel-orer reuction significantly outperform polynomial reuce-orer moeling when solutions for small frequencies are require. We also note that the EKS approach can be applie to wave problems in the time-omain. Furthermore, our approach is not limite to secon-orer schemes in the interior. In fact, since the optimal iscrete PML has spectral accuracy (see Druskin an Remis (213); Druskin et al (213)), it woul be preferable to use optimal gris (Asvaurov et al (2)) or spectral methos for interior part. Acknowlegements We thank Dr. Guangong Pan for proviing us with results for his BiCGStab finite-ifference solver. References Asvaurov, S, Druskin, V., Knizhnerman, L. [2] Application of the ifference Gaussian rules to the solution of hyperbolic problems. J. Comp. Phys., 158, Berenger, J. P. [1994] A perfectly matche layer for the absorption of electromagnetic waves. J. Comput. Phys., 114, Druskin, V., Guettel, S., Knizhnerman, L. [213] Near-optimal perfectly matche layers for inefinite Helmholtz problems. MIMS perprint , University of Manchester. Druskin, V., Knizhnerman, L. [1998] Extene Krylov subspaces: approximation of the matrix square root an relate functions. SIAM J. Matrix Anal. Appl., 19(3), Druskin, V., Remis, R. [213] A Krylov stability-correcte coorinate stretching metho to simulate wave propagation in unboune omains. SIAM J. Sci. Comput., 35(2), B376 B4. Jagels, C., Reichel, L. [29] The extene Krylov subspace metho an orthogonal Laurent polynomials. Linear Algebra Appl., 431, Pan, G., Abubakar, A., Habashy, T. [212] An effective perfectly matche layer esign for acoustic fourth-orer frequency-omain finite-ifference scheme. Geophys. J. Int., 188, Zaslavsky, M., Druskin, V., Knizhnerman, L. [211] Solution of 3D time-omain electromagnetic problems using optimal subspace projection. Geophysics, 76, F339 F351.