SIAM J. SCI. COMPUT. c© 2009 Society for Industrial and Applied MathematicsVol. 31, No. 5, pp. 3760–3780

SOLUTION OF LARGE SCALE EVOLUTIONARY PROBLEMSUSING RATIONAL KRYLOV SUBSPACES WITH OPTIMIZED

SHIFTS∗

VLADIMIR DRUSKIN† , LEONID KNIZHNERMAN‡, AND MIKHAIL ZASLAVSKY†

Abstract. We consider the computation of u(t) = exp(−tA)ϕ using rational Krylov subspacereduction for 0 ≤ t < ∞, where u(t), ϕ ∈ RN and 0 < A = A∗ ∈ RN×N . The objective of thiswork is the optimization of the shifts for the rational Krylov subspace (RKS). We consider thisproblem in the frequency domain and reduce it to a classical Zolotaryov problem. The latter yieldsan asymtotically optimal solution with real shifts. We also construct an infinite sequence of shiftsyielding a nested sequence of the RKSs with the same (optimal) Cauchy–Hadamard convergence rate.The effectiveness of the developed approach is demonstrated on an example of the three-dimensionaldiffusion problem for Maxwell’s equation arising in geophysical exploration.

Key words. matrix function, matrix exponential, Zolotaryov problem, time-domain Maxwell’ssystem, rational approximation, model reduction

AMS subject classifications. 30C85, 30E10, 41A05, 41A20, 65M60, 86–08

DOI. 10.1137/080742403

1. Introduction. Many boundary value problems can be reduced to the com-putation of

(1.1) u = f(A)ϕ,

where u, ϕ ∈ RN and A = A∗ ∈ RN×N . We shall assume throughout the paper that‖ϕ‖ = 1.

We are particularly interested in the solution of the evolutionary equation

(1.2) Au + ut = 0, u|t=0 = ϕ,

with A > 0 for 0 ≤ t <∞, or equivalently in computing the vector function

(1.3) u(t) = exp(−tA)ϕ.

Our main application arises from geophysical deep hydrocarbon exploration in whichcase (1.2) is the semidiscretized time-domain Maxwell system (in the diffusion ap-proximation). In practice, A is a large ill-conditioned matrix (a finite-difference orfinite-element disretization of the spatial operator). This problem is multiscale innature and requires computation for very small and large diffusion times.

Problem (1.1) can be solved with use of the so-called spectral Lanczos decom-position method (SLDM) that has been known since the 1980s (see, e.g., papers[38, 49, 13], and see [18, 29] for a more up-to-date reference list). The SLDM canbe described as follows. Let us perform n steps of the Lanczos recursion (pro-vided there is no early termination) with matrix A and initial vector ϕ. The

∗Received by the editors December 2, 2008; accepted for publication (in revised form) June 15,2009; published electronically October 9, 2009.

http://www.siam.org/journals/sisc/31-5/74240.html†Schlumberger Doll Research, 1 Hampshire St., Cambridge, MA 02139 (druskin1@slb.com,

mzaslavsky@slb.com).‡Consultant to Schlumberger Doll Research, Geroev Panfilovtsev St., House 1, Building 5, Flat

132, Moscow, 125480 Russia (lknizhnerman@gmail.com).

3760

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3761

Lanczos recursion, whose coefficients are stored in a symmetric tridiagonal matrixH ∈ Rn×n, produces an orthonormal basis Q ∈ RN×n of the Krylov subspaceKn = span{ϕ,Aϕ, . . . , An−1ϕ}. The SLDM approximate solution un ∈ Km can bewritten as

un = Qf(H)e1,

where e1 ∈ Rn is the first unit vector. In exact arithmetic H = Q∗AQ and Q∗ϕ = e1,so the SLDM can be viewed as the Galerkin method on the Krylov subspace. A prioriSLDM convergence estimates (for exact arithmetic) were obtained in [14]. Theseestimates show that the SLDM converges at least with the same speed as the bestuniform polynomial approximation of f(λ) on the spectral interval of A (see also[19, 43] for related results). In particular, for (1.3),

(1.4) ‖u− un‖ = O

[√t‖A‖n

exp(− n2

‖A‖t)]

provided n ≤ 0.5t‖A‖ [14]; similar estimates can be found in [30]. Moreover, SLDMconverges strictly monotonically in the Euclidean norm [12].

Estimate (1.4) shows that the SLDM requires n = O(√t‖A‖), which can be too

restrictive for large t‖A‖. It is well known that the best rational approximants toexp(−λ) have a linear convergent rate1 on [0,+∞], i.e., applied to A substituted for λthey have uniform linear convergence for all nonnegative A. This makes it attractiveto consider instead of polynomial Krylov subspaces Km their rational counterparts,i.e., rational Krylov subspaces (RKS) [42]. A common name for the RKS basedapproximations of (1.1) is the rational Krylov subspace reduction (RKSR). Let usconsider a subdiagonal RKS in the generic form

(1.5) Un = span{b, Ab, . . . , An−1b}, b =n∏j=1

(A+ sjI)−1ϕ,

provided n does not exceed the degree of the minimal polynomial of ϕ, i.e., the nonzeromonic polynomial p of lowest degree such that p(A)ϕ = 0 [44]. In [15] the so-calledextended Krylov subspace method was suggested for computation of Stieltjes–Markovmatrix functions, where only rational approximants with zero poles were used. Theso-called restricted denominator method using the Krylov subspace with a singleoptimally chosen shift for computation of Stieltjes–Markov and exponential matrixfunctions was considered in [36, 16, 8].

We assume that the RKS is computed using iterative methods for which there areno computational advantages in solving multiple linear systems with the same shifts(because of extensive memory requirements for the discretization of large scale geo-physical electromagnetic problems), so we suppose that sj do not coincide (extensionto subspaces allowing multiplicity of sj will be discussed in Remark 6).

Such RKSs are widely used in model reduction, in particular, for computation oftransfer functions of linear problems; see for details literature reviews [5, 17]. A keyquestion in the construction of the RKS is the optimal choice of shifts sj .

The denominator q (i.e., the shifts sj) can be chosen from the best rational ap-proximant. For exp(−λ), such approximants are well known (see, e.g., [10, 22, 3]).

1Though not superlinear, unlike (1.4).

3762 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

However, we need to compute not just one, but a parametric family of matrix functions(1.3). Generally, matrix functions can be computed via contour integration involvingthe resolvent. Trefethen with collaborators in a series of papers [47, 51, 28, 46] devel-oped a number of special quadrature approaches to compute corresponding integralsfor some families of matrix functions (see also [35]). Evaluation of such quadraturesrequires the solution of a generally complex shifted linear system for every node (moreprecisely, one linear system for a couple of complex conjugate nodes). This approachallows one to compute (1.3) for positive bounded time intervals, but the convergencespeed (independent on A’s spectral interval) is still significantly affected by the timeinterval’s condition number [51].

This paper is organized as follows. In section 2 we give general background onthe RKSR. In section 3 we prove that the RKSR error for (1.1) is bounded by twicethe error of the best uniform rational approximation (with the same denominator q)of f on A’s spectral interval; i.e., we reduce the problem of optimal choice of the RKSto the problem of finding the best rational approximants.

In section 4 we consider the Fourier transformed (1.3), i.e., the problem for theresolvent (A + sI)−1ϕ with parameter s on the entire imaginary axis. For rationalapproximation of this parametric problem we modify an approach developed in [31].The key to the above approach is approximating the resolvent by the two-parameterskeleton approximation introduced in [48, 39]. We reduce the minimization of theerror of the skeleton approximation to the third Zolotaryov problem in the complexplane. In [31] (for a slightly different problem of the computation of the resolvent of anunbounded operator on a bounded interval of the imaginary axis) we obtained asymp-totically optimal (in the Cauchy–Hadamard sense) pure imaginary sj in a closed formvia elliptic integrals. Surprisingly, for the problem considered here, i.e., the optimalrational approximation of the resolvent of a bounded A on the entire imaginary axis,a classical Zolotaryov solution with real sj gives an asymptotically optimal solutionthat also according to [6] satisfies the necessary optimality conditions. It is impor-tant because real shifts have a clear advantage over complex ones for the solutionof linear systems with matrices A + sjI (especially with the preconditioned Krylovmethods).

When one increases the dimension of the RKS, the larger set of the Zolotaryovshifts does not include the smaller one, which is inconvenient when an appropriate sub-space dimension is not known a priori and should be determined adaptively to achievedesirable convergence level. To circumvent this drawback, we modify Zolotaryov’s so-lution to construct an infinite sequence of shifts (equidistributed with respect to themeasure generated by the Zolotaryov points) yielding a nested sequence of the RKSswith the same asymptotically optimal (in the Cauchy–Hadamard sense) convergencespeed. The resulting L∞ RKSR error of (A + sI)−1ϕ for s on the entire imaginaryaxis behaves as

O

[exp

(−π

2n[1 + o(1)]2 log 4λmax

λmin

)]

for large A’s condition numbers λmaxλmin

.In section 5 we extend the obtained results to the time domain. The extension

is based on the Plancherel identity, which equates the L2-error of computation of(A + sI)−1ϕ on the entire imaginary axis and the one of exp(−tA)ϕ on the realpositive semiaxis.

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3763

In section 6 we illustrate the developed approach by a numerical example, wherewe compare our approach with the one developed in [16]. The latter is superiorwhen used for a single time point or moderate time interval, but our algorithm issignificantly more efficient for large time intervals.

In section 7 (concluding remarks) we discuss connection to a recent work byBeckermann and Reichel [7], extension to subspaces with multiple poles and some opentopics, such as connection with the H2-optimal reduced order models and possibleextension to Stieltjes–Markov functions.

In the appendices we give proofs of some of the results announced in section 4.

2. Formulation of the RKSR for general functions. Let us assume that Ais positive definite and its spectrum is a subset of a segment [λmin, λmax] with 0 <λmin < λmax. Assume also that a function f is continuous on [λmin, λmax]. We choosen distinct complex numbers sj outside [−λmax,−λmin] symmetrically with respect tothe real axis. We assume that n does not exceed the degree of the minimal polynomialof (A, b). Such an assumption allows us to construct the RKS (1.5) with dimUn = n.Obviously, Un = span{(A + s1I)−1ϕ, . . . , (A + snI)−1ϕ}. By construction, sl eitherare real or have a complex conjugate counterpart, thus, b ∈ RN .

To approximate (1.1), we will use the Galerkin projection on Un. Let G = Gn ={g1, . . . , gn} be the matrix of an orthonormal basis of Un. Then the RKSR’s approx-imant to u is (well) defined as

(2.1) un = Gf (V )G∗ϕ,

where

V = G∗AG.

In the limiting case of infinite shifts, Un becomes equivalent to Kn, and the RKSRcoincides with the SLDM. The RKSR is efficient when n� N . For moderate n, (2.1)can be computed, for example, via the spectral decomposition of V . There are manyways to construct G. They are known by generic name as the rational Arnoldi method(see, e.g., [42, 17]). In our numerical experiments we implement a well-known simplevariant of rational Arnoldi that can be described as follows. Set g1 = (A+s1I)

−1ϕ‖(A+s1I)−1ϕ‖ .

Then for l > 1 the vector gl is obtained by the Gram–Schmidt orthogonalization of(A + slI)−1gl−1 to gj , j = 1, . . . , l − 1. Usually, the most computationally expensivepart of the rational Arnoldi is the solution of shifted linear systems.

3. A general error bound via rational approximation. Using the spectraldecomposition, we obtain

(3.1) f(A)ϕ =N∑j=1

f(ωj)ϕjzj,

where ωj ∈ R are the eigenvalues (say, enumerated in the increasing order) of Aand zj ∈ RN are the corresponding normalized eigenvectors ϕj = z∗jϕ (

∑Nj=1 ϕ

2j =

‖ϕ‖2 = 1). Similarly,

(3.2) un = Gf (V )G∗ϕ =n∑j=1

f(θj)φjyj ,

3764 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

where (θj , yj) with θj ∈ R, yj ∈ RN are the Raleigh–Ritz pairs of A on U , φj = y∗jϕ,and

∑nj=1 φ

2j = 1.

A well-known property of the Ritz values for self-adjoint operators is the two-sidedinequality

(3.3) λmin < θj < λmax, 1 ≤ j ≤ n.

The following result is the counterpart of the exactness of the SLDM for polyno-mial functions [14].

Lemma 3.1. Put

q(λ) =n∏l=1

(λ + sl).

For any polynomial p with deg p ≤ n− 1 one has

(3.4) p(A)q(A)−1ϕ = Gp(V )q(V )−1G∗ϕ.

Proof. Introduce the vectors

vj = G(V + sjI)−1G∗ϕ, 1 ≤ j ≤ n.

Due to the linearity it is sufficient to establish the equality

(3.5) vj = (A+ sjI)−1ϕ, 1 ≤ j ≤ n.

By construction, vj is the Galerkin approximation of (A+ sjI)−1ϕ on Un, and (3.5)follows from Galerkin’s interpolation properties [25, 26].

It is known that the SLDM error is bounded by twice the error of the best uniformpolynomial approximation on A’s spectral interval [14]. A similar bound can beestablished for the RKSR.

Proposition 3.2. Let p be a polynomial of degree not exceeding n− 1, then

(3.6) ‖f(A)ϕ−Gf(V )G∗ϕ‖ ≤ 2 maxλ∈[λmin,λmax]

∣∣∣∣f(λ) − p(λ)q(λ)

∣∣∣∣ .Proof. In view of (3.4), (3.1), and (3.2), we obtain

‖f(A)ϕ−Gf(V )G∗ϕ‖ =∥∥[f(A) − p(A)q(A)−1

]ϕ−G

[f (V ) − p(V )q(V )−1

]G∗ϕ

∥∥=

∥∥∥∥∥∥N∑j=1

[f(ωj) − p(ωj)

q(ωj)

]ϕjzj −

n∑j=1

[f(θj) − p(θj)

q(θj)

]φjyj

∥∥∥∥∥∥ .Using the triangle inequality and due to (3.3), we get

‖f(A)ϕ−Gf(V )G∗ϕ‖ ≤√√√√ N∑

j=1

ϕ2j

∣∣∣∣f(ωj) − p

q(ωj)

∣∣∣∣2 +

√√√√ n∑j=1

φ2j

∣∣∣∣f(θj) − p

q(θj)

∣∣∣∣2

≤ 2 maxλ∈[λmin,λmax]

∣∣∣∣f(λ) − p(λ)q(λ)

∣∣∣∣ .

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3765

If we choose p to be the optimal polynomial, we obtain

(3.7) ‖u− um‖ ≤ 2 minp(λ)∈C[λ], deg p≤n−1

maxλ∈[λmin,λmax]

∣∣∣∣f(λ) − p(λ)q(λ)

∣∣∣∣ .Remark 1. Passing to a limit, we can eliminate in the assertions of this section

the assumption that sj are distinct. Also, Proposition 3.2 essentially remains validfor nonsymmetric A and for f sufficiently smooth on the numerical range A (insteadof [λmin, λmax]); [11, theorem 2 and formula (1)] is used to this end.

Bound (3.7) indicates that for every f the RKSR yields a rational approximationwith a nearly optimal numerator and the prescribed denominator q. A good choice ofq for a given f can be the denominator of its best (n− 1)/n rational approximant on[λmin, λmax]. For exp(−λ) such approximants are well known [10, 40, 22, 3]. In thecase of good spectral adaptation, the RKSR can even overperform (3.7).

However, the problem is more complicated for parametric exponential (1.3), be-cause it requires approximation for λ ∈ [λmin, λmax] and all t ≥ 0. For this case theRKSR produces a family of rational approximants with time-dependent numeratorsnearly optimal for every t. Our objective is to find a denominator that will be goodfor all nonnegative t.

4. Optimization of the RKS and the third Zolotaryov problem in thecomplex plane. In this section we will optimize the poles sj of the RKSR in the fre-quency domain, i.e., for the approximation of (A+sI)−1ϕ for s ∈ iR. Proposition 3.2allows us to reduce this problem to rational approximation of 1/(λ+ s).

It will be designed with the help of the so-called skeleton approximation intro-duced in [48] and then used in [24, 27]. This approximation for the function 1/(λ+ s)was investigated in [39]. Given two families of distinct complex parameters sj and λj(1 ≤ j ≤ n), the underlying approximant is defined as

(4.1) fskel(λ, s) =(

1λ+s1

, . . . , 1λ+sn

)M−1

⎛⎜⎝1

s+λ1,

...1

s+λn

⎞⎟⎠ ,

where M = (Mkl) is the n× n matrix with the entries Mkl = 1/(λk + sl). Note thatfskel is an (n− 1)/n rational function of λ and of s separately.

It can be easily shown that any rational function p(λ)/q(λ) with deg p ≤ n−1 canbe exactly presented via the skeleton approximation. In particular, if λj = θj for j =1, . . . , n, then fskel(A, s)ϕ = G (V + sI)−1

G∗ϕ [31]; i.e., the skeleton approximationgives an explicit representation of the RKSR solution.

Theorem 3 from [39] asserts that

(4.2) ε(λ, s) = 1 − (λ + s)fskel(λ, s) =n∏j=1

λ− λjλ+ sj

·n∏j=1

s− sjs+ λj

=r(λ)r(−s)

with

r(z) =n∏j=1

z − λjz + sj

;

i.e., λj and sj are the interpolating points.

3766 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

Lemma 4.1. The L2-error of the resolvent on the entire imaginary axis can bebounded via the error of the skeleton approximation as(4.3)√∫ +i∞

−i∞‖(A+ sI)−1ϕ−G(V + sI)−1G∗ϕ‖2 · |ds| ≤ 2c min

λ1,...,λn

maxλ∈[λmin,λmax] |r(λ)|mins∈iR∪{∞} |r(s)| ,

where

(4.4) c =

√∫ +i∞

−i∞

∣∣∣∣ 1λmin + s

∣∣∣∣2 · |ds|.Proof. We consider fskel with some arbitrary λj and the same sj as in (1.5).

Obviously, ε(λ,s)λ+s is the absolute error of the approximation of 1

λ+s by fskel, which isan [n− 1/n]-rational function of λ. Substituting (4.2) into (3.6), we deduce

‖(A+ sI)−1ϕ−G(G∗AG+ sI)−1G∗ϕ‖ ≤ 2 mindeg(p)≤n−1

maxλ∈[λmin,λmax]

∣∣∣∣ 1λ+ s

− p(λ)q(λ)

∣∣∣∣≤ 2 max

λ∈[λmin,λmax]

∣∣∣∣ 1λ+ s

− fskel(λ, s)∣∣∣∣ = 2 max

λ∈[λmin,λmax]

∣∣∣∣ ε(λ, s)λ+ s

∣∣∣∣≤ 2λmax + s

maxλ∈[λmin,λmax] |r(λ)||r(s)|

from which by integration we obtain√∫ +i∞

−i∞‖(A+ sI)−1ϕ−G(V + sI)−1G∗ϕ‖2

ds ≤ 2cmaxλ∈[λmin,λmax] |r(λ)|

mins∈iR∪{∞} |r(s)| .

Using the optimal λj in the above bound we derive (4.3).Remark 2. Augmenting Un with the initial vector ϕ (such subspaces are consid-

ered, for example, in [7]), one can obtain a sharper bound (without c) than (4.3) interms of the residual:

maxs∈iR∪{∞}

∥∥ϕ− (A+ sI)G(V + sI)−1G∗ϕ∥∥ ≤ 2 min

λ1,...,λn

maxλ∈[λmin,λmax] |r(λ)|mins∈iR∪{∞} |r(s)| .

We prefer not to work with the augmented subspace, because it would generate an[n/n]-rational approximant, i.e., not the [n− 1/n] one as fskel.

We wish to choose sj minimizing the bound of Lemma 4.1 by solving the mini-mization problem

(4.5) σn = minλ1,...,λn,s1,...,sn

maxλ∈[λmin,λmax] |r(λ)|mins∈iR∪{∞} |r(s)| .

Looking at (4.5), one can easily recognize the third Zolotaryov problem in the extendedcomplex plane [50, 20].

It was discovered in [6] that this problem has a solution with real parameterssatisfying at least necessary optimality conditions. This solution can be obtained byrestricting

(4.6) sj = λj , j = 1, . . . , n,

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3767

then |r(s)| = 1 for s ∈ iR, so instead of (4.5) we obtain the problem

(4.7) σn = mins1,...,sn

maxλ∈[λmin,λmax]

|r(λ)|.

The latter is a classical Zolotaryov problem on the real line, which has an explicitsolution in terms of elliptic integrals (see [33] or [39, section 4]). We introduce thenumber

(4.8) ρ = exp[−π

4· K

′(μ)K(μ)

],

where K and K ′ (see [1, Ch. 17]) are, respectively, the principal

K(κ) =

1∫0

dt√(1 − t2)(1 − κ

2t2)

and complementary

K ′(κ) = K(√

1 − κ2)

complete elliptic integrals of modulus κ, 0 < κ < 1,

μ =

(1 −√

δ

1 +√δ

)2

and δ =λmin

λmax.

We shall also need the Jacobi elliptic function

dn(v,κ) =√

1 − κ2 sn(v,κ)2,

where sn is another elliptic function defined by

sn(v,κ) = sinψ, v =

ψ∫0

dζ√1 − κ

2 sin2 ζ

(see [1, Ch. 16]).Theorem 4.2 (Zolotaryov). Problem (4.7) has a unique solution

(4.9) sj = λmax dn(

2(n− j) + 12n

K ′(δ),√

1 − δ2), j = 1, . . . , n,

at that

(4.10) σn ≤ 2ρn.

It is known that sj defined by (4.9) lie in [λmin, λmax].For δ → +0 (i.e., for λmax/λmin → +∞) we have (see [1, (17.3.26)])

μ = 1 − 4√δ +O(δ), K(μ) =

12

log2√δ

+ o(1), K ′(μ) =π

2+ o(1),

3768 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

Table 4.1

The true and approximate convergence factors for a few values of λmax/λmin.

δ−1 = λmax/λmin ρ ρappr

101 0.262196267917709 0.262435220633760102 0.438830450281231 0.438831958487455103 0.551573019030466 0.551573028922366104 0.627698561404623 0.627698561473588105 0.682108908983458 0.682108908983963106 0.722802332906693 0.722802332906697107 0.754335555734798 0.754335555734798

whence

ρ = ρappr + o(1), ρappr = exp(π2

2

/log

δ

4

)= exp

(−π

2

2

/log

4λmax

λmin

);

this enables one to obtain an asymptotics to the ρ in terms of elementary functions.Table 4.1 illustrates the asymptotics’ quality.

As was already mentioned, the parameter set providing σn satisfies the necessaryoptimality conditions [6]. It is not clear if it is the true global minimum of theapproximation error function

maxλ∈[λmin,λmax] |r(λ)|mins∈iR∪{∞} |r(s)| .

However, any other set of complex parameters sj , λj , j = 1, . . . , n, cannot give morethan a two-fold decrease of the error function compared to the real parameters givenby (4.6) and (4.9). Namely, the following lower bound is valid.

Theorem 4.3. The double inequality

(4.11) σn ≥ ρn ≥ 0.5σn

holds.A proof is presented in Appendix A.To obtain sj, according to Theorem 4.2 one needs to know a suitable n a priori.

Otherwise, it is desirable to have infinite sequence of sj (yielding a nested sequenceof the RKSs) that can be used until convergence of un. To construct such a sequence,we shall implement equidistributed sequences [9, Ch. VIII]. A possibility of usingsequences equidistributed with respect to a proper measure in the potential theoryproblems was mentioned in [21, section 3]. Other algorithms to compute rationalfunctions with the best asymptotical convergence factor and an extendable parameterset have been presented in [45, 32, 4].

We can assume without loss of generality that λmax = 1. The equilibrium Borelmeasure α of the segment [δ2, 1] with respect to the domain C\R− is defined throughits values on segments as

α([a, b]) = α(b) − α(a), δ2 ≤ a ≤ b ≤ 1,

with the function α satisfying

(4.12) α(δ2) = 0, α′(y) =1

2K ′(δ)√

(y − δ2)y(1 − y), δ2 < y < 1.

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3769

This measure can be viewed as the limit of the distribution of s2jλ2

maxfrom Theorem 4.2

for n→ ∞, see (B.1) from Appendix B.Theorem 4.4. Let tj ∈ [0, 1[, j = 1, 2, . . ., be an equidistributed sequence (EDS).

Define the interpolation points sj by

(4.13) sj =√yj , α(yj) = tj , j = 1, 2, . . . .

Then the rational functions

rn(z) =n∏l=1

z − slz + sl

, n = 1, 2, . . . ,

obey

(4.14) limn→∞

n

√max

λ∈[λmin,λmax]|rn(λ)| = ρ.

A proof of this theorem is postponed to Appendix B. By construction sj givenby (4.13) also satisfy limiting condition (B.1).

Theorem 4.4 asserts that interpolation points (4.13) generate rational fractionswhich are optimal in the Cauchy–Hadamard sense.

In Figure 4.1 we compare Zolotaryov fractions with ones based on equidistributedsequences. We used a simple number-theoretical generator of equidistributed se-quences based on Weyl’s criterion [9].

5. The time-domain problem. The exponential can be presented via inverseFourier transform as

(5.1) exp(−λt) =1

2πi

∫ +i∞

−i∞exp(st)(λ + s)−1ds, t > 0;

similarly, (1.3) can be presented via action of a matrix resolvent (A+ sI)−1ϕ as

exp(−tA)ϕ =1

2πi

∫ +i∞

−i∞exp(st)(A+ sI)−1ϕds, t > 0.

So, we reduced the time-domain problem on the entire positive semiaxis to the com-putation of a resolvent (the frequency domain problem) on the entire imaginary axisconsidered in the previous section. The Plancherel identity yields∫ ∞

0

‖exp(−tA)ϕ−G exp(−tV )G∗ϕ‖2dt(5.2)

=12π

∫ +i∞

−i∞

∥∥(A+ sI)−1ϕ−G(V + sI)G∗ϕ∥∥2 · |ds|.

This equality allows us to formulate a time-domain equivalent of Lemma 4.1.Proposition 5.1. Let us take r with sj the same as in the definition of the RKS

(see section 2). Then the RKSR error satisfies the inequality√∫ ∞

0

‖exp(−tA)ϕ−G exp(−tV )G∗ϕ‖2dt ≤

√2πc minλ1,...,λn

maxλ∈[λmin,λmax] |r(λ)|mins∈iR∪{∞} |r(s)| ,

with c given by (4.4).

3770 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

100

0 20 40 60 80 100

n

function (5.11), δ=10-2

function (5.5), δ=10-2

function (5.11), δ=10-4

function (5.5), δ=10-4

function (5.11), δ=10-8

function (5.5), δ=10-8

Fig. 4.1. The error distribution maxz∈[λmin,λmax] |Πnl=1

z−slz+sl

| as a function of n for the two

families of interpolation points given by Theorems 4.2 and 4.4, λmin = δ, λmax = 1, and threevalues of δ.

Remark 3. Integral (5.1) is strongly oscillatory. So if one needed to compute (1.3)just for a single time point or for a bounded positive time interval (with a moderatecondition number), then an error estimate via the right-hand side of (5.2) could bevery loose. Accurate estimates in such cases can be obtained by deformations of theintegration contour in (5.1) to avoid oscillations (see, e.g., [51]).

Remark 4. Using the optimal sj given by Theorem 4.2, we asymptotically (as thecondition number tends to infinity) have

(5.3) ‖exp(−tA)ϕ−G exp(−tV )G∗ϕ‖L2[0,+∞] ≤ 4 exp

(−π

2n[1 + o(1)]2 log 4λmax

λmin

)

uniformly in n. Even though (5.3) gives linear convergence, for a positive definite Awith large enough ‖A‖t it is obviously better than a superlinear estimate (1.4).

Let us compare (5.3) with the error estimate

O

[exp

(−2πn√

tmax/tmin + 1

)]

of a contour integration method of [51] for the computation of the exponential functionon a time interval [tmin, tmax], where n is the quadrature order (equal to the order

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3771

of the rational approximant). Our bound is given for the entire time domain butaffected by λmax

λmin. On the contrary, the estimate of [51] is uniform with respect to

λmaxλmin

but dependent on tmax/tmin. Relative efficiency of each method depends onthe problem. For example, for the solution of the Maxwell equations in unboundeddomains for geophysical exploration (which will be considered in section 6) a consistentspatial discretization yields λmax

λmin∼ tmax

tmin, in which case our bound will be superior for

problems with large condition numbers.Remark 5. It is well known that the poles of the optimal rational approximation

of exp(−tλ) for a single t are complex [10, 40], and the same is true for known “good”approximations on moderate positive time intervals; see, e.g., [51]. Here we caneffectively use real poles (shifts) given by Theorem 4.2 or Theorem 4.4 because of theapproximation on the entire time domain. The same reasoning as in Remark 3 can beapplied for explanation of this phenomenon. Real poles also appear in the H2-optimalreduce order models with self-adjoint A; see Remark 8.

6. Numerical experiments. As an important practical application we considerthe time-domain (forward) electromagnetic problem arising in geophysical oil explo-ration. It can be reduced to magnetic field formulation of the time-domain Maxwellequations in R3 (displacement currents are assumed to be negligible)

(6.1) ∇× ( μσ)−1∇× H +∂H∂t

= 0, H|t=0 = H0,

with zero boundary conditions at infinity. Here H(t) is the time-dependent vectormagnetic field, μ is the magnetic permeability, which is assumed to be constant, andσ is a (variable in space and constant in time) uniformly bounded positive electricalconductivity distribution. For the accurate solution of the inverse problem we needthe forward solutions on a time interval [tmin, tmax] with very large tmax/tmin (104

and more) to separate and resolve near and far zone inhomogeneities of geologicalformations.

We use the Maxwell equation solver described in [52]. Finite-difference dis-cretization of (6.1) on Lebedev’s grid yields nonnegative symmetric matrix A ≈∇ × ( μσ)−1∇×. The operator A has a null-space, but A’s restriction on the di-vergence free subspace is positive definite, so we require ∇ ·H0 = 0. For consistencyof the approximation the minimal spacial grid step and the size of the computa-tional domain should be proportional, respectively, to minimal and maximal diffusionlengths, i.e., respectively to

√tmin and

√tmax, and thus,

λmax/λmin ∼ tmax/tmin.

We considered a problem with 3 · 105 unknowns and condition number λmax/λmin ≈4 · 106. It is assumed that the 3D geological structure (anomaly) is imbedded ina horizontally layered background medium, and the inverse of the finite-differenceoperator for the latter (computed with the help of the Fourier method) is used as thepreconditioner. The solution of the shifted linear systems yields the main contribution(more than 90%) to the total cost of the rational Arnoldi in our experiments. Theoriginal solver in [52] was intended for the traditional frequency domain problems,i.e., for the solution of complex symmetric linear systems that was done with the helpof the preconditioned QMR solver. Because of real shifts, in our approach we arriveat real symmetric systems. This has allowed us to use the CG instead of the QMR.It yields four times fold speed up for the same number of internal iterations due to

3772 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

Fig. 6.1. A model of layered conductive medium with a resistive inclusion.

real arithmetic and in addition often requires significantly fewer iterations due to nearbreakdowns of the QMR, so the observed total speed up (only due to the real shifts)varies from four to ten times. According to our observations the RKSR convergencecurve plateaus at the level of the error of the linear solver. So for consistency, theentire rational Arnoldi algorithm (including the solution of the shifted linear systems)is performed with double precision.

We considered the medium shown in Figure 6.1 and compared our method againsta variant of the restricted denominator (RD) method proposed by van den Eshofand Hochbruck [16]. The authors of [16] use a rational Krylov subspace of the formspan{b, Ab, . . . , An−1b}, b = (A+γI)−nϕ (i.e., an ordinary Krylov subspace generatedwith a shifted and inverted matrix). It is targeted to accurate approximation of thesolution for a particular value of t (we took the value t = 1) and does not requireknowledge of A’s spectrum. In contrast, our approach is targeted for the semi-infinitetime interval, however weakly depends on A’s condition number. A choice of a shift γwas proposed by Andersson [2] who proved that γ =

√2n is an asymptotically optimal

value (see a relative result in [41]). Adaptive determination of n according to theobtained accuracy would be computationally ineffective since for a new n one needsto reconstruct Un right from the beginning (which is a disadvantage compared to thesequence of nested subspaces in our approach). Thus, to obtain γj we set a priorin = 60 for our numerical examples. In Figures 6.2, 6.3, and 6.4 we plotted the relativeerror of solution measured at the receiver at t = 1, t = 10, and t = 100, respectively.As one can expect, for t = 1 the RD converges much faster than our approach (EDScurve). But according to Figure 6.3 for t = 10 the RD approach is slower. Moreover,for t = 100 the RD stagnates the first 150 iterations. At the same time our approachshows almost the same (close to that given by Theorem 4.4 ) convergence rate forall the three values of t: t = 1, t = 10, and t = 100 (until reaching the ratio of thecomputer roundoff level to the value of the solution in the receiver that varies from

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3773

0 50 100 15010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

EDSRDTheoretical slope

Fig. 6.2. t = 1. The RD converges significantly faster than our algorithm (EDS).

0 50 100 15010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

EDSRDTheoretical slope

Fig. 6.3. t = 10. The RD becomes slower than the EDS.

10−12 to 10−10). In Figure 6.5 for n = 30 and n = 70 we plotted ‖un − u‖ for bothalgorithms as a function of time. The behavior of the error of our approach is moreuniform on the entire time interval, while the RD approach is more accurate just in avicinity of t = 1 and significantly loses accuracy away from this point. As expected,this comparison clearly shows the superiority of our approach for large intervals.

3774 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

0 50 100 15010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

EDSRDTheoretical slope

Fig. 6.4. t = 100. The RD stagnates, while the EDS shows almost the same convergence rateas for t = 1 and t = 10.

0.001 0.01 0.1 1 10 10010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

EDSRDn=30

n=70

Fig. 6.5. The EDS shows much more uniform error distribution than the RD.

7. Concluding remarks. We solve a semidiscrete parabolic equation with aself-adjoint stiffness matrix on the entire semi-infinite time interval by projecting ontoan RKS. This problem is equivalent to two-parameter approximation of exp(−λt) byrational functions of λ with prescribed (time-independent) poles for λ on the spec-trum of A and all t ≥ 0. The numerators of such approximants are time-dependent

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3775

polynomials of λ (nearly optimal for every t), and the denominator is defined byshifts of the RKS to be optimized. We consider an equivalent problem in the fre-quency domain and optimize its upper bound. The optimization is reduced to theclassical Zolotaryov problem. It gives us real shifts with an optimal geometric con-vergence rate. The effectiveness of the developed approach is demonstrated on anexample of the 3D diffusion problem for Maxwell’s equation arising in geophysicalexploration.

When we were preparing this paper, we learned about simultaneous independentresearch on functions of nonsymmetric matrices by Beckermann and Reichel [7]. Thisimportant insightful paper considers matrix function approximations by both thepolynomial and rational Krylov subspaces. Its RKS part is mainly tailored to theMarkov–Stieltjes functions (see Remark 9); however, there are some intersectionswith our results, in particular, a counterpart of Proposition 3.2.

Remark 6. In some cases (e.g., when A is obtained from disretization of a 2Delliptic operator) direct methods can be efficiently used for the solution of the linearsystems with shifted A. Then it can be beneficial to store the LU factorization ofA + sjI and compute the action of (A + sjI)−1 several times, i.e., to allow multiplepoles in the RKS. As it was already mentioned in Remark 1, multiplicity can betreated as a limiting case of distinct poles, so most of the results obtained here canbe extended for multiple poles.

One way to introduce multiplicity is to use a cyclical subspace of a fixed period nand extendible multiplicity m, i.e., Unm = span{(A+ s1I)−1ϕ, . . . , (A+ snI)−1, (A+s1I)−2ϕ, . . . , (A + snI)−2, . . . , (A + s1I)−mϕ, . . . , (A + snI)−mϕ} with sj from The-orem 4.2. The estimate of the (Cauchy–Hadamard) geometric convergence rate (viathe skeleton approximation) for such a cyclical subspace yields n

√σn. According to

Theorem 4.2, n√σn ≤ ρ n

√2; i.e., for large enough n the cyclical approach yields the

geometric convergence rate close to the optimal one. However, n needed for goodconvergence grows with the increase of A’s condition number as do the storage re-quirements for all factorized A+ sjI, j = 1, . . . , n.

Another approach is to use a sequence of Krylov subspaces of a fixed multiplicitym (chosen to absorb the cost of factorization) and extendible length n with shiftssj from Theorem 4.4, i.e., Unm = span{(A + s1I)−1ϕ, . . . , (A + s1I)−m, . . . , (A +snI)−1ϕ, . . . , (A+ snI)−mϕ}. The RKSR error on such a subspace is bounded by thequantity maxλ∈[λmin,λmax] |rn(λ)|m, and thus, it gives the optimal Cauchy–Hadamardrate ρ. A possible drawback is longer “plateaus” than the ones seen in Figure 4.1.

We prefer the latter approach because it automatically yields the optimal Cauchy–Hadamard convergence rate and does not require storage of multiple factorizedA+sjI.Optimization of shifts for such subspaces is also considered in [7].

Remark 7. Formally, the rational approximants used for generation of the RKSshifts can be directly used for the approximation of f(A)ϕ with the same error boundas the RKSR according to Proposition 3.2 as was suggested in [7]. In exact arithmeticone would be able to directly compute fskel(A, s)ϕ and analytically transform it to thetime domain. However, using fskel in the form of the partial fraction decompositionas in (4.1) is equivalent to the discrete inverse Laplace transform, and as such it isunstable in the case of real shifts. The RKSR successfully fights this instability thanksto orthogonalization in the rational Arnoldi.

Remark 8. The results of [6] and Theorem 4.3 indicate that Theorem 4.2 givesan optimal or asymptotically optimal choice of sj for the RKSR approximation of(A + sI)−1ϕ on iR without information about distribution of the spectral measure

3776 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

of A within [λmin, λmax]. If such information is available, the optimal choice can beobtained with the help of the H2-optimal conditions for reduced order models wellknown in the control theory from the 1960s [34]. The H2-optimal sj minimize theL2[iR] error of the transfer function obtained with the help of RKSR. A necessarycondition of the H2-optimality is

(7.1) sj = θj , j = 1, . . . , n,

where θj are the eigenvalues of V . This condition shows that the H2-optimal shiftsmust also be real positive. The H2-optimal sj can be found iteratively by computinga sequence of Krylov subspaces [26]; i.e., to compute an H2-optimal n-dimensionalRKS one needs to compute several Krylov subspaces of the same dimension. Thatmay not be efficient, however, if the main RKSR cost is the solution of the linearsystems with matrices A+ sjI.

It was shown in [31] that if

λj = θj

for j = 1, . . . , n, then

fskel(A, s)ϕ = G (V + sI)−1G∗ϕ.

In this case, conditions (7.1) and (4.6) coincide. Let us consider the set of all operatorsA with the spectrum on [λmin, λmax] and the corresponding set of the RKSR’s solutionssatisfying H2-optimality condition (7.1). It is quite realistic to conjecture that thereexists an RKSR solution in the latter set that has the same poles as in Zolotaryov’sTheorem 4.2. The rational approximant generated by this RKS would be optimal inthe sense of section 5 for the set of all H2-optimal approximants for operators with thespectrum on [λmin, λmax]. This interesting connection deserves further investigation.

Remark 9. Semidiscretization of elliptic equations invariant along one (z) coor-dinate and supplied with a Neumann boundary condition at z = 0 leads to thecomputation of

(7.2) A−1/2 exp(−z√A)ϕ

for z ∈ [0,+∞] [15]. Similar to the parabolic problem, the Plancherel identity equatesthe L2-error of A−1/2 exp(−z√A)ϕ on [0,+∞] to the one of the resolvent on the entireimaginary axis, and again one can use optimal sj from Theorems 4.2 and 4.4.

Possibly, it is not just a coincidence that sj given by Theorem 4.2 are also the in-terpolation points of the best [n/2, n/2] rational approximants to z−1/2 on [λmin, λmax]if n is even [40]. So for this case

∥∥GV −1/2G∗ϕ−A−1/2ϕ∥∥ is bounded by the error of

the best rational approximant of order n/2.Functions given by (7.2) are related to Markov functions considered in [7], where

the authors use the Blaschke product for shift optimization. A possible connectionbetween the Blaschke product and the skeleton approximation is yet to be investi-gated.

Acknowledgments. The authors are thankful to A. Abubakar, A. I. Aptekarev,B. Beckermann, A. B. Bogatyryov, M. Botchev, V. S. Buyarov, T. Habashy, V. I.Lebedev, V. Simoncini, V. N. Sorokin, S. P. Suetin, L. Reichel, and E. E. Tyrtyshnikovfor bibliographical support and/or useful discussions. The authors are also gratefulto the two referees for valuable comments.

SOLUTION OF EVOLUTIONARY PROBLEMS USING RKS 3777

Appendix A. Proof of Theorem 4.3.Actually, as the maximum modulus theorem prompts, (4.5) is the third Zolotaryov

problem for the condenser (E,D) (see [20, section 2]) whose compact (in C) platesare E = [λmin, λmax] and D = {z ∈ C | z ≤ 0} ∪ {∞} (the left half-plane).

Let us prove first that the Riemann modulus ρ−1 (see [37, p. 334]) of the condenser(E,D) is determined by the formula

(A.1) ρ = exp[−π

4· K

′(L2)K(L2)

],

where the number

L =

⎡⎣1 + δ

1 − δ+

√(1 + δ

1 − δ

)2

− 1

⎤⎦−1

obeys the double inequality 0 < L < 1. It can be elementarily checked that the linearfractional transformation

z �→ λmax

(a

z + 1+ b

), a =

11

1−L − 12

, b = −a2,

conformally and bijectively translates the open unit disk, supplied with the slit[−L,L], onto the domain C\(E ∪D). The Riemann modulus of the former domainis given by formulas [37, (47), (48), p. 294], which lead us to (A.1).

It then follows from theorem 1 in [20] that σn ≥ ρn.Finally, one can easily check that L2 = μ, whence ρ = ρ in view of (4.8) and

(A.1). This confirms (4.11) due to (4.10).

Appendix B. Proof of Theorem 4.4.First, using the characteristic functions of intervals as trial functions, one can

deduce from Weyl’s theorem [9, Ch. VIII, section 4, Theorem 3] that

(B.1)1n

n∑j=1

ε(y − yj) → α as n→ ∞

in the weak sense, where ε is a unit measure concentrated at 0.Second, put n = 2k and take into consideration the (k − 1)/k rational function

Rk(y) =p(y)q(y)

, deg p = k − 1, deg q = k,

interpolating the function

(B.2) y−1/2, δ2 ≤ y ≤ 1,

at the nodes y1, y2, . . . , yn. Since function (B.2) is Markov with a generating measureincreasing on ] − ∞, 0] and since the measure (4.12) is equilibrium, we can derivefrom [23, Theorem 4] that Rk converges to y−1/2 on [δ2, 1] at the optimal in theCauchy–Hadamard sense rate (the same at which Zolotaryov’s approximants do).

Third, put x =√y,

h(x) =xp(x2) + q(x2)

2, deg h = 2k,

3778 V. DRUSKIN, L. KNIZHNERMAN, AND M. ZASLAVSKY

and

E(y) = 1 −√yp(y)q(y)

,

so E(yj) = 0 for j = 1, . . . , n. We derive

h(−x)h(x)

=E(y)

2 − E(y)= rn(x),

because the rational functions h(−x)/h(x) and rn(x) are of the same type (the polesare minus the roots) and have the same roots. Thus,

(B.3) limn→∞, n even

n

√max

λ∈[λmin,λmax]|rn(λ)| ≤ ρ.

Fourth, since |x−sn+1| and 1/|x+sn+1| are bounded uniformly in n, (B.3) implies

(B.4) limn→∞

n

√max

λ∈[λmin,λmax]|rn(λ)| ≤ ρ.

Finally, the combination of (B.4) and (4.11) gives (4.14).

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