RESTARTING ITERATIVE PROJECTION

RESTARTING ITERATIVE PROJECTION METHODS FOR HERMITIAN NONLINEAR EIGENVALUE PROBLEMS WITH MINMAX PROPERTY MARTA M. BETCKE∗ AND HEINRICH VOSS† Abstract. In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. The variational property allows for the eigenvalues of such problems to be computed safely one after another and avoiding repetitions, as it is the case for linear Hermitian eigenvalue problems. This can be achieved applying iterative projection methods, which make use of the minmax induced enumeration of the eigenvalues in the inner iteration. However, to numerically exploit the minmax principle, it is necessary to include all the previously computed eigenvectors in the search subspace. This subspace growth is an obvious limitation if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed. In this paper we propose to overcome this problem employing a local numbering of eigenvalues. This local numbering only requires a presence of one eigenvector in the search subspace, effectively eliminating the search subspace growth and therewith the super-linear increase of the computational costs. Using this local numbering we derive a new restart technique for iterative projection methods and integrate it into nonlinear Arnoldi and Jacobi Davidson methods. The efficiency of our new method is demonstrated on a real-life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire. Key words. nonlinear eigenvalue problem, iterative projection method, Arnoldi method, JacobiDavidson method, minmax characterization, restart, purge and lock AMS subject classification. 65F15, 15A18, 35P30, 49R50, 65N25

1. Introduction. Over the last decade simulation of acoustic properties has been increasingly gaining importance in the engineering design process, in particular in automobile industries. The objective of such simulations, is to gain insight how to alter the design of a vehicle or its parts in order to minimize the noise exposure to the passengers as well as to the environment. The major effort associated with optimization of a noise and vibration performance is the frequency response computation. As the underlying finite element models are of formidable size, for the computational purposes, the problem is usually first projected on a modal basis consisting of structural modes in the relevant frequency range. Nonetheless, the computation of such a modal basis is a nontrivial task. Firstly, the resulting eigenvalue problem is typically a quadratic eigenvalue problem. Moreover depending on a problem at hand a large number of structural modes not necessarily at the end of the spectrum may be required to obtain a satisfactory accuracy over the relevant frequency range. In this work we consider the nonlinear eigenvalue problem (NEP) T (λ)x = 0,

(1.1)

where T (λ) ∈ Cn×n is a family of large and sparse Hermitian matrices for every λ in an open real interval J ⊂ R. We furthermore assume that the eigenvalues of (1.1) in J can be characterized as minmax values of a Rayleigh functional [24]. ∗ n´ ee Markiewicz, Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK ([email protected]). † Institute of Numerical Simulation, Hamburg University of Technology, D-21071 Hamburg, Germany ([email protected]). 1 A preliminary version of this paper appeared in the proceedings of ALGORITMY 2005 [10].

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M. M. BETCKE AND H. VOSS

Variational principle is a very desirable, yet quite stringent requirement on a nonlinear eigenvalue problem. Let us illustrate this on an example of a quadratic eigenvalue problem (QEP) T (λ) = λ2 M + λC + K, where M, K, C are symmetric positive definite. Such problem can have real or conjugate complex eigenvalues p± (x) =

−c(x) ± d(x) , 2m(x)

p with m(x) = x∗ M x, c(x) = x∗ Cx, k(x) = x∗ Kx and d(x) = c(x)2 − 4k(x)m(x). The variational principle in [24] applies to a subset of the real spectrum, namely + − − to λ+ k = p+ (xk ) if λk is greater than supd(x)>0 p− (x) and to λk = p− (xk ) if λk is less than inf d(x)>0 p+ (x). An extreme example of such a problem is the overdamped case [4] where d(x) > 0 for every x 6= 0. Then all eigenvalues are real and admit a minmax or maxmin characterization. In particular, gyroscopic eigenproblems Kx + iλGx − λ2 M x = 0 with K = K ∗ > 0, M = M ∗ > 0 and G = −G∗ are always overdamped. The problem of computing a moderate number of eigenpairs of a nonlinear eigenvalue problem at the beginning of the spectrum has been extensively studied. For minmax admitting problems a nonlinear Arnoldi method was suggested in e.g. [19] and the Jacobi Davidson method in [3]. For more general nonlinear eigenproblems iterative projection methods were considered in [2, 8, 9, 11, 17, 18, 20, 21, 22]. However, the approach in [3, 19] hits its limitations if a large number of eigenvalues (in particular in the interior of the spectrum) of (1.1) is needed. To algorithmically exploit the minmax property, one has to project the problem under consideration onto a sequence of search spaces of growing dimensions. For a large number of eigenvalues this naturally requires an excessive amount of storage and computing time. In this work we propose a new restart technique which allows to project the NEP (1.1) only to search spaces of a limited dimension throughout the iteration. As a result we obtain a new method, Nonlinear Restarted Arnoldi (NRA), capable of computation of a large number of eigenpairs, in the interior of the spectrum of a nonlinear eigenvalue problem. The name of the method should not be taken to imply a subspace expansion. The same argument applies if Jacobi-Davidson or in fact any other subspace expansion type is used and hence the same restart technique works with any of such iterative projections methods. The paper is organized as follows. In Section 2 we describe an acoustic problem which gives rise to a nonlinear eigenvalue problem. In Sections 3 and 4 we recapitulate the variational characterization for nonoverdamped nonlinear eigenproblems and the iterative projection method framework for their solution. The new restart technique is presented in Section 5 along with a strategy for dealing with spurious eigensolutions which are an intrinsic part of the interior eigenvalue computation. The resulting Nonlinear Restarted Arnoldi method for interior eigenvalue computation is summarized in Section 6 and the performance of the method is analyzed in Section 7 on two gyroscopic eigenvalue problems. Section 8 concludes with a summary and outlook for future research. 2. Noise radiation from rolling tires. Sound radiation from the rolling tires has been identified as a major source of noise generated by vehicles moving at speeds above 50 km/h [13, 12]. Therefore, much effort has been directed into development of

Restarting Iterative Projection Methods for NEPs

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Fig. 2.1. Acoustic model of a tire.

methods allowing for a simulation of the effect of tire-road surface interaction with the objective to improve the design of the tires, e.g. by optimization of the tread pattern. Simulations of the sound radiation require the solution of the Helmholtz equation ∆u + k 2 u = 0 on the exterior of the domain occupied by the tire (cf. Figure 2.1) for a particular set of wave numbers k = ωk /c, where c denotes the speed of sound. The frequencies ωk P are obtained by Fourier analysis of the transient excitation forces f (t) ≈ k fk eiωk t such as an impact of the roughness of the road surface or the tread pattern. As the domain is infinite, the Sommerfeld radiation condition ∂u ≤ C − iku r2 ∂r is assumed for large r. On the outer boundary of the tire the normal velocities of the tire for the particular frequency ωk are prescribed as an excitation. The description of the transient vibration of the tire employs the Arbitrary Lagrangian Eulerian framework [12], which decouples the rotation of the tire from its deformation. Discretization by finite elements accounting for the anisotropic behavior of the material and the axial load [7] yields the transient equation of motion Mu ¨ + Gu˙ + Ku = f (t),

(2.1)

with a symmetric positive definite mass matrix M = M T , a skew-symmetric gyroscopic matrix G = −GT and a symmetric stiffness matrix modified by the inertia forces due to the stationary rolling, K = K T . In order to make the computation of the system response tractable for a range of frequencies ωk , (2.1) is projected on a modal basis of the associated quadratic eigenvalue problem −M λ2 x + iGλx + Kx = 0, consisting of eigenvectors with eigenfrequencies close to the sought frequency ωk (up to 2000 Hz).

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3. Variational characterization of eigenvalues. We consider a nonlinear eigenvalue problem (1.1) T (λ)x = 0, where T (λ) ∈ Cn×n is a family of Hermitian matrices for every λ in an open real interval J. As in the linear case, T (λ) = λI − A, we call the parameter λ ∈ J an eigenvalue of T (·), whenever (1.1) has a nontrivial solution x 6= 0, which we call an eigenvector corresponding to λ. It is well known that all eigenvalues of a linear Hermitian problem Ax = λx are real. Moreover, if they are ordered, λ1 ≤ λ2 ≤ · · · ≤ λn , it is possible to characterize them by the minmax principle of Poincar´e λk = min max1 v ∗ Av, V∈Sk v∈V

(3.1)

where Sk denotes the set of all k dimensional subspaces of Cn and V 1 := {v ∈ V : kvk2 = 1} is the unit sphere in V. It turns out, that a similar result holds also for a certain type of nonlinear eigenvalue problems. In the following we assume that, for every fixed x 6= 0, the real function f (λ; x) := x∗ T (λ)x is continuous in J. Suppose, that the equation f (λ; x) = 0

(3.2)

has at most one solution in J, then (3.2) implicitly defines a functional p on some subset D of Cn \ {0}. We refer to p as a Rayleigh functional, since it generalizes the notation of the Rayleigh quotient in the variational characterization of the eigenvalues of the linear problem. We furthermore require that f (λ; x)(λ − p(x)) > 0

for every λ ∈ J and x ∈ D with λ 6= p(x),

(3.3)

which is a natural generalization of the definiteness assumption for linear pencils. Under these assumptions a variational characterization in terms of the Rayleigh functional has been considered by various authors. To mention a few Duffin [4, 5] and Rogers [16] proved the variational principle for the finite dimensional overdamped problems, i.e. problems for which the Rayleigh functional p is defined on the entire space Cn \ {0}. Nonoverdamped problems were considered by Werner and the second author [23, 24]. The key to the variational principle is an adequate enumeration of the eigenvalues. In general, the natural enumeration, i.e. the first eigenvalue is the smallest one, followed by the second smallest one etc. is not appropriate. Instead, the number of an eigenvalue λ of the nonlinear problem (1.1) is inherited from the number of the eigenvalue 0 of the matrix T (λ) based on the following consideration: Let λ ∈ J be an eigenvalue of the nonlinear problem (1.1), then µ = 0 is an eigenvalue of the linear problem T (λ)y = µy. Therefore there exists k ∈ N such that 0 = max

min w∗ T (λ)w.

W∈Sk w∈W 1

In this case we call λ a kth eigenvalue of (1.1).


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REMARK 3.1. We note that if T (λ) is differentiable w.r.t. λ and T 0 (λ) is positive definite, then replacing T (λ)y = µy with the generalized eigenvalue problem T (λ)y = µT 0 (λ)y yields the same enumeration. With this enumeration the following minmax characterization of the eigenvalues of the nonlinear eigenproblem (1.1) was proved in [23, 24]: Theorem 3.2. For every x ∈ D ⊂ Cn , x 6= 0 let the real equation (3.2) have at most one solution p(x) ∈ J, and assume that the definiteness condition (3.3) is satisfied. Then the following assertions hold: (i) For every k ∈ N there is at most one kth eigenvalue of problem (1.1) which can be characterized by λk = min

W∈Sk , W∩D6=∅

sup

p(w).

w∈W∩D

Hence, there are at most n eigenvalues of (1.1) in J. (ii) If λ ∈ J and k ∈ N such that (1.1) has a kth eigenvalue λk ∈ J. Then it holds that      >   >  = = λ λ ⇐⇒ µk (λ) := max min1 w∗ T (λ)w 0. W∈Sk w∈W   k   < < The correspondence between the kth eigenvalue λk of T (·) and the kth largest eigenvalue of the matrix T (λk ) suggests the safeguarded iteration for computing the kth eigenvalue of a nonlinear problem. An outline of the safeguarded iteration is given in Algorithm 1. Its convergence properties were proved in [15], and are recapitulated in Theorem 3.3. Algorithm 1 Safeguarded Iteration Require: 1: Initial approximation ν1 to the kth eigenvalue λk of (1.1) Execute: 2: for i = 1, 2, . . . until convergence do 3: Determine eigenvector x corresponding to the kth largest eigenvalue of T (νi ) 4: Evaluate νi+1 = p(x) 5: end for Theorem 3.3. (i) If λ1 := inf x∈D p(x) ∈ J, then the safeguarded iteration converges globally to λ1 . (ii) If λk ∈ J, k ≥ 1 is a kth eigenvalue of (3.1) which is simple then the safeguarded iteration converges locally and quadratically to λk . (iii) If T 0 (λ) is positive definite for λ ∈ J and x in step 3. of Algorithm 1 is chosen to be an eigenvector corresponding to the k largest eigenvalue of the generalized eigenproblem T (νi )x = κT 0 (νi )x then the convergence is even cubic. Certainly, the safeguarded iteration, is not capable of solving large nonlinear eigenvalue problems. However, it is well suited as an inner iteration in a projection method since its convergence properties and for small dimension also its complexity

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are comparable to those of the inverse iteration. Moreover, it has a significant advantage over the inverse iteration, namely it aims at an eigenvalue with a specific number. As a consequence, it is less likely to miss an eigenvalue, which is desirable whenever the knowledge of all the eigenvalues in an interval is relevant for the underlying application. 4. Iterative projection methods for nonlinear eigenproblems. For sparse linear eigenvalue problems Ax = λx iterative projection methods are well-established and a very efficient tool. The key idea is to reduce the dimension of the eigenproblem by projecting it to a subspace of a much smaller dimension. The reduced problem is then handled by a fast technique for dense problems. Of course, this idea can only be successful if the subspace used for projection has good approximating properties w.r.t. some of the wanted eigenpairs, which translates to eigenvalues of the projected matrix being good approximations to the wanted eigenvalues of the large sparse matrix. In iterative projection methods the search subspace is expanded iteratively in a way promoting the approximation of the wanted eigenvalues. Prominent representatives are the Lanczos, Arnoldi, rational Krylov, and Jacobi-Davidson methods. The generalizations of the above mentioned methods to nonlinear eigenvalue problems were discussed in [2, 3, 8, 9, 11, 17, 18, 19, 20, 21, 22]. Two representative examples are the nonlinear Arnoldi or Jacobi Davidson methods. Algorithm 2 outlines a framework for iterative projection methods in case when (1.1) is Hermitian and it admits a variational characterization of its eigenvalues. We recall that those properties enable us to enumerate the eigenvalues as discussed in Section 3. Hence, the goal becomes to compute eigenvalues in an interval [λjmin , λjmax ]. There are many details that have to be considered when implementing an iterative projection method, as outlined in Algorithm 2, concerning the choice of the initial basis, solving the projected nonlinear eigenvalue problem, when to change and how to choose the preconditioner, when and how to restart, and how to continue after an eigenpair was accepted. The comprehensive review is out of scope of this work, thus in the remainder of this section we restrict ourselves to only the essentials necessary for motivation and derivation of the local restart technique in Section 5. For more detailed discussion we refer the reader to [19, 20, 22]. 4.1. Initialization. In order to preserve the numbering of the eigenvalues, the initial basis V has to contain the eigenbasis corresponding to all the eigenvalues preceding λjmin . This is either known or it can be determined from the linear eigenvalue problem T (λjmin )y = µy, see discussion in Section 4.4. 4.2. Solution of projected NEP. While applying the iterative projection methods to general nonlinear eigenvalue problems with the objective to approximate more than one eigenvalue, it is crucial to prevent the method from converging to the same eigenvalue repeatedly. In the linear case this is readily done by the Krylov subspace solvers. These construct an orthogonal basis of the ansatz space not aiming at a particular eigenvalue. As a result one gets approximations to extreme eigenvalues without replication (at least after reorthogonalization is employed). If several eigenvalues are to be computed by the Jacobi-Davidson method, an incomplete Schur factorization is determined in order to prevent the method from reapproaching an already previously computed eigenvalue (cf. [6]). Unfortunately, a similar normal form

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Algorithm 2 Iterative Projection Method Require: 1: First wanted eigenvalue number j := jmin 2: Initial basis V , V ∗ V = I 3: Choose initial shift σ 4: Initial preconditioner K ≈ T (σ)−1 , σ close to first wanted eigenvalue, λjmin Execute: 5: while j ≤ jmax do ˜ y), λ ˜ is the j smallest eigenvalue of the 6: Compute the eigenpair (λ, projected problem TV (λ)y := V ∗ T (λ)V y = 0 ˜ x ˜ x 7: Compute the Ritz pair (λ, ˜ := V y) and its residual r = T (λ)˜ ˜ x 8: if (λ, ˜) converged then ˜ x 9: return approximate eigenpair (λj , xj ) := (λ, ˜) 10: j := j + 1 11: optionally choose a new shift σ and recompute K ≈ T (σ)−1 , if indicated 12: optionally restart ˜ x ˜ x 13: Choose approximation (λ, ˜) to the next eigenpair, r = T (λ)˜ 14: end if 15: Expand V by a new direction v 16: Orthonormalize v = v − V V ∗ v, v = v/kvk, V = [V, v] 17: Reorthogonalize V if necessary 18: Update projected problem TV (λ) := V ∗ T (λ)V 19: end while does not exist for nonlinear eigenvalue problems. However, if the nonlinear eigenvalue problem admits a minmax characterization, the induced eigenvalue ordering can be used to remedy this problem. There are many methods for solving small and dense nonlinear eigenvalue problems. For polynomial eigenvalue problems a method at hand is linearization, where the enumeration of eigenvalues in the sense of Section 3 can be deduced from the natural ordering of the eigenvalues of the linearized problem. For general nonlinear eigenvalue problems we advocate the use of the safeguarded iteration for its efficiency and simplicity of enforcing the minmax induced eigenvalue ordering. Obviously, for the minmax induced numbering to hold in the inner iteration of an iterative projection method, the projected problems have to allow a variational characterization of their eigenvalues as well. Let T (λ) be a family of Hermitian matrices allowing minmax characterization of their eigenvalues in an open interval J and let the columns of V ∈ Cn form a basis of the current search space V of Cn , then it is easily seen that the projected problem TV (λ)y := V ∗ T (λ)V y = 0

(4.1)

inherits this property, i.e. its eigenvalues in J are minmax values of the restriction of the Rayleigh functional p of T (·) to D ∩ V. Although, in general the enumeration of the eigenvalues of the original problem and the projected problem may differ. 4.3. Subspace expansion. In general two approaches to subspace expansion can be found in the literature: Jacobi-Davidson [3] and nonlinear Arnoldi [20] type expansion. Both schemes approximate inverse iteration, which is known to provide a direction with high approximating potential to the eigenpair it is aiming for.

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4.3.1. Jacobi-Davidson. In Jacobi-Davidson the search subspace is expanded by an orthogonal direction t ⊥ xk obtained from the following correction equation T 0 (λk )xk x∗k xk x∗ I− ∗ 0 T (λk ) I − ∗ k t = −rk , t ⊥ xk , (4.2) xk T (λk )xk xk xk where (λk , xk ) is a currently available approximation to the eigenpair and rk = T (λk )xk its residual. (4.2) is equivalent to T (λk )t − αT 0 (λk )xk = −rk if α is chosen such that t ⊥ xk . For the correction t we have t = −xk + αT (λk )−1 T 0 (λk )xk . Therefore, the expanded space contains the direction of the inverse iteration and the latter converges cubically for symmetric nonlinear eigenproblems if the eigenvalue approximation is updated with the Rayleigh functional. Hence, if (4.2) is solved exactly we can expect asymptotically cubic convergence. However, it is a common experience that solving (4.2) only approximately is sufficient to maintain a reasonably fast convergence. In practice it is necessary to employ a preconditioner for solving the correction equation (4.2). Since the operator T (λk ) is restricted to map the subspace orthogonal to xk into itself the preconditioner has to be modified accordingly. 4.3.2. Nonlinear Arnoldi. The Nonlinear Arnodi method draws upon the residual inverse iteration [14], which suggests the following expansion v = T (σ)−1 T (λ)x of the search space V , where σ is a fixed parameter close to the wanted eigenvalues. This direction retains the linear convergence rate of the method in the general case and the quadratic one for Hermitian problems. For a linear eigenproblem T (λ) = A − λB this expansion corresponds to the Cayley transformation with pole σ and zero λ. The Cayley transform can be recast (A − σB)−1 (A − λB) = I + (σ − λ)(A − σB)−1 B and since the Krylov spaces are shift-invariant the resulting projection method is nothing else but the shift-invert Arnoldi method. Moreover, if the linear system T (σ)v = T (λ)x is too expensive to solve for v we may choose as a new direction v = K −1 T (λ)x with K ≈ T (σ). For the linear problem this corresponds then to an inexact Cayley transformation or a preconditioned Arnoldi method. Therefore the resulting projection method is called nonlinear Arnoldi although no Krylov space is constructed and no Arnoldi recursion holds. 4.4. Restart. As the subspace expands in the course of the algorithm the increasing storage and computational cost for solving the projected eigenvalue problems may make it necessary to restart the algorithm and purge some of the basis vectors. Restarting with a subspace V, which along with all the already converged eigenvectors xjmin , . . . , xk contains the initial search space chosen to enforce the correct enumeration at the start then obviously keeps the enumeration of the eigenvalues, and we can continue as above to determine the subsequent eigenpairs. Notice that we only


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restart if an eigenvector has just converged since a restart destroys information on the eigenvectors and in particular on the currently iterated one. If J contains the first eigenvalue λ1 = minx∈D p(x), then by Theorem 3.3 the safeguarded iteration for the projected nonlinear problem (4.1) converges globally, i.e. for any initial vector x ∈ V ∩ D, to the smallest eigenvalue of (4.1). If xj denotes an eigenvector corresponding to the jth eigenvalue λj of the original problem (1.1), and if xj ∈ V for j = 1, . . . , k, then λj is a jth eigenvalue of the projected problem (4.1), as well. Hence, expanding the search space V iteratively, and determining the (k + 1)th eigenvalue of the projected problems, one gets a sequence of upper bounds of λk+1 which (hopefully) converges to λk+1 . Thus, the eigenvalues of (1.1) can be determined quite safely one after the other by the iterative projection method starting with an approximation to x1 . If λ1 = inf x∈D p(x) 6∈ J we can modify this approach in the following way. The proof of the minmax characterization (cf. Theorem (3.2)) in [24] shows that the minimum is attained by the invariant subspace W of T (λk ) spanned by the eigenvectors corresponding to its k largest eigenvalues. Hence, if the current search space V satisfies W ⊂ V then it is easily seen that the kth eigenvalue of the projected problem (4.1) is λk , i.e. again the numeration of the eigenvalues is not altered in the projected problem, and the eigenvalues can be determined successively. In either case, for the numbering to be preserved, the search subspace after restart has to contain all the previous eigenvectors and in the latter case also appropriate initial vectors, hence the restart requires global information. 4.5. Convergence criterion. Backward error is a common criterion to assess convergence of an eigenpair of a linear problem. In this case it simply amounts to computing the residual norm. For general nonlinear eigenvalue problems no simple relation between the residual and the backward error is known. Our argument for using the residual norm in the nonlinear case is based on the perturbation argument ˇ x while the eigenpair approaches convergence. Each eigenpair (λ, ˇ) of a nonlinear ˇ problem T (·), is an eigenpair of the matrix T (λ) or if T is differentiable also the ˇ = µT 0 (λ)x. ˇ generalized eigenvalue problem T (λ)x During the iteration the residual of the nonlinear eigenpair is becoming closer and closer to the residual of the eigenpair of the linear problem, for which backward error is an established convergence criterion. 5. A local restart technique. The iterative projection methods described in the last section hit their limitations if a large number of eigenvalues or a set of some subsequent eigenvalues in the interior of the spectrum is required. In order to preserve the numbering the dimension of the search space has to be at least as large as the number of eigenvalues in J preceding the sought one. Therefore the size of the projected problem is growing with the number of the wanted eigenvalues, which results in increasing time consumed by the nonlinear solver and increasing storage requirements. 5.1. Local numbering of eigenvalues. To overcome these difficulties while retaining the benefits of the minmax characterization we propose to replace the global numbering of the eigenvalues by a local one, i.e. a numbering obtained w.r.t. some chosen eigenvalue. In contrast to the global variant, for the local numbering to hold, it is not necessary to include the entire set of preceding eigenvectors or the invariant subspace of T (λk ), mentioned in the last paragraph of Section 4.4, into the search subspace after a restart. ˆ x ˆ ∈ J and x Assume that we are given an eigenpair (λ, ˆ), λ ˆ ∈ Cn , of the nonlinear

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eigenproblem (1.1). We refer to such an eigenpair as an anchor. In the following to ˆ is a simple eigenvalue, but all the avoid unnecessary technicalities we assume that λ results can be generalized to multiple eigenvalues. Let V be a subspace of Cn that contains x ˆ, and let the columns of V form a basis of V. ˆ is also an eigenvalue of the projected problem We observe that since x ˆ ∈ V, λ TV (λ)y := V ∗ T (λ)V y = 0. Furthermore, as the projected problem TV (·) along with the original problem T (·) ˆ a local number, ` = `(V), satisfies the conditions of Theorem 3.2 we can assign to λ in the following way: ˆ is the `th eigenvalue of the nonlinear problem TV (λ)y := V ∗ T (λ)V y = 0 λ ⇔ ˆ = 0 is the `th largest eigenvalue of the linear problem V ∗ T (λ)V ˆ y = µ(λ)y. ˆ µ(λ) The remaining eigenvalues of TV (·) can now be given numbers relative to the anchor number, `(V). We call such a relative numbering local. EXAMPLE 5.1. Let V := span{x1 , x3 , x7 , x8 , x10 } where xi is an eigenvector of (1.1) corresponding to the ith eigenvalue λi . Then the projected problem TV (λ)y = 0 has exactly the eigenvalues λi , i = 1, 3, 7, 8, 10 in J. For the anchor x ˆ := x7 it holds that ` = 3, and the local numbers of the subsequent eigenvalues λ8 and λ10 are 4 and 5, respectively. ˆ can be determined REMARK 5.2. Numerically, the local number of the anchor λ, ˆ ˆ as the number of the eigenvalue of the linear problem TV (λ)y = µ(λ)y with the smallest absolute value: if µ1 ≥ µ2 ≥ . . . are its eigenvalues then `(V) := arg

min

k=1,...,dim V

|µk |.

5.2. Local restart framework. The local numbering can be used to target the eigenvalue of the original large sparse problem in the following way. Let V0 := span{ˆ x, v} and v ∈ Cn be an approximation to the next eigenvector. Starting with V0 we determine approximations to the eigenvalue subsequent to the ˆ projecting the nonlinear eigenvalue problem (1.1) to a sequence of subspaces anchor λ V0 ⊂ V1 ⊂ V2 ⊂ . . . . The subspaces Vk+1 are generated in a course of an iterative projection method aiming at the (`(Vk ) + 1)th eigenvalue in the kth iteration step. Explicitly stating the dependence of ` on Vk we emphasize that the number `(Vk ) of the anchor may change in the course of the iteration. After convergence of the eigenvalue we may continue the iterative projection method aiming at the (`(Vk ) + 2)th eigenvalue or we may replace the anchor by the newly converged eigenpair. Since the current search space contains useful information about further eigenvalues it is advisable to continue expanding the search space until the convergence has become too slow or the dimension exceeds a given bound. Once we have the local numbering there is no necessity any more to include all the eigenvectors corresponding to the preceding eigenvalues in J or the invariant ˆ corresponding to its nonnegative eigenvalues into the search space subspace of T (λ) after a restart. All that we need to set up a new search subspace is the eigenvector ˆ and an approximation v to the next eigenvector (or x ˆ corresponding to an anchor λ a random vector if such an approximation is not at hand). This leads to the restart framework in Algorithm 3.


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Algorithm 3 Restart Framework Require: 1: Preconditioner K ≈ T (σ)−1 for a suitable pole σ, 2: Anchor pair (λi , xi ), an (approximate) eigenpair of T (·) 3: optionally v an approximation to xi+1 otherwise v := rand Execute: 4: V := [xi , v] 5: j := 1 6: while restart condition not satisfied do 7: repeat 8: Determine the local number of the anchor, `(V ) ˜ `+j , y˜`+j ) of TV (·) 9: Compute (` + j)th eigenpair (λ 10: Expand V aiming at the (` + j)th eigenpair (λ`+j , x`+j ) ˜ `+j , V y˜l+j ) =: (λi+j , xi+j ) converged 11: until eigenpair (λ 12: if (λi+j , xi+j ) ∈ {(λi+k , xi+k ), 0 ≤ k < j} then 13: Locate suspect eigenvalue θ, and its Ritz vector xθ 14: if (θ, xθ ) converged then 15: Recover missed out eigenpair (θ, xθ ) and adjust numbering 16: j := j + 1 17: else 18: Expand V aiming at the Ritz pair (θ, xθ ) 19: end if 20: else 21: j := j + 1 22: end if 23: end while

5.3. Spurious eigenvalues. In Example 5.1, the search subspace V has been chosen to contain eigenvectors only, and therefore successive eigenvalues of T (·) have consecutive local numbers. However, in a course of iteration, the search subspace will also contain further vectors besides the eigenvectors which can adversely affect the local numbering. Only for the sake of the following argument let us assume that the nonlinear eigenvalue problem (1.1) is overdamped such that the Rayleigh functional p is defined on Cn \ {0} and hence there exists a basis V = {v1 , . . . , vn } of Cn of eigenvectors of T (·) to the n eigenvalues λi , corresponding to p, in ascending order. ˆ x Let the anchor pair be (λ, ˆ) with local number `. Assume that the method with search space V has already computed the eigenvalues λ` ≤ λ`+1 ≤ · · · ≤ λ`+m of T (·). After expanding V to V˜ := span{V, v} by some vector v, each of λ` , . . . , λ`+m remains an eigenvalue of the new projected problem TV˜ (λ)z = 0. However, TV˜ (λ)z = 0 has ˜ z˜) and it may happen that λ` < λ ˜ < λ`+m . an additional eigenpair (λ, From the minmax characterization of the eigenvalues of T (·) it follows that there exist at least m + 2 eigenvalues not exceeding λ`+m . If λ` were the smallest eigenvalue of T (·) i.e. λ` = λ1 , then it would be clear that at least one eigenvalue is missing in the interval [λ` , λ`+m ]. However, with an anchor in the interior of the spectrum it is possible for the additional Ritz ˜ := V˜ z˜, that its representation with respect to Pvector, x the eigenbasis V of T (·), x ˜ = i αi vi , contains components αi vi such that some of the corresponding eigenvalues are smaller than λ` and others larger than λ`+m . We call such a Ritz value a spurious eigenvalue of T (·). The presence of a spurious eigenvalue

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obviously causes an increase of the local number of all the eigenvalues following it. Our argument took advantage of the existence of an eigenbasis of Cn , i.e. we assumed that the nonlinear eigenvalue problem (1.1) is overdamped. It is clear however, that spurious eigenpairs may also appear for nonoverdamped problems since there may also exist Ritz vectors which are linear combinations of eigenvectors corresponding to eigenvalues on either side of the interval of interest. The additional complication for nonoverdamped problems is that the linear combination can also contain vectors which are not in the domain of definition of the Rayleigh functional p, which can have the same effect. Occurrence of spurious eigenvalues is inherent to interior eigenvalue computation. It also happens for linear problems, when no transformation is applied to the eigenproblem to map the eigenvalues from the interior to the lower or upper end of the spectrum. Hence, in order to algorithmically exploit the local numbering we need to find a way to recognize when the local numbering has been obscured by spurious eigenvalues and how to effectively restore it. 5.4. Restoring local numbering. In the course of the computation it may happen that the algorithm converges to an eigenvalue twice, i.e. it returns λi < λi+1 < · · · < λi+k ≈ λi+k+1 for some k ≥ 1. Then two possibilities arise, either the eigenvalue λi+k is a multiple (at least double) eigenvalue or the algorithm repeatedly converged to a single eigenvalue λi+k . If λi+k is a multiple eigenvalue, which can be checked as described in Corollary 5.3 below, we continue Algorithm 3 to compute the (i + k + 2)th eigenvalue. Corollary 5.3. If the angle between the eigenvectors xi+k and xi+k+1 is different from 0 (in the numerical practice, larger than a prescribed small threshold) or if λi+k is the (`(V ⊥ ) + k)th eigenvalue of the projected problem V ⊥∗ T (λ)V ⊥ y = 0, where V ⊥ denotes a basis of the orthogonal complement of xi+k+1 in V and ` denotes the local number of λi , then λi+k is a multiple (at least a double) eigenvalue. However, if in this way λi+k is not shown to be a double eigenvalue, then the algorithm mistakingly returned the eigenpair (λi+k , xi+k ) instead of (λi+k+1 , xi+k+1 ). From the discussion of spurious eigenvalues in Section 5.3 we infer that the local number of (λi+k , xi+k ) has been raised by 1, i.e. for the current search space V the projected problem TV (·) possesses an additional eigenvalue θ ∈ (λi , λi+k ) such that the Ritz pair (θ, xθ ) 6= (λi+j , xi+j ) for j = 0, . . . , k. This may have happened for one of the following two reasons: • An eigenvalue of the original problem (1.1) in the interval (λi , λi+k ) might have been previously missed out because the corresponding eigenvector x ˇ was not sufficiently present in the initial search space span{xi , v} and might not have been amplified sufficiently in the course of the expansions of V until computing λi+k . Afterwards the component of x ˇ in the search space V has grown large enough to produce the additional eigenvalue approximation θ ∈ (λi , λi+k ), and Algorithm 3 yields the eigenvalue approximation λi+k the second time with a different local number. • It might be the case that no eigenvalue of (1.1) is missing in (λi , λi+k ) but the newly produced eigenvalue θ of the projected problem (4.1) is a spurious one, i.e. a linear combination of eigenvectors of (1.1) corresponding to eigenvalues less than λi and eigenvalues greater than λi+k and possibly some vectors


13

outside of the domain of definition of the Rayleigh functional if the problem is not overdamped. In both cases we identify the additional eigenvalue θ (henceforth we shall refer to such eigenvalues as “suspects”) and its local number ` + j, and we expand the search space aiming at (θ, xθ ), where xθ denotes the Ritz vector corresponding to θ. Then for the projected problem with the extended subspace Vˆ TVˆ (λ)ˆ y=0

(5.1)

either of the following situations happens: • Problem (5.1) has exactly k + 1 eigenvalues λi , . . . , λi+k ∈ [λi , λi+k ], i.e. the additional eigenvalue has left the interval of interest. • There are still k +2 eigenvalues λi , . . . , λi+k , θˆ ∈ [λi , λi+k ], and it follows from the minmax characterization that θˆ ≤ θ. In this case we repeat the expansion of the subspace until the sequence of additional eigenvalues has been moved out from the interval [λi , λi+k ] or has converged to a previously missed out regular eigenvalue. We then adjust the enumeration of the eigenvalues and continue the iterative projection method. • There are still k + 2 eigenvalues λi , . . . , λi+k , θˆ ∈ [λi , λi+k ], and it holds that θˆ > θ. By the minmax theory this is only possible if the spurious eigenvalue we were aiming at has left the interval [λi , λi+k ], and another spurious eigenvalue has entered it. In this case we proceed as in the previous point, targeting the ˆ This scenario is however unlikely, as the direction new spurious eigenvalue θ. which has been added to the subspace was chosen to target eigenvalues close to θ, which is not promoting eigendirections outside of the interval [λi , λi+k ], which are those contributing to spurious eigenvalues. REMARK 5.4. If more than one additional eigenvalue exist in [λi , λi+k ], Algorithm 3 will return a replicate eigenvalue with a number smaller than i + k. All such spurious eigenvalues can be identified and they can be treated in the way described above one after the other. Lines 12-22 of Algorithm 3 summarize the strategy for dealing with spurious values in pseudocode. 5.5. Automated local restart. For problems, for which the cost to setup a restart, i.e. time for computing the preconditioner, generating the new search space and the projected problem, is relatively high in comparison to the remaining computations, we can further improve the performance allowing the algorithm to balance those expenses automatically. Let tr denote the time for the setup of a restart, and let tke be the time needed for computing the kth eigenvalue of problem (1.1), where k denotes the local number after Pk a restart. Then the total time for computing the first k eigenvalues is tkt = tr + j=1 tje , and hence the running average time for computing one eigenvalue since last restart is t¯ke = tkt /k. Notice, that as we compute more and more eigenvalues the setup time per eigenvalues decreases. Let α ≥ 1 and Nv ∈ N0 be parameters depending on the given problem, and we initialize nv := Nv . After computing the kth eigenpair since a restart we adjust nv in the following way min{Nv , nv + 1} if tke ≤ α · t¯ke nv ← nv − 1 else

14


Whenever nv < 0 we restart the method. The presented strategy compares the time required for convergence of an eigenvalue with the running average time and triggers a restart when the eigenvalue convergence is repeatedly slower by factor α than in average. In particular, if Nv = 0 and α = 1 we restart the algorithm straightaway when the time for convergence to an eigenvalue exceeds the average time for computing the eigenvalues since the last restart. 6. Nonlinear Restarted Arnoldi. Integrating the local restart with the iterative projection method framework, we arrive at the nonlinear Arnoldi Method summarized in Algorithm 4. Again, the name Nonlinear Restarted Arnoldi, should not be interpreted as restriction of choice of subspace expansion. In contrast to the globally restarted method in Algorithm 2, the subspace V at no stage needs to contain all the preceding eigenvectors, thanks to the local numbering of eigenvalues. An initial anchor pair can be determined for instance with an iterative projection method expanding the search space by KT (σ)xk where σ is a fixed shift close to the wanted eigenvalues, K ≈ T (σ)−1 is a preconditioner, and xk are the iterates of the projection method aiming at the eigenvalue closest to σ. Obviously, no anchor eigenpair is required if inf x∈D p(x) ∈ J and one is looking for eigenvalues at the lower end of the spectrum as the natural enumeration can be used in the first interval. After a restart one of the just computed eigenpairs can serve as an anchor. More general, for nonlinear eigenvalue problems where the minmax induced ordering and the natural (say ascending) ordering coincide (e.g. minmax admitting quadratic eigenvalue problem), it is also possible to use one of the bounds of the interval of interest [a, b] ⊂ R and enumerate the eigenvalues relatively to this bound. In such a case, we do not fix the anchor, but use the first eigenvalue of the projected nonlinear problem TV (·) larger or equal a as a temporary anchor, meaning that we do iterate it to convergence before moving on the to next eigenpair. Corollary 6.1 is a direct consequence of Theorem 3.2 and shows how to locate a temporary anchor. ˜ m , ym ) be the first eigenvalue of the projected nonlinear Corollary 6.1. Let (λ ˜ m−1 < a ≤ λ ˜ m and from problem TV (·) in the interval [a, b]. Then by assumption λ Theorem 3.2 it follows ˜ m−1 a>λ

⇔

˜m a≤λ

⇔

µm−1 (a) =

max

min y ∗ TV (a)y > 0

W∈Sm−1 y∈W 1

µm (a) = max min1 y ∗ TV (a)y ≤ 0. W∈Sm y∈W

Thus, the local number, m, of the first eigenpair of TV (·) in the interval [a, b] is the number of the largest nonpositive eigenvalue, µ(a) ≤ 0, of TV (a). Since the eigenvalues of the projected problem are upper bounds for the eigenvalues of the original problem, selecting those eigenvalues of the projected problem that are larger or equal the lower bound a, will not cause missing out eigenvalues in the interval [a, b]. We remark that a very similar strategy can be applied when the anchor pair is not available to the required precision i.e. its residual norm is above a set tolerance. We incorporated both those important cases in the pseudocode in Algorithm 4. We might want to keep nlocked eigenpairs in addition to the anchor pair at the restart, to minimize the occurrence of spurious eigenvalues. However the benefits have to be traded off against increased cost of the solution of the projected problems due to larger search subspace dimensions.


15

7. Numerical experiments. We consider a conservative gyroscopic eigenvalue problem originating for instance from the sound radiation problem in Section 2. It is well known that all its eigenvalues are real and occur in pairs ±λ, the corresponding eigenvectors are complex conjugate, and the positive eigenvalues 0 < λ1 ≤ · · · ≤ λn satisfy the minmax characterization [5] λi = min max1 p(w), W∈Si w∈W

where p(x) is the positive solution of the quadratic equation x∗ T (λ)x = −λ2 x∗ M x + iλx∗ Gx + x∗ Kx = 0. All the tests were performed with QARPACK Matlab package [1] on a desktop machine with two quadcore Intel Xeon X5550, 2.67GHz processors and 48 GB RAM. The LU decompositions and the subsequent system solves for small problems were performed using Matlab built in LU and for large problems with the PARDISO routines. The projected problems were solved by linearization. The results are uniformly presented in terms of elapsed CPU times. We preconditioned the Arnoldi method by the LU factorization of K − σ 2 M where σ is a shift not too far away from the wanted eigenvalues We chose to neglect the gyroscopic term in T (σ) since its influence on the action of preconditioner is small, and moreover K − σ 2 M can be obtained in real arithmetic. We updated the LU factorization at each restart. 7.1. Qualitative properties of the method. We start with showing some qualitative behavior or our method on a small example of a wheel, composed of solid elements with Mooney-Rivlin material, see Figure 7.1. The wheel is pressed on the track and is rotating at a rate of 50Hz. It is discretized with 450 brick elements with 720 nodes yielding after application of boundary conditions, 1728 degrees of freedom.

Fig. 7.1. Solid rubber wheel

Fig. 7.2. Continental AG 205/55R16-91H tire

For the purpose of comparison we computed all the eigenpairs in the interval [0, 16820] by a globally restarted nonlinear Arnoldi method. This corresponds to the

16


smallest 200 eigenvalues. In all experiments an eigenvalue was regarded as converged if its residual norm was smaller than 10−4 . The preconditioner was recomputed, whenever the ratio τ of two successive residual norms in the last step before convergence exceeded 0.1. To prevent the search subspace from getting arbitrarily large we used global restarts which were triggered whenever the search subspace dimension exceeded 230, a hard threshold on the subspace dimension. In the global restart technique we restart the nonlinear Arnoldi method with an orthogonal basis of the subspace spanned by all eigenvectors computed so far. The total computing time, and the time spent on solving the projected nonlinear problems for the wheel problem are summarized in Table 7.1. In the next experiment we used the same global restart technique, but this time the restart was triggered whenever the dimension of the subspace exceeded the number of the last converged eigenvalue by 30, a soft threshold on the subspace dimension. In this way the average dimension of the search spaces and therefore the time for solving the projected problems were reduced, see Table 7.1. Plotting the total computing time and the time for solving the projected nonlinear problems in Figure 7.3 reveals a superlinear growth. In fact, the CPU time spent on solving the projected eigenproblems itself grows super-linearly, determining the general trend. Table 7.1 Global restarts with hard and soft threshold

gl. rest. hard thres. gl. rest. soft thres.

total CPU [s] 11883 2941

PNEP CPU [s] 11721 2782

#LU 8 39

#Restarts 5 41

Next, we computed the smallest 200 eigenvalues with the local restart strategy described in Section 5. A restart was triggered whenever the search space dimension exceeded 80 or the convergence rate τ became slower than 0.5. The experiment was repeated 10 times and the averaged elapsed computing times are shown in Figure 7.3. The super-linear time growth has been effectively eliminated thought the local restart strategy. CPU times for local restart, λ∈ [1.175e+04,1.682e+04]

CPU times for global and local restart, eigenvalues in [0, 1.682e+04]

350

0.8

total CPU time, global NEVP Solver CPU time, global total CPU time, local NEVP Solver CPU time, local

0.7

250 CPU time[s]

CPU time[h]

0.6 0.5 0.4 0.3

200 150 100

0.2

50

0.1 0

total CPU time, dim=80 PNEVP CPU time, dim=80 total CPU time, auto PNEVP CPU time, auto

300

0

50 100 150 Number of computed eigenvalues

200

Fig. 7.3. Global versus local restart for first 200 eigenvalues

0

0

20 40 60 80 Number of computed eigenvalues

Fig. 7.4. 101 . . . 200

100

Local restart for eigenvalues

The outstanding advantage of the local restart strategy is its ability of computing eigenvalues in an interval in the interior of the spectrum without the need of computing

17


all the preceding eigenpairs. In the general case all that is needed is an anchor pair, which can be determined for instance by a residual inverse iteration with a shift close to the interval bound. Using the same local restart strategy we computed all the eigenpairs in (λ100 = 11748, 16820] which corresponds to the eigenvalues 101 to 200 using the eigenpair (λ100 , x100 ) as the initial anchor. Again for reproducibility of the results we repeated the experiment 10 times. As expected the averaged computing time has been approximately halved from 700 to 350[s], Figure 7.4. To illustrate typical behavior of the method in more detail, in Figure 7.5 for just one run of the experiment we plotted histograms of the computed eigenvalues and of the occurrences of the spurious eigenvalues in each of the intervals between the consecutive restarts. The corresponding eigenvalue convergence history throughout first 500 iterations is depicted in Figure 7.6. The not encircled dots pin down the occurrence of the spurious values (cf. histogram in Figure 7.5) during the iteration. Eigenvalue convergence

Spurious value occurrence 10 No of spurious values in an interval

No of eigenvalues in an interval

10

8

6

4

2

0

200

400 600 800 Iteration intervals

8

6

4

2

0

1000

200

400 600 800 Iteration intervals

1000

Fig. 7.5. Histogram of (a) eigenvalue convergence, (b) spurious value occurrence per interval between consecutive restarts in one run of computation of eigenvalues in (11748, 16820]

4

1.5

NRA: Eigenvalue convergence history

x 10

E

D

C

D

E

E

1.45 1.4

Real(λ)

1.35 1.3 1.25 1.2 converged values eigenvalues anchors

1.15 1.1

0

50

100

150

200

250 Iteration

300

350

400

450

500

Fig. 7.6. Eigenvalue convergence history throughout first 500 iterations of NRA+RII while computing eigenvalues in (11748, 16820] of the gyroscopic wheel problem

The automated restart strategy described in Section 5.5, can be used to let the algorithm balance the limit on the search subspace size on fly. Using the automated

18


restart with α = 1 and Nv = 1 further reduced the average CPU time to about 130[s], see Figure 7.4. Figures 7.3 and 7.4 demonstrate that using the local restart technique, the cost for computing one eigenvalue is approximately the same throughout the iteration, no matter what is the eigenvalue number. Thus we conclude that the new local restart technique effectively eliminates the super-linear CPU time growth with the number of computed eigenvalues and hence constitutes an efficient method for computing eigenvalues in the interior of the spectrum. 7.2. Arnoldi vs Jacobi Davidson subspace expansion. For the purpose of performance comparison of residual inverse iteration (RII) and Jacobi Davidson (JD) type subspace expansions, we computed all the eigenpairs of the wheel problem in the interval (11747, 16820] using both subspace expansions. Each of the experiments was repeated 10 times and the performance figures were averaged through 10 runs. The balancing was switched off, to focus on the effect of subspace expansion only. In both cases NRA consistently found all eigenvalues. NRA+RII needed on average 1087 iterations and 15.4 LU factorizations, while NRA+JD 878.5 and 12, respectively. Nonetheless, the RII variant is still slightly faster in terms of the total CPU time, see Figure 7.7. This is due to an JD expansion step being more expensive than an RII step.

CPU time [s]

CPU times for gyroscopic QEVP in interval [11747,16820] 350 total CPU time, NRA+RII PNEVP CPU time, NRA+RII 300 total CPU time, NRA+JD PNEVP CPU time, NRA+JD 250 200 150 100 50 0

0

20

40 60 Number of eigenvalues

80

100

Fig. 7.7. CPU time of NRA+RII and NRA+JD for computation of eigenvalues in (11748, 16820] of the gyroscopic wheel problem without automatic balancing

7.3. Large sparse gyroscopic EVP. Our second example is a tire 205/55R1691H (see Figure 7.2) provided by Continental AG. The tire is discretized with 39204 brick elements and 42876 nodes. The nodes at the wheel rim are fixed resulting in 124992 degrees of freedom. To account for a complex structure of the tire, 19 different materials are used in the finite element model. The model includes the stress approximating the air pressure in the tire. The tire is pressed on the track and is rotating at a rate corresponding to a vehicle speed of 50 km/h. The tolerance for the residual norm was chosen as 10−6 , the maximal search subspace was 300. After each restart, only the anchor vector and the next approximation were kept in the subspace. The preconditioner was recomputed after at most 300 iterations, subject to residual norm ratio of at most 0.9 and automated balancing with Nv = 1 and α = 1. We used residual inverse iteration for subspace expansion and PARDISO as a preconditioner.


19

NRA: CPU times for tire, λ∈ [0, 10000] total CPU time PNEVP CPU time

6

CPU time[h]

5 4 3 2 1 0

0

50

100 Number of eigenvalues

150

Fig. 7.8. CPU time for NRA+RII for eigenvalues in [0, 10000] of the gyroscopic tire 205/55R16-91H problem

We computed all the eigenvalues in the interval [0, 10000]. NRA needed 2036 iterations and 8 LU factorizations to find all 184 eigenvalues in this interval. Figure 7.8 shows the total CPU time and the CPU time for solving projected nonlinear eigenvalue problems. In this problem, we observe a slight increase of CPU time, as we progress to the eigenvalues higher in spectrum. Here the reason is an increased occurrence of spurious eigenvalues immediately after the restarts. Furthermore, we observe that while in the initial iterations our automated balancing strategy effectively governs the restarts, higher in spectrum its effectiveness is diminished. The reason is that, due to the increased occurrence of spurious values, the running average time for computing an eigenpair gets elongated already in the iterations immediately after the restart. This mixes with the effect of the CPU time growth due to the subspace expansion and reduces the effectiveness of balancing and hence causes it to frequently fall back on a hard threshold on subspace dimension to trigger the restart. The key to the optimal performance is to balance the subspace growth with the occurrence of spurious eigenvalues. An optimal strategy may include adapting the number of eigenvectors kept in the local basis along with the anchor, nlocked , in dependence of e.g. occurrence of spurious eigenvalues in the previous interval. 8. Conclusion. We presented a local restart technique for iterative projection methods for solution of nonlinear Hermitian eigenvalue problems admitting a minmax characterization of their eigenvalues. We showed how the proposed technique can effectively eliminate a super-linear search subspace growth experienced when computing a large number of eigenvalues. Properly initialized, the method can be employed for computing eigenvalues in the interior of the spectrum. Iterative projection methods here considered work directly on the nonlinear eigenvalue problem without increasing its size and possibly destroying its structure by prior linearization. In this setting we do not have a transformation, like shift-invert for linear problems, mapping the eigenvalues close to a chosen shift to the exterior of the spectrum. In the absence of such transformation, spurious eigenvalues are intrinsic to interior eigenvalue computations and we proposed an effective strategy for dealing with such spurious values. We incorporated the proposed technique in the nonlinear Arnoldi and Jacobi Davidson methods. We illustrated various aspects of the method on numerical examples. The efficiency of the new method was demonstrated for a large gyroscopic eigenvalue

20


problem modeling the dynamic behavior of a rotating tire. In the future we intend to extend the local restart technique to problems with complex eigenvalues, which distribution in the complex plain is close to a line. The first steps towards such generalization are problems with a dominant real or imaginary eigenvalue parts. We implemented one such technique in QARPACK and the results look promising.

REFERENCES [1] M.M. Betcke. QARPACK Quadratic Arnoldi Package. University College London. [2] M.M. Betcke. Iterative Projection Methods for Symmetric Nonlinear Eigenvalue Problems with Applications. PhD thesis, Institute of Numerical Simulation, Hamburg University of Technology, 2007. [3] T. Betcke and H. Voss. A Jacobi–Davidson–type projection method for nonlinear eigenvalue problems. Future Generation Comput. Syst., 20(3):363 – 372, 2004. [4] R.J. Duffin. A minimax theory for overdamped networks. J. Rat. Mech. Anal., 4:221 – 233, 1955. [5] R.J. Duffin. The Rayleigh–Ritz method for dissipative and gyroscopic systems. Quart. Appl. Math., 18:215 – 221, 1960. [6] D.R. Fokkema, G.L.G. Sleijpen, and H.A. van der Vorst. Jacobi-Davidson style QR and QZ algorithms for the partial reduction of matrix pencils. SIAM J. Sci. Comput., 20:94 – 125, 1998. [7] P. Hellnwein, C.H. Liu, G. Meschke, and H.A. Mang. A new 3-D finite element model for cord reinforced rubber composites - applications to analysis of automobil tires. Finite Elements in Analysis and Design, 14:1 – 16, 1993. [8] E. Jarlebring and H. Voss. Rational Krylov for nonlinear eigenproblems, an iterative projection method. Applications of Mathematics, 50:543 – 554, 2005. [9] B.-S. Liao, Z. Bai, L.-Q. Lee, and K. Ko. Nonlinear Rayleigh-Ritz iterative methods for solving large scale nonlinear eigenvalue problems. Taiwanese J. Math., 14:869, 2010. [10] M. Markiewicz and H. Voss. A local restart procedure for iterative projection methods for nonlinear symmetric eigenproblems. In A. Handlovicova, Z. Kriva, K. Mikula, and D. Sevcovic, editors, Algoritmy 2005, 17th Conference on Scientific Computing, Vysoke Tatry Podbanske, Slovakia 2005, pages 212 – 221, Bratislava, Slovakia, 2005. Slovak University of Technology. [11] K. Meerbergen. Locking and restarting quadratic eigenvalue solvers. SIAM J. Sci. Comput., 22:1814 – 1839, 2001. [12] U. Nackenhorst. The ALE–formulation of bodies in rolling contact. Theoretical foundations and finite element approach. Comput. Meth. Appl. Mech. Engrg., 193:4299 – 4322, 2004. [13] U. Nackenhorst and O. von Estorff. Numerical analysis of tire noise radiation – a state of the art review. In Inter-noise 2001. The 2001 International Congress and Exhibition on Noise Control Engineering, The Hague, The Netherlands, 2001. [14] A. Neumaier. Residual inverse iteration for the nonlinear eigenvalue problem. SIAM J. Numer. Anal., 22:914 – 923, 1985. [15] V. Niendorf and H. Voss. Detecting hyperbolic and definite matrix polynomials. Linear Algebra Appl., 432:1017 – 1035, 2010. [16] E.H. Rogers. A minimax theory for overdamped systems. Arch. Ration. Mech. Anal., 16:89 – 96, 1964. [17] A. Ruhe. A rational Krylov algorithm for nonlinear matrix eigenvalue problems. Zapiski Nauchnyh Seminarov POMI, 268:176 – 180, 2000. [18] A. Ruhe. Rational Krylov for large nonlinear eigenproblems. In J. Dongarra, K. Madsen, and J. Wasniewski, editors, Applied Parallel Computing. State of the Art in Scientific Computing, volume 3732 of Lecture Notes on Computer Science, pages 357 – 363, Berlin, 2006. Springer Verlag. [19] H. Voss. An Arnoldi method for nonlinear symmetric eigenvalue problems. In Online Proceedings of the SIAM Conference on Applied Linear Algebra, Williamsburg., http://www.siam.org/meetings/la03/proceedings/VossH.pdf, 2003. [20] H. Voss. An Arnoldi method for nonlinear eigenvalue problems. BIT Numerical Mathematics, 44:387 – 401, 2004. [21] H. Voss. A Jacobi–Davidson method for nonlinear eigenproblems. In M. Buback, G.D. van Albada, P.M.A. Sloot, and J.J. Dongarra, editors, Computational Science – ICCS 2004, 4th


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International Conference, Krak´ ow, Poland, June 6–9,2004,Proceedings, Part II, volume 3037 of Lecture Notes in Computer Science, pages 34–41, Berlin, 2004. Springer Verlag. [22] H. Voss. A Jacobi–Davidson method for nonlinear and nonsymmetric eigenproblems. Computers & Structures, 85:1284 – 1292, 2007. [23] H. Voss. A minmax principle for nonlinear eigenproblems depending continuously on the eigenparameter. Numer. Lin. Algebra Appl., 16:899 – 913, 2009. [24] H. Voss and B. Werner. A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems. Math. Meth. Appl. Sci., 4:415–424, 1982.

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Algorithm 4 Nonlinear Restarted Arnoldi Require: 1: Eigenvalue interval [a, b], if λj ∈ J not monotone, required a = λi , b = λimax ˆ := a, x 2: Anchor vector x ˆ : (λ ˆ) ≈ (λi , xi ), ith eigenpair of T (·), required if λj ∈ J not monotone 3: optionally Initial basis V , V ∗ V = I 4: Choose initial shift σ close to the lower interval bound, a 5: Compute initial preconditioner K ≈ T (σ)−1 6: optionally v an approximation to xi+1 , otherwise v := rand Execute: 7: if no anchor vector x ˆ then 8: anchor f lag := f alse 9: Initial search subspace V := [ˆ x := rand, v] 10: Local offset j := 0 11: else 12: anchor f lag := true 13: Initial search subspace V := [ˆ x ≈ xi , v] ˆ x 14: if anchor pair (λ, ˆ) sufficiently accurate then 15: Local offset j := 1 16: else 17: Local offset j := 0 18: end if 19: end if 20: while λi+j 6= b do 21: while restart condition not satisfied do 22: repeat 23: if anchor f lag then ˆ 24: `(V ) := local number of the anchor λ 25: else 26: `(V ) := local number of the first eigenvalue in the interval [a, b] 27: end if ˜ `+j , y˜`+j ) of TV (·) 28: Compute (` + j)th eigenpair (λ 29: Expand V aiming at the (` + j)th eigenpair (λ`+j , x`+j ) ˜ `+j , V y˜l+j ) =: (λi+j , xi+j ) converged 30: until eigenpair (λ 31: if (λi+j , xi+j ) ∈ {(λi+k , xi+k ), 0 ≤ k < j} then 32: Locate suspect eigenvalue θ, and its Ritz vector xθ 33: if (θ, xθ ) converged then 34: Recover missed out eigenpair (θ, xθ ), adjust ordering 35: Increase local offset j := j + 1 36: else 37: Expand V aiming at the Ritz pair (θ, xθ ) 38: end if 39: else 40: Increase local offset j := j + 1 41: end if 42: end while{restart condition not satisfied} 43: Global anchor number becomes i := i + j − 1 − nlocked ˆ := λi 44: Reset the anchor to a recently computed eigenvalue, λ 45: anchor f lag := true 46: Reset local offset j := nlocked + 1 ˜ i+j 47: Choose new shift σ close to approximation of the next eigenvalue λ −1 48: Recompute preconditioner K ≈ T (σ) ˜ i+j )˜ 49: Reset search subspace V := orthonormalize([xi , . . . , xi+nlocked , KT (λ xi+j ]) 50: end while{λi+j 6= b}