Download SPECTRAL APPROXIMATION OF TIME WINDOWS IN THE

SPECTRAL APPROXIMATION OF TIME WINDOWS IN THE
SOLUTION OF DISSIPATIVE LINEAR DIFFERENTIAL EQUATIONS
K. BURRAGE∗, Z. JACKIEWICZ†, AND B. D. WELFERT‡
Abstract. We establish a relation between the length T of the integration window of a linear
differential equation x0 + Ax = b and a spectral parameter s∗ . This parameter is determined by
comparing the exact solution x(T ) at the end of the integration window to the solution of a linear
system obtained from the Laplace transform of the differential equation by freezing the system matrix.
We propose a method to integrate the relation s∗ = s∗ (T ) into the determination of the interval of
rapid convergence of waveform relaxation iterations. The method is illustrated with a few numerical
examples.
Key words. Linear differential systems, time window, spectral approximation, waveform relaxation.
AMS(MOS) subject classi£cations. 65L05.
1. Introduction. Consider the differential system
(1.1)
x0 (t) + Ax(t) = b(t),
t ∈ [0, T ],
x(0) = 0,
with solution
(1.2)
x(t) =
Z
t
e−(t−s)A b(s)ds.
0
The matrix A ∈ Cn×n in (1.1) is constant and assumed to be positive de£nite,
with eigenvalues λi such that 0 < <(λn ) ≤ . . . ≤ <(λ1 ). The integral form of
(1.1) is
(1.3)
x(t) + IAx(t) = Ib(t),
t ∈ [0, T ],
where I is the (linear) integral operator de£ned by Iu(t) =
The equivalent of (1.1) in the spectral domain is
(1.4)
Rt
0
u(s)ds.
(sI + A)X(s) = B(s),
where X(s) and B(s) denote the Laplace transforms L{x(t)} and L{b(t)} of
x(t) ∈ Cn and b(t) ∈ Cn , respectively. Fixing s = s∗ in the matrix sI + A then
yields (s∗ I + A)X(s) ' B(s), i.e., (s∗ I + A)x(t) ' b(t) back in the temporal
domain. We thus de£ne y(t) ∈ R n by
(1.5)
(s∗ I + A)y(t) = b(t),
∗ Department of Mathematics, University of Queensland, Brisbane 4072, Australia (e-mail:
[email protected]).
† Department of Mathematics & Statistics, Arizona State University, Tempe, Arizona 85287-1804
(e-mail: [email protected]). The work of this author was partially supported by the
National Science Foundation under grant NSF DMS–9971164.
‡ Department of Mathematics & Statistics, Arizona State University, Tempe, Arizona 85287-1804
(e-mail: [email protected]).
1
2
K. BURRAGE ET AL.
so that
(1.6)
(s∗ I + IA)y(t) = Ib(t).
Our goal is to determine s∗ such that the solution x(t) of (1.1) approximates,
in some sense, the solution y(t) of (1.5). In the context of waveform relaxation
applied to (1.1) we thereby hope that if M ' A is a preconditioning matrix for
the system Ax(t) = b(t) then s∗ I + M is a suitable preconditioning matrix for
(1.5) and thus for (1.1).
A comparison of (1.3) and (1.6) shows that
(I + IA)(x(t) − y(t)) + (I − s∗ I)y(t) = 0
and, consequently, that y(t) may be a reasonable approximation of x(t) if
s∗ I ' I. On small time windows x(t) can be approximated by a constant x
so that
s∗ T x ' s∗ Ix(T ) ' Ix(T ) = x(T ) ' x
yields
(1.7)
s∗ '
1
.
T
The approximation (1.7) is also consistent with large time windows estimates, at
least when b(t) = b is constant. Indeed, the steady state solution of (1.1) is then
x(∞) = A−1 b which is equal to the solution y(∞) = y of (1.5) obtained when
setting s∗ = 0.
The estimate (1.7) was £rst suggested by Leimkuhler [11] for estimating
windows of convergence in waveform relaxation methods applied to (1.1). He
based his analysis on the size of spectral radius of the matrix sI+A for <(s) > s ∗ .
He noted that (1.7) is a simpli£cation, a fact later con£rmed, especially for larger
time windows, by extensive numerical experiments conducted by Burrage et al.
[1], [2]. Jackiewicz et al. [10] proposed instead an estimate of the form
(1.8)
s∗ =
C
T
for some constant C. They related C to the ²-contour of the pseudospectrum
[14] of A, namely C = − ln ², but determine the appropriate value of ² only
numerically by comparing the pseudospectra of a discrete version of the integral
operator de£ned by (1.2) and of the Laplace transform (sI + A) −1 of its kernel
e−tA .
The relation (1.8) between the size of the time window and the spectral parameter s∗ lays at the heart of a recent strategy developed by Burrage et al. [3]
for accelerating the convergence of waveform relaxation iterations. It is therefore
important to make this relation more precise, and in particular to £nd out whether
it can be extended to larger time windows.
SPECTRAL APPROXIMATION OF TIME WINDOWS
3
In the following T > 0 is assumed to be £xed. Our strategy for determining
s∗ = s∗ (T ) is based on the minimization of the norm kx(T ) − y(T )k2 of the difference between the solution x(T ) of (1.1) at the end of the interval of integration
and the solution y(T ) of (1.5) (which depends on s∗ ); i.e., we seek s∗ such that
kx(T ) − y(T )k2 → min .
(1.9)
We £rst start in Section 2 by considering speci£c right-hand sides b(t) and provide
a detailed analysis under the assumption that A is hermitian.
Section 3 deals with general right-hand sides. We show that the solution
of the minimization problem (1.9) satis£es a nonlinear equation which is wellposed and guaranteed to have a solution for small enough time windows and we
suggest a method to integrate the computation of s∗ (t) and the resulting window
of fast convergence of waveform relaxation iterations for each 0 < t ≤ T into
the ODE solution process. We illustrate the results numerically in Section 4 using
an example arising from a discretization of a one-dimensional boundary value
problem for the convection-diffusion equation via the method of lines.
The relationship between s∗ and T derived in this paper puts the determination of the size of the window of rapid convergence of waveform relaxation
iterations on a solid theoretical and practical ground and it is in our opinion the
most serious attempt up to date to address this important problem of great practical importance.
2. Minimization with particular right-hand side. In this section we consider speci£c right-hand side functions of the form b(t) = f (t)b with b ∈ C n ,
b 6= 0, and f (t) a scalar function. We focus on two choices for f (t): monomials
tk with k ≥ 0 integer and (combinations of) exponentials eat .
2.1. Monomial right-hand side. The choice b(t) = tk b with k ≥ 0 integer
1 (k)
and b ∈ Cn , b 6= 0, is driven by the observation that b(t) ' tk k!
b (0) for some
k ≥ 0 is a reasonable approximation of b(t) on small windows, provided b(t) is
analytic around t = 0.
We shall denote by Rk,0 (z) and Rk,1 (z) the (k, 0)− and (k, 1)−Padé approximations of ez at z = 0, respectively [8, p. 48]. It is easy to verify (e.g. by
induction) that the solution (1.2) of (1.1) at t = T can then be expressed as
(2.1)
x(T ) =
Z
T
e−(T −s)A sk b ds =
0
T
ϕk (−T A) b(T )
k+1
with ϕk de£ned by
(2.2)
ϕk (z) = (k + 1)! z −(k+1) (ez − Rk,0 (z))
for z 6= 0 and ϕk (0) = 1 by continuity. The functions ϕk (z), k = 0, . . . , 4, are
shown in Fig. 1 and shall play an important role in the remainder of this section.
Our £rst theorem shows that if s ∗ = k+1
T the quantity y(T ) is exactly equal
to the approximation of x(T ) obtained by replacing e−T A in the expression of
4
K. BURRAGE ET AL.
1
0.9
0.8
0.7
0.6
ϕ0 to ϕ4
0.5
0.4
0.3
0.2
0.1
0
−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
z
F IG . 1. Functions ϕk (z), 0 ≤ k ≤ 4, for −20 ≤ z ≤ 0.
ϕk (−T A) in (2.1) by a (k, 1)-Padé approximation. This can for example be
the result of solving (1.1) using an appropriate Runge-Kutta or linear multistep
method.
k+1
. Then
T HEOREM 2.1. Assume that b(t) = tk b, b 6= 0, and let s∗ =
T
y(T ) =
(2.3)
T
ψk (−T A) b(T )
k+1
with
(2.4)
ψk (z) = (k + 1)! z −(k+1) (Rk,1 (z) − Rk,0 (z)) .
Proof. From (2.4), Lemma A.1 (see Appendix), and (1.5) the right-hand side
of (2.3) reduces to
µ
¶−1
k+1
T
TA
−(k+1) (−T A)
RHS =
I+
b(T )
(k + 1)! (−T A)
k+1
(k + 1)!
k+1
¶−1
µ
k+1
I + A b(T ) = y(T )
=
T
provided s∗ =
k+1
T .
5
SPECTRAL APPROXIMATION OF TIME WINDOWS
is numerically acceptable
Theorem 2.1 shows that the choice s∗ = k+1
T
when b(t) = tk b. The following result then compares y(T ) to the exact solution
(2.1) of (1.1) rather than an approximation. The quantity µ2 (A) represents the
logarithmic norm of A with respect to the norm k · k2 [5, pp 18-19]. The positive
H
de£niteness of A and of its hermitian part A H = A+A
implies in particular that
2
(2.5)
−<(λn ) ≤ µ2 (−A) = max λ(−AH ) = − min λ(AH ) < 0
(see [5, p. 19]). Note that the leftmost inequality in (2.5) becomes an equality
when A is normal.
k+1
T HEOREM 2.2. Assume that b(t) = tk b, b 6= 0, and let s∗ =
. Then
T
(2.6)
kAk2
kx(T ) − y(T )k2
≤
|φk (T µ2 (−A))|
ky(T )k2
|µ2 (−A)|
z
ϕ0k (z) for z < 0.
with φk (z) = k+1
Proof. From (2.1), (1.5), and using Lemma B.1(b) (see Appendix) we obtain
x(T ) − y(T )
=
=
T
ϕk (−T A) b(T ) − y(T )
k+1
¶
µ
TA
ϕk (−T A) y(T ) − y(T )
I+
k+1
TA 0
ϕ (−T A) y(T ).
k+1 k
R
¿From Lemma B.1(c) we have ϕ0k (z) = (k + 1)! Ω tk etk z dtk . . . dt0 > 0, where
Ω is the region 0 ≤ tk ≤ . . . ≤ t0 ≤ 1 of Rk+1 . Therefore
°
°Z
°
°
−tk T A
0
°
dtk . . . dt0 °
kϕk (−T A)k2 = (k + 1)! ° tk e
°
=
Ω
≤
(k + 1)!
Z
Z
2
tk ke−tk T A k2 dtk . . . dt0
Ω
tk etk T µ2 (−A) dtk . . . dt0
≤
(k + 1)!
=
ϕ0k (T µ2 (−A)).
Ω
The last inequality follows from the fact that the logarithmic norm µ2 (A) provides
the optimal exponential bound exp(µ2 (A)t) for k exp(At)k2 , see [5, p. 18]). The
bound (2.6) then follows from
kx(T ) − y(T )k2 ≤
which completes the proof.
T kAk2 0
ϕ (T µ2 (−A)) ky(T )k2
k+1 k
6
K. BURRAGE ET AL.
The function φk (z) introduced in Theorem 2.2 is negative for z < 0. We can
also verify using Lemma B.1(a) and (b) that
µµ
¶
¶
z
z
k+1
k+1
0
ϕ (z) =
φk (z) =
1−
ϕk (z) +
k+1 k
k+1
z
z
z
ϕk (z) − ϕk (z) + 1 = ϕk−1 (z) − ϕk (z)
=
k+1
for k ≥ 0. Since
ϕk (z) = 1 +
z
+ O(z 2 )
k+2
z→0
as
and
k+1
+O
ϕk (z) = −
z
µ
1
z2
¶
as
z → −∞
it follows that
φk (z) '
z
(k + 1)(k + 2)
as
z→0
and
φk (z) '
1
z
as
z → −∞.
Hence, the solution y(T ) of (1.5) is a £rst order approximation of x(T ) on short
and long time windows, i.e.,
(
O(T )
as T → 0,
kx(T ) − y(T )k2
=
−1
ky(T )k2
O(T ) as T → ∞.
The hermitian case. A £ner analysis can be carried out when A is hermitian
and positive de£nite. We denote by A = U ΛU H the Schur decomposition of A
with Λ = diag(λi )1≤i≤n , 0 < λn ≤ . . . ≤ λ1 , and U a unitary matrix. ¿From
(2.1) and (1.5) we obtain
°
°2
° T
°
∗
−1
2
°
kx(T ) − y(T )k2 = °
ϕk (−T A) b(T ) − (s I + A) b(T )°
°
k+1
2
°
°2
° T
°
∗
−1 °
= °
° k + 1 ϕk (−T Λ) c − (s I + Λ) c°
2
¯2
n ¯
X
¯
¯ T
1
2
¯
¯
=
¯ k + 1 ϕk (−T λi ) − s∗ + λi ¯ |ci |
i=1
with c = U H b(T ).
7
SPECTRAL APPROXIMATION OF TIME WINDOWS
0
φ
φ5
4
−0.05
−0.1
φ1
φ
−0.15
2
−0.2
−0.25
φ0
−0.3
−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
z
F IG . 2. Functions φk (z) from Theorem 2.1, 0 ≤ k ≤ 4, for z ≤ 0.
T HEOREM 2.3. Assume that A is hermitian, positive de£nite, and that T >
0. For each k ≥ 0 there exists a value µk with 0 < λn ≤ µk ≤ λ1 and such that
kx(T ) − y(T )k2 corresponding to b(t) = tk b, b 6= 0, considered as a function of
s ≥ 0 is minimal for
s∗k =
(2.7)
ϕk−1 (−T µk ) k + 1
,
ϕk (−T µk )
T
with the convention that ϕ−1 (z) ≡ ez . Moreover, the following interlacing property holds:
(2.8)
0<
λn
T
λ
e n−
1
≤ s∗0 ≤
1
2
k
k+1
≤ s∗1 ≤
≤ ... ≤
≤ s∗k ≤
.
T
T
T
T
Proof. For £xed k ≥ 0, T > 0 and µ > 0 de£ne the functions f (µ, s) =
Pn
2
1
− s+µ
and F (s) = i=1 |f (λi , s)| |ci |2 . The function F is de£ned and continuously differentiable for all s ≥ 0. From Lemma B.1(a) we have
T
k+1 ϕk (−T µ)
f (µ, 0) = −
−T µϕk (−T µ) + k + 1
ϕk−1 (−T µ)
=−
<0
(k + 1)µ
µ
and
f (µ, ∞) =
T
ϕk (−T µ) > 0.
k+1
8
K. BURRAGE ET AL.
Pn f (λi ,s)
2
Consequently, the function F 0 (s) = 2 i=1 (s+λ
2 |ci | admits at least one zero
i)
s = s∗k > 0. Using Lemma B.1(a) again, the relation F 0 (s∗k ) = 0 can be written
as
µ
¶
n
n
X
ϕk−1 (−T λi ) k + 1
T (s∗k + λi )ϕk (−T λi ) − k − 1 2 X
∗
αi s k −
|ci | =
(k + 1)(s∗k + λi )3
ϕk (−T λi )
T
i=1
i=1
=0
T ϕk (−T λi )
2
3 |ci |
(k+1)(s∗
k +λi )
Pn
6= 0 and i=1 αi
with αi =
≥ 0. Note that αi > 0 if ci 6= 0 so that b 6= 0
(z)
is
implies c
> 0. By Lemma B.1(d) the function ϕϕk−1
k (z)
non-decreasing for z < 0. Therefore s∗k satis£es
!
Ã
n
X
ϕk−1 (−T λ1 ) k + 1
ϕk−1 (−T λi ) k + 1
αi
∗
Pn
≤ sk =
ϕk (−T λ1 )
T
ϕk (−T λi )
T
j=1 αj
i=1
(2.9)
ϕk−1 (−T λn ) k + 1
≤
ϕk (−T λn )
T
so that (2.7) holds for some λn ≤ µk ≤ λ1 . The fact that F (s) reaches its
minimum at s∗k follows from
F 0 (0) = 2
n
X
T λi ϕk (−T λi ) − k − 1
(k + 1)λ3i
i=1
|ci |2 = −2
n
X
ϕk−1 (−T λi )
i=1
(k + 1)λ3i
|ci |2 < 0
and F 0 (∞) = 0+ . Finally, the interlacing property (2.8) is a consequence of
k
ϕk−1 (−∞)
=
ϕk (−∞)
k+1
and
ϕk−1 (0)
= 1.
ϕk (0)
For k = 0 we obtain
s∗0 =
ϕ−1 (−T µ0 ) 1
µ0
ϕ−1 (−T λn ) 1
λn
=
.
≥
= T λn
−1
−1
ϕ0 (−T µ0 ) T
ϕ0 (−T λn ) T
e
eT µ 0
This completes the proof.
If ci = cj then it is easy to verify that the weights αi in the proof of Theorem
2.1 satisfy 0 ≤ α1 ≤ . . . ≤ αn so that, in such a case,
µk ' λ n .
For small time windows T the value s∗k given by (2.7) becomes
(2.10)
s∗k '
k+1
T
for k ≥ 0. On the other hand for large time windows T (e.g. T λn À 1) we obtain
(2.11)
s∗k '
k
T
9
SPECTRAL APPROXIMATION OF TIME WINDOWS
for k > 0 and
s∗0 ' λn e−T λn .
(2.12)
R EMARK 2.1. If cp+1 = . . . = cn = 0 for some 0 < p < n
(i.e., c = U H b(T ) is `2 -orthogonal to the £rst n − p eigenvectors of A) then
αp+1 = . . . = αn = 0 and s∗0 ' λp e−T λp instead for large T . Such a situation
may occur in particular when b is randomly chosen. Indeed, such a vector is typically “oscillatory” and likely to be orthogonal to the “smooth” eigenvectors of
the matrix A associated with the smallest eigenvalues.
Note however that round-off errors in the numerical determination of the
vector c may prevent cn from vanishing exactly and still may yield, for larger
values of T and in the case k = 0, αn À αn−1 if cn ≈ cn−1 .
R EMARK 2.2. Other strategies for £nding an optimal s∗ may be used. For
example
it is possible,
°R
° for a given window [0, T ], to minimize the “average error”
° T
°
° 0 (x(t) − y(t))dt°. If xk and yk denote the solutions of (1.1) and (1.5) when
b(t) = tk b, respectively, note however that
¶
Z T
Z T µZ t
−(t−s)A k
∗
−1 k
(xk (t) − yk (t))dt =
e
s b ds − (s I + A) t b dt
0
=
0
Z
T
0
µ
1
k+1
1
=
k+1
ÃZ
1
=
k+1
ÃZ
0
µ
¶
Z t
¯t
¯
e−(t−s)A sk+1 b ¯ − A
e−(t−s)A sk+1 b ds
0
0
¶
−(s∗ I + A)−1 tk b dt
!
T
(tk+1 b − Axk+1 (t)) dt − (s∗ I + A)−1 T k+1 b
0
T
0
x0k+1 (t) dt
− yk+1 (T )
!
=
1
(xk+1 (T ) − yk+1 (T )) ,
k+1
i.e., the optimization will lead to s∗ = s∗k+1 rather than s∗ = s∗k .
2.2. Exponential right-hand side. The choice of exponential right-hand
sides b(t) = eat b with a ∈ C is essentially guided by the fact that many physical
processes are driven by exponentially growing forcing terms.
We shall restrict ourselves to values of a such that the matrix A + aI remains
positive de£nite, i.e., in particular <(a + λ n ) > 0. From (1.2) we obtain
Z T
Z T
x(T ) =
e−(T −s)A+asI b ds =
e−(T −s)(A+aI) ds b(T )
0
(2.13)
0
= T ϕ0 (−T (A + aI)) b(T )
which is exactly the solution (2.1) of (1.1) with k = 0 and A replaced by A + aI.
The expression (2.13) is now to be compared to
y(T ) = (s∗ I + A)−1 b(T ) = ((s∗ − a)I + A + aI)
−1
b(T ).
10
K. BURRAGE ET AL.
¿From Section 2.1 the optimal choice of s∗ satis£es, in the hermitian case and
with a ∈ R (so that A + aI is also hermitian),
(2.14)
s∗ − a =
µ
ϕ−1 (−T µ) 1
= Tµ
ϕ0 (−T µ) T
e −1
for some a + λn ≤ µ ≤ a + λ1 . On short time windows eT µµ−1 ' T1 so that
s∗ ' a + T1 while s∗ → a exponentially fast as T increases, provided
0. ¢
¡ iωt a >−iωt
1
e −e
b
We now consider a right-hand side b(t) = sin(ωt)b = 2i
where b 6= 0 is a constant vector (e.g. a Fourier mode for general functions b(t)).
The solution of (1.1) at t = T is now
x(T )
¢
T ¡
ϕ0 (−T (A + iωI))eiωT − ϕ0 (−T (A − iωI))e−iωT b
=
2i
¢
1 ¡
(A + iωI)−1 (eiωT I − e−T A ) − (A − iωI)−1 (e−iωT I − e−T A ) b
=
2i
¡
¢
1
= (A2 + ω 2 I)−1 (A − iωI)(eiωT I − e−T A ) − (A + iωI)(e−iωT I − e−T A ) b
2i
¢
¡
= (A2 + ω 2 I)−1 sin(ωT )A − ω cos(ωT )I + ωe−T A b
while the solution y(T ) of (1.5) is
y(T ) = sin(ωT ) (s∗ I + A)
−1
b.
Note that the optimization process to determine the optimal value of s∗ breaks
down when ωT = π since y(T ) then becomes independent of s∗ . For values
T ¿ ωπ the approximation b(t) ' ωt holds and we expect the optimal choice of
s∗ to follow the recommendations from Section 2.1 with k = 1.
3. Minimization with general right-hand side. We now consider a general right-hand side b(t) and derive a nonlinear equation for the optimal value s ∗
obtained in the case k · k = k · k2 .
Let As∗ = s∗ I + A. We write
(3.1)
(3.2)
∗
x(T ) − y(T ) = A−1
s∗ (s x(T ) + Ax(T ) − b(T ))
∗
0
= A−1
s∗ (s x(T ) − x (T )) .
Since x(T ) does not depend on s∗ we have also
d
d
−2
(x(T ) − y(T )) = − ∗ A−1
∗ b(T ) = As∗ b(T ).
ds∗
ds s
Therefore
µ
¶
d
2
H d
kx(T
)
−
y(T
)k
=
2<
(x(T
)
−
y(T
))
(x(T
)
−
y(T
))
2
ds∗
ds∗
´
³
H
−2
(3.3)
= 2< (s∗ x(T ) − x0 (T )) A−H
s∗ As∗ b(T )
SPECTRAL APPROXIMATION OF TIME WINDOWS
11
vanishes for
(3.4)
¢
¡
−2
< x0 (T )H A−H
s∗ As∗ b(T )
¢.
s = ¡
−2
< x(T )H A−H
s∗ As∗ b(T )
∗
The expression (3.4) de£nes s ∗ as the solution of a nonlinear equation. The following result shows that for suf£ciently small windows this equation admits at
least one solution s∗ > 0.
T HEOREM 3.1. Assume that b(t) is continuous on an interval [0, T + ] for
some T + > 0. Then there exists an interval [0, T − ] with 0 < T − ≤ T + such that
the equation (3.4) is well-posed and admits a solution s∗ > 0 for all 0 < T ≤
T −.
Proof. From (1.1) we have x0 (0) = b(0). We £rst assume that b(0) 6= 0
and show that both numerator and denominator on the right-hand side of (3.4) are
positive for any s∗ > 0 provided T > 0 is small enough. By the continuity of
∗
x0 on [0, T + ] and the positive de£niteness of A s∗ and A−1
s∗ for any s > 0 there
+
exists 0 < T1 ≤ T such that
³¡
¢
¡
¢´
¢H −1 ¡ −1
−2
0
< x0 (t)H A−H
As∗ As∗ b(T ) > 0
(A−1
s∗ As∗ b(T ) = <
s∗ x (t)
¢
¡
−2
for all t such that 0 ≤ t ≤ T ≤ T1 . In particular < x0 (T )H A−H
s∗ As∗ b(T ) > 0
¢
¢
¡
R T ¡ 0 H −H −2
−2
and < x(T )H A−H
s∗ As∗ b(T ) = 0 < x (t) As∗ As∗ b(T ) dt > 0 for all
T ∈ (0, T1 ] and s∗ > 0.
We next show that the minimum of kx(T ) − y(T )k2 is solution of (3.4).
Similarly as (3.3) we obtain
¯
¯
¡
¢
d
−2
2¯
= −2< x0 (0)H A−H
kx(0)
−
y(0)k
2¯
0 A0 b(0)
∗
ds
s∗ =0
³¡
¢H
¡
¢´
= −2< A−1 b(0) A−1 A−1 b(0) < 0
because A and A−1 are positive de£nite. By continuity there exists T 2 > 0 such
d
that ∗ kx(T ) − y(T )k22 < 0 for all T ∈ [0, T2 ].
ds
Since x(0) = 0, x0 (0) 6= 0 and x0 is continuous on [0, T + ] there also exists
d
0 < T3 ≤ T + such that dT
kx(T )k22 > 0 for all T ∈ (0, T2 ]. Therefore
¡
¢
¡
¢ 1 d
kx(T )k22 + < x(T )H Ax(T ) > 0
< x(T )H b(T ) =
2 dT
for all T ∈ (0, T3 ]. Since As∗ ' s∗ I as s∗ → ∞ we obtain
¡
¢
¯
¯
< x(T )H b(T )
d
2¯
>0
kx(T ) − y(T )k2 ¯
' 2
2
ds∗
(s∗ )
s∗ →∞
for all T ∈ (0, T3 ]. This shows that for any T such that 0 < T ≤ T − =
min(T1 , T2 , T3 ) the quantity kx(T ) − y(T )k2 reaches its minimum at a critical
point 0 < s∗ < ∞ which satis£es (3.4).
12
K. BURRAGE ET AL.
The result can also be shown to hold in the case b(0) = 0 using a continuity
argument based on b(t) 6= 0 for any t > 0 suf£ciently small.
Observe that the equation (3.4) can be interpreted as a weak form of the
condition y(T ) − x(T ) = 0. Indeed, we obtain from (1.5), (3.1) and (3.4)
³
´
¡
¢
H −H −2
∗
0
(3.5) < (y(T ) − x(T ))H A−1
s∗ y(T ) = < (s x(T ) − x (T )) As∗ As∗ b(T )
= 0.
On long time windows the right-hand side of (3.4) may vanish or become negative, in particular when b(T ) itself vanishes as in the case of sinusoidal right-hand
side (see Section 2.2). In this case y(T ) vanishes as well and (3.5) can no longer
be used to determine s∗ .
3.1. Short time window estimate. For right-hand sides b(t) of the form
b(t) = f (t)b
where f (t) is a continuous scalar function and b ∈ Rn , b 6= 0, we obtain from
(1.2)
ÃZ
!
Z T
T
x(T ) =
(I + O(T − t)) f (t)b dt =
f (t)dt (1 + O(T )) b
0
0
and
x0 (T ) = f (T )b − Ax(T ) = f (T ) (1 + o(1)) b
for small time windows. Then (3.4) reduces to
(3.6)
f (T )
s∗ ' R T
f (t)dt
0
for small T independently of A (although the time interval on which (3.6) remains
a good approximation does depend in general on A). In particular for f (t) = t α
with α > 0 we obtain the estimate s∗ ' α+1
T , which agrees with the results of
aT
Section 2 for integer values of α. For f (t) = eat we also obtain s∗ ' eae
aT −1
which¡is identical
to (2.14) with µ = a. The choice f (t) = sin(ωt) yields s∗ '
¢
ω cot ωT
.
2
3.2. Numerical solution of (3.4). From a practical point of view s∗ can be
ef£ciently computed from (3.4) for increasing values of T once x(T ) and x 0 (T )
have been determined (for example using an adaptive solver), until the right-hand
side of (3.4) fails to remain positive. The nonlinear equation (3.4) is of the form
s∗ = F (s∗ ). In all our numerical tests a Picard iteration s∗p,m = F (s∗p,m−1 ) for
m > 0 starting with the solution of (3.4) obtained at the previous time step T (and
with a predicted s∗1,0 = h10 for the £rst step) was successfully used to solve (3.4)
at time T = h0 + . . . + hp−1 , see Algorithm 1.
Pq−1
Algorithm 1: Adaptive determination of s∗ = s∗ (T ) for T = p=0 hp .
SPECTRAL APPROXIMATION OF TIME WINDOWS
13
x0 = 0, h0 > 0 given, t0 = 0, s∗1,predicted = h10
for p = 1, 2, . . . , q
compute hp−1 and xp ' x(tp−1 + hp−1 ) using an (adaptive) ODE solver
tp = tp−1 + hp−1
x0p = b(tp ) − Axp
s∗p,0 = s∗p,predicted
for m = 1, 2, . . . , M
d = A−H
A−2
b(tp )
s∗
s∗
p,m−1
p,m−1
<( x 0 H d )
s∗p,m = < xpH d
( p )
if s∗p,m ≤ 0 stop
if |s∗p,m − s∗p,m−1 |
≤ T OL break
end(m)
s∗p = s∗p,M
s∗p+1,predicted = s∗p
end(p)
One or two Picard iteration(s) (m ≤ 2) are generally suf£cient to get satisfactory results because of the near independence of the right-hand side of (3.4)
on s∗ for small s∗ (As∗ ' A) and large s∗ (As∗ ' s∗ I so that the limit of the
right-hand side for s∗ → ∞ is independent of s∗ ). As a result, assuming that
one Picard iteration is only required, the cost of computing s∗ corresponds to the
solution of three linear systems related to d = A−H
A−2
b(tp ). This cost is quite
s∗
s∗
p,0
p,0
acceptable, especially if the matrices of these systems are sparse. This is the case
if the problems corresponding to partial differential equations are discretized in
space by £nite element or £nite difference methods, compare example (4.1) in
Section 4. On the other hand, the use of spectral or pseudospectral methods leads
to differential systems with dense matrices A for which the solution process is less
ef£cient. However, because of very fast, spectral convergence of these methods
the number of spectral coef£cients or collocation points to resolve the solution to
the required accuracy is not very large. As a consequence, the cost of computing s∗ is also acceptable in such cases since the dimension of the matrix As∗p,0 is
usually not too large, compare the example (4.3) at the end of Section 4.
R EMARK 3.1. The simpli£ed expression d = b(tq ) in Algorithm 1 was
also tested. It did not signi£cantly affect the numerical results while making the
process more ef£cient.
3.3. Effect on (3.4) of using a numerical integration scheme. Suppose the
problem (1.1) is solved by applying q steps of a k-step multistep method
(3.7)
k
X
j=0
αj xp−j = h
k
X
j=0
βj (bp−j − Axp−j )
14
K. BURRAGE ET AL.
where xj ' x(jh) and bj = b(jh) for 0 ≤ j ≤ q such that qh = T . The relation
(3.7) can be written as (α ⊗ I) X + (hβ ⊗ A) X = hβ ⊗ B or
¡
(3.8)
with

Sα
 αk


α=
(3.9)




Sβ
(3.10)  βk


β=



...
..
.
¢
(hβ)−1 α ⊗ I X + (I ⊗ A) X = B,

α0
..
.
..
.
..
..
.
.
αk
...
..
.
.
...
α0

β0
..
.
..
.
..
..
.
.
..

 µ

Sα
=

U
α


βk
..
.
...
β0

 µ

Sβ
=

U
β


0
Lα
¶
∈ R(q+1)×(q+1) ,
0
Lβ
¶
∈ R(q+1)×(q+1) ,
and

0
x1
..
.





X=
 xk−1
 xk

 .
 ..
xq
(3.11)






,






b0
b1
..
.





B=
 bk−1
 bk

 .
 ..
bq






.





The symbol ‘⊗’ denotes the Kronecker product. The matrices Sα and Sβ correspond to the starting procedure. In (3.8) we have assumed that Sβ is nonsingular,
β0 6= 0 and that the same time-step h is used in the starting procedure. MultiplyH
ing (3.8) by eH
q+1 ⊗ I with eq+1 = (0, . . . , 0, 1) of dimension q + 1, we obtain
¡
¢
−1
α ⊗ I X + Axq = bq = b(T ),
eH
q+1 (hβ)
¡
¢
−1
i.e, the quantity eH
α ⊗ I X is the discrete equivalent of x0 (T ). This
q+1 (hβ)
leads to
´
³
¡
¢H
−2
−1
< X H eH
α ⊗ I A−H
q+1 (hβ)
s∗ As∗ b(T )
¡
¢
(3.12)
.
s∗ =
−H −2
< xH
q As∗ As∗ b(T )
SPECTRAL APPROXIMATION OF TIME WINDOWS
15
A simpler estimate for the expression (3.12) can be derived if we assume that the
(£rst q steps of the) computed solution X is of the form




0
0
 zy 
 z 




X =  .  = u ⊗ y,
u =  . ,
(3.13)
 .. 
 .. 
zq y
zq
for some z > 1 and some nonzero vector y ∈ Rn .
T HEOREM 3.2. Assume that the IVP (1.1) is solved using a k-step method
(3.7) of order r and that the numerical solution can be approximated by a vector
of the form (3.13). Then the optimal s∗ given by (3.12) obtained after q steps
reduces to
µ
¶
1
1
ln(z)
r+1
∗
+ (z − 1)
+ O
(3.14)
.
s =
h
h
z q−k+1
Proof. From (3.9), (3.10), (3.13) and the properties of symbols σ of lower
triangular Toeplitz matrices (see Lemma C.1) we obtain
¡ H
¢
eq+1 (hβ)−1 α ⊗ I X
¶
µ
0
∗
u⊗y
= eH
−1
q+1
∗ (hLβ ) L
α



zk
¡
¢
.
1
−1
=  eH
Lα  ..  + O z k−1  ⊗ y
q−k+1 (hLβ )
h
zq
·
¸
¡
¢ 1 ¡
¢
=
z q σ(hLβ )−1 Lα z −1 + O z k−1 y
h
" Ã
#
!
¡ −1 ¢
σ Lα z
1 ³¡ −1 ¢q−k+1 ´
1 ¡ k−1 ¢
q
=
z
y
+ O z
+ O z
hσLβ (z −1 ) h
h
¸
·
¢
1 ¡
α(z)
+ O z k−1 y
=
zq
hβ(z) h
Pk
Pk
k−j
k−j
where α(z) =
and β(z) =
are the characteristic
j=0 αj z
j=0 βj z
polynomials of the multistep method (3.7). It then follows from xq = z q y and
(3.12) that
µ
¶
1
1
α(z)
+ O
(3.15)
s∗ =
.
hβ(z) h
z q−k+1
The estimate
(3.16)
¡
¢
α(z)
= ln(z) + O (z − 1)r+1
β(z)
16
K. BURRAGE ET AL.
follows from [9, p. 227] (see also [7, Theorem 2.4, p. 370]), β(z) = β(1) +
O(z − 1) = O(1) and ln(z) = O(z − 1) for z → 1. Combining (3.15) and (3.16)
£nally yields (3.14).
Note that the leading term in the estimate (3.14) is independent of the particular
numerical method used.
1
The substitution of z = ²− q with 0 < ² < 1 into (3.15) leads to
µ
¶
(ln ²)r+1
ln ²
1− k−1
∗
q
+O ²
(3.17)
,
+
s '−
T
qr T
an expression already found in [10] but without proper mention of how ² relates
to the IVP (1.1). From (3.13) we have
1
z q−1 kyk
1
kxq−1 k
= q
= = ²q
kxq k
z kyk
z
so that
(3.18)
²=
µ
kxq−1 k
kxq k
¶q
.
Thus ² is a measure of the growth in the solution from step q − 1 to step q. Note
that other expressions of ² can be obtained from (3.13), such as
µ
¶ q
kx1 k q−1
²=
(3.19)
.
kxq k
In practice, however, the numerical solution X cannot be written exactly in the
form (3.13) so that the expressions (3.18) and (3.19) are not equivalent. The formula (3.18) is preferable since it minimizes the in¤uence of the starting procedure
and tends to better re¤ect changes in the solution as q increases.
In the case b(t) = tk b for some b 6= 0 we have xq ' (qh)k+1 y for some y
(dependent on b and A) so that (3.18) yields
¶q µ
¶(k+1)q
µ
1
(q − 1)k+1 hk+1 kyk
(3.20)
=
1
−
' e−(k+1)
²'
q k+1 hk+1 kyk
q
for larger values of q and − ln ² ' k + 1. This is consistent with the results of
Section 2.1. Note however that the estimate (3.20) is obtained for larger values of
q, i.e., large T , while (3.18) is based on the approximation (3.13) of the solution
X, which does not generally hold on large windows.
4. Numerical example. The one-dimensional convection-diffusion equation
(4.1)

x + axu − xuu = 0, 0 < u < 1, t ≥ 0,


 t
x(0, u) = 0, 0 < u < 1



x(t, 0) = x(t, 1) = f (t)
17
SPECTRAL APPROXIMATION OF TIME WINDOWS
4
10
0.5
0
−0.5
2
10
s
−1
−1.5
−2
0
10
−2.5
−3
−3.5
−2
10
−4
−2
10
0
10
2
10
10
T
F IG . 3. Levels of log10
(n + 1)2
kx(T )−y(T )k2
ky(T )k2
for the second-order central differentiation matrix A =
tridiag(−1, 2, −1) and the right-hand side b(t) = (n + 1)2 [1, 0 . . . , 0, 1]H of dimension
n = 24 (λn ' π 2 ), function s =
inverted triangles).
ϕ−1 (−T λn ) 1
ϕ0 (−T λn ) T
(white line) and values obtained from (3.4) (white
4
10
2
1
0
2
10
s
−1
−2
−3
0
10
−4
−5
−6
−2
10
−4
10
−2
0
10
10
2
10
T
F IG . 4. Levels of log10
kx(T )−y(T )k2
ky(T )k2
for the second-order central differentiation matrix A =
(n + 1)2 tridiag(−1, 2, −1) and the right-hand side b(t) = t(n + 1)2 [1, 0 . . . , 0, 1]H of dimension
ϕ (−T λ )
n = 24 (λn ' π 2 ), function s = ϕ0 (−T λn ) T1 (white line) and values obtained from (3.4) (white
n
1
triangles).
18
K. BURRAGE ET AL.
1
1
= 25
.
(a ≥ 0) is discretized using the method of lines with spatial step h = n+1
A backward difference scheme is used for the £rst order space derivative (convection) while the usual central difference approximation is used for the second
order space derivative (diffusion). This leads to a differential system of the form
(1.1) with


 
d −1
−e


 .. 
.
.
.
.

 
.
.
1  e
 , b(t) = f (t)  .  ,
A= 2
(4.2)


2
. 
.
.
h 
h 
..
. . −1 
 .. 
e
d
1
d = 2 + ah and e = −1 − ah, so that A is positive de£nite.
We £rst determine kx(T
) − ¢y(T )k 2 when a = 0. The matrix A is then
¡
symmetric. Since cn = U H b(T ) n 6= 0 we use µk = λn ' π 2 . The solution
x(T ) and x0 (T ) is obtained using the M ATLAB adaptive stiff ode solver ode15s.
)−y(T )k2
Fig. 3 shows the levels of log10 kx(T
in the (T, s) plane over
ky(T )k2
−4
2
−2
4
the range [10 , 10 ] × [10 , 10 ] for f (t) = 1, together with the curve
(−λn T ) 1
obtained from (2.7) with k = 0 and µ0 = λn , and the discrete
s = ϕϕ−1
0 (−λn T ) T
estimates computed from (3.4) using Algorithm 1 with only one Picard iteration
(every fourth data point is shown). There is an excellent agreement between the
discrete points, the theoretical curve and the actual position of the minima. Fig. 4
shows similar results for the boundary condition f (t) = t (k = 1). As expected
from the analysis of Section 2.1 the optimal value s∗ now behaves as T1 for larger
values of T , compare (2.11). There is again excellent agreement between theoretical estimates and actual values. Fig. 5 displays the product C = s∗ T versus
T for both cases b(t) = b (bottom curve/points) and b(t) = tb (top curve/points).
Both sets of discrete estimates start with s∗ T = 1 according to the initialization
used in Algorithm 1. Note that for the case b(t) = tb this initialization does not
correspond to the correct limit s∗ T = 2 given by the theoretical estimate (2.10).
As the solution is being computed its behavior is increasingly taken into account
by the numerical procedure 1 and the discrete points quickly move to a position
closer to the theoretical curve de£ned by (2.7). Note also that in both cases the
computed estimates remain between the theoretical bounds given by (2.8).
Numerical results for b(t) = tk b with k > 1 are similar to the case k = 1
modulo a shift up by (k−1) unit(s) for the curves in Fig. 5. Although the estimates
in Section 2.1 were derived for integer values of k, numerical experiments for
non-integer values of k yield similar results.
We next consider the problem (4.1) with a = 100. The resulting matrix A in
(4.2) is non-symmetric (and non-normal because of boundary effects). Figs. 6 and
)−y(T )k2
and should be compared to Figs. 3 and 4,
7 show the value log10 kx(T
ky(T )k2
respectively. The curve of minima is essentially shifted left by log 10 100 = 2 units
1
in the case b(t) = b but no signi£cant change is observed in
as soon as T ' 100
the case b(t) = tb. Since A is not hermitian the formula (2.7) is no longer valid,
but it is tempting to generalize the results of Section 2.1 to the nonsymmetric
19
SPECTRAL APPROXIMATION OF TIME WINDOWS
sT
2
1
0 −4
10
−2
0
10
2
10
10
T
F IG . 5. Product C = s∗ T versus T for the examples of Fig. 3 (right-hand side b(t) = b) (5)
and Fig. 4 (right-hand side b(t) = tb) (4) together with the corresponding theoretical estimates (2.7)
with µk = λn ' π 2 for k = 0, 1.
4
10
0.5
0
−0.5
2
10
s
−1
−1.5
−2
0
10
−2.5
−3
−3.5
−2
10
−4
10
−2
0
10
10
2
10
T
F IG . 6. Levels of log10
(n + 1)2
kx(T )−y(T )k2
ky(T )k2
for the second-order central differentiation matrix A =
tridiag(−1, 6, −5) and the right-hand side b(t) = (n + 1)2 [5, 0 . . . , 0, 1]H of dimension
n = 24 and values obtained from (3.4) (white inverted triangles).
20
K. BURRAGE ET AL.
4
10
2
1
0
2
10
s
−1
−2
0
−3
10
−4
−5
−2
10
−4
10
−2
0
10
2
10
10
T
F IG . 7. Levels of log10
kx(T )−y(T )k2
ky(T )k2
for the second-order central differentiation matrix A =
(n + 1)2
tridiag(−1, 6, −5) and the right-hand side b(t) = t(n + 1)2 [5, 0 . . . , 0, 1]H of dimension
n = 24 and values obtained from (3.4) (white triangles).
sT
2
1
0 −4
10
−2
0
10
10
2
10
T
F IG . 8. Product C = s∗ T versus T for the nonsymmetric examples of Fig. 6 (right-hand side
b(t) = b) (5) and Fig. 7 (right-hand side b(t) = tb) (4).
example using instead the smallest singular value µ0 = σn ' 167 of A for the
21
SPECTRAL APPROXIMATION OF TIME WINDOWS
sT
2
1
0 −4
10
−2
0
10
10
2
10
T
F IG . 9. Product C = s∗ T = − ln ² versus T with ² determined from (3.18) for the symmetric
examples of Fig. 3 (right-hand side b(t) = b) (5) and Fig. 4 (right-hand side b(t) = tb) (4) together
with the corresponding theoretical estimates (2.7) (continuous lines). The numerical solution x q is
obtained using the explicit Euler method with time step h = 10−4 (hρ(A) ' 0.25 < 2; not all data
points are shown). Compare with Fig. 5.
case k = 0 in (2.7). This modi£cation yields however a curve which matches the
computed minima for lower values of T only (σn T < 1).
Fig. 8 displays the corresponding estimates of the product s∗ T obtained
from Algorithm 1 and can be compared to Fig. 5. Note that the bounds given by
(2.8) seem to remain valid in this nonsymmetric case.
Similar results where s∗ is now obtained from (3.17) with ² speci£ed by
(3.18) are plotted in Figs. 9 and 10) for the cases b(t) = b and b(t) = tb, respectively. The numerical solution xq is determined by using the explicit Euler
method with h = 10−4 (so that it remains stable). These results compare quite
well with Figs. 5 and 8. We have also checked that they were not signi£cantly
affected by the use of a different integrator such as the trapezoidal method applied
to (1.1) or the trapezoidal rule applied to (1.2).
The problem (4.1) can also be discretized using a pseudospectral approximation
x≈
n+1
X
αj `j (x)
j=1
where `j , j = 0, . . . , n + 1, are Lagrange polynomial basis functions based on
a set 0 = x0 < · · · < xj < · · · < xn+1 = 1 of nodes, compare [4], [6].
22
K. BURRAGE ET AL.
sT
2
1
0 −4
10
−2
0
10
10
2
10
T
F IG . 10. Same as Fig. 9 for the nonsymmetric case a = 100 with no theoretical estimate
(hρ(A) ' 0.65 < 2; not all data points are shown). Compare with Fig. 8.
sT
2
1
0 −4
10
−2
0
10
10
2
10
T
F IG . 11. Same as Fig. 10 for a Chebyshev pseudospectral discretization.
This also leads to a problem of the form (1.1) but A is now a full, non symmetric
matrix, even in the purely diffusive case a = 0. Speci£cally, using M ATLAB
SPECTRAL APPROXIMATION OF TIME WINDOWS
23
notations, A is the matrix obtained from the £rst–order differentiation matrix D =
(`0j (xi ))0≤i,j≤n+1 as
(4.3)
½
A = (aD − D 2 )(2 : n, 2 : n),
b(t) = −f (t)((aD − D 2 )(2 : n, 1) + (aD − D 2 )(2 : n, n + 1)).
Figure 11 shows how the product s∗ T computed with Algorithm 1 varies with
iπ
T for collocation polynomials `j ’s based on the (shifted) zeros xi = sin2 2(n+1)
of Chebyshev polynomials, and the functions f (t) = 1 (5) and f (t) = t (4).
Although A is not symmetric it is again interesting to note that the bounds obtained for the symmetric case remain valid here. Additional experiments with
other collocation points lead to similar results provided A is dissipative (i.e., has
eigenvalues in the left half complex plane). The general trend is that the higher
the clustering of nodes xi around the endpoints the more stable the discretization
is, and the lower the curve s∗ T = φ(T ) is.
5. Conclusions. We have presented a new approach to determine the relation between the time window [0, T ] of integration of a linear initial value problem
and a quantity s∗ obtained by “freezing” the spectral parameter s in the Laplace
transform of the kernel operator of the solution. This relation is of the form
s∗ T = C where C is optimized by comparing the solutions of the initial value
problem and the solution of the “frozen” spectral equation. This relation generalizes previous results by Leimkuhler [11] and Jackiewicz et. al. [10]. Theoretical
and numerical results show that in shorter time windows C can be approximated
by a constant which depends on the data of the problem itself.
We also proposed an algorithm which adaptively adjusts the optimal choice
of s∗ as the time window is increased. Applications to the preconditioning of
differential linear systems and to the determination of optimal time windows in
waveform relaxation methods are currently under investigation. Future work will
also address the extension of the theory developed in this paper to the case the
matrix A appearing in (1.1) is time dependent and to nonlinear problems. In particular, the analysis of Section 3 can be extended to time dependent case, although
possibly with a deterioration of the convergence rate of the Picard iteration in Algorithm 1 and a reduction of the time window on which convergence takes place.
On the other hand, the more speci£c results of Section 2 seem more dif£cult to
generalize unless additional assumptions such as commuting properties between
A(t) and its antiderivatives are made.
Acknowledgements. The authors wish to express their gratidude to the anonymous referees for their useful comments.
24
K. BURRAGE ET AL.
APPENDIX
A. Relationship between Padé approximations Rk,0 (z) and Rk,1 (z) to
ez at z = 0.
L EMMA A.1. For k ≥ 0 we have
Rk,1 (z) = Rk,0 (z) +
(A.1)
z k+1
(k + 1)!
µ
z
k+1
1−
¶−1
.
Proof. The explicit form of Rk,1 (z) is not needed to prove the lemma but
can be found, for example, in [8, p. 48]. The numerator of
k+1
1
z
=
Rk,0 (z) +
z
(k + 1)! 1 − k+1
(1 −
z
k+1 )
³P
1
k
zj
j=0 j!
z
− k+1
´
+
z k+1
(k+1)!
1 1
is a polynomial of degree at most k + 1 whose z k+1 coef£cient is − k+1
k! +
1
=
0,
i.e.,
is
of
degree
at
most
k.
On
the
other
hand,
we
have,
for
z
→
0,
(k+1)!
k+1
Rk,0 (z) +
X zj
z k+1
1
=
+ O(z k+2 ).
z
(k + 1)! 1 − k+1
j!
j=0
Therefore the right-hand side of (A.1) is a rational approximation to ez of order
k + 1 at z = 0 such that the degree of the numerator and denominator are k
and 1, respectively, thus is the (k, 1)-Padé approximation by uniqueness of such
approximation (see [8, Theorem 3.11 p. 48]).
B. Properties of the functions ϕ(z) introduced in (2.2).
L EMMA B.1. The functions ϕk de£ned in (2.2) satisfy the relations (we
de£ne ϕ−1 (z) = ez ):
(a) ϕk (z) = k+1
(ϕk−1 (z) − 1) for z 6= 0 and k ≥ 0;
¢
¡ z k+1
0
(b) ϕk (z) = 1 − z R ϕk (z) + k+1
z for z 6= 0 and k ≥ 0;
(c) ϕk (z) = (k + 1)! Ωk etk z dtk . . . dt0 > 0 for z ∈ C and k > 0, where
Ωk = {(tk , tk−1 , . . . , t0 ) : 0 ≤ tk ≤ · · · ≤ t0 ≤ 1} ⊂ Rk+1 ;
(d)
³
ϕk−1 (z)
ϕk (z)
´0
≥ 0 for z ≤ 0 and k ≥ 0.
Proof. A direct calculation yields
ϕk (z) = (k + 1)! z
−(k+1)
µ
zk
e − Rk−1,0 (z) −
k!
z
¶
=
k+1
(ϕk−1 (z) − 1)
z
25
SPECTRAL APPROXIMATION OF TIME WINDOWS
for z 6= 0 and k > 0. It is easy to check that (a) also holds for k = 0 with the
given de£nition of ϕ −1 . The relation (b) can be shown for example by using (a):
³
´
ϕ0k (z) = (k + 1)! z −(k+1) (ez − Rk−1,0 (z)) − (k + 1)z −(k+2) (ez − Rk,0 (z))
¶
µ
k+1
k+1
z
=
(ϕk−1 (z) − ϕk (z)) =
ϕk (z) + 1 − ϕk (z)
z
z
k+1
µ
¶
k+1
k+1
= 1−
ϕk (z) +
.
z
z
The integral form (c) of ϕk (z) can easily be Rchecked for k = 0 and is proved for
1
we obtain
k > 0 by using the induction and (a). Since Ωk−1 dtk−1 . . . dt0 = k!
for z 6= 0
!
à Z
k+1
tk−1 z
e
dtk−1 . . . dt0 − 1
k!
ϕk (z) =
z
Ωk−1
Z
etk−1 z − 1
= (k + 1)!
dtk−1 . . . dt0
z
Ωk−1
µZ tk−1
¶
Z
tk z
e dtk dtk−1 . . . dt0
= (k + 1)!
0
Ωk−1
=
(k + 1)!
Z
etk z dtk . . . dt0 .
Ωk
R
Since ϕk (0) = 1 = (k + 1)! Ωk dtk . . . dt0 the result holds also for z = 0.
The formula (c) implies that ϕk (z) > 0 for z ∈ R. Moreover, the p-th
derivative of ϕk becomes
Z
(p)
ϕk (z) = (k + 1)!
tpk etk z dtk . . . dt0 > 0
Ωk
for z ∈ R. For the functions f , g : Ωk → R de£ne the scalar inner product by
Z
hf, gi =
f (tk , . . . , t0 )g(tk , . . . , t0 )dtk . . . dt0 .
Ωk
Applying the Cauchy-Schwarz inequality to the functions
f (tk , . . . , t0 ) = e
tk z
2
,
g(tk , . . . , t0 ) = tk e
tk e
tk z
tk z
2
we obtain
(ϕ0k (z))2
((k + 1)!)2
=
≤
2
hf, gi =
Z
µZ
Ωk
etk z dtk . . . dt0
Ωk
Z
Ωk
dtk . . . dt0
¶2
t2k etk z dtk . . . dt0 =
ϕk (z)ϕ00k (z)
.
((k + 1)!)2
26
K. BURRAGE ET AL.
This leads to
2
(ϕ0k (z)) ≤ ϕk (z)ϕ00k (z)
for z ∈ R. Hence,
µ
ϕ0k (z)
ϕk (z)
¶0
=
ϕ00 (z)ϕk (z) − (ϕ0k (z))2
≥0
ϕ2k (z)
ϕ0k (z)
ϕk (z)
and, as a result, the function
other hand (a) and (b) yield
is non-decreasing on R for all k ≥ 0. On the
k
k
(ϕk−1 (z) − 1) = ϕk−1 (z) −
ϕk (z)
z
k+1
³
´
ϕ0k−1 (z)
k+1
k (z)
for z 6= 0 and k > 0. Consequently, the function ϕϕk−1
1 − ϕk−1
(z) = k
(z)
³
³
´0
´0
(z)
k (z)
≤ 0 and ϕϕk−1
≥ 0 for z ≤ 0. The
is non-increasing, i.e., ϕϕk−1
(z)
(z)
k
property (d) can easily be veri£ed in the case k = 0.
ϕ0k−1 (z) = ϕk−1 (z) −
C. Symbols of lower triangular Toeplitz matrices. The symbol of a lower
triangular Toeplitz matrix
 r0

..
..
 = toeplitz(r0 , . . . , rn−1 ) ∈ Rn×n
.
.
R=
..
. r0
rn−1
is de£ned as the (polynomial) function
n−1
X
Ã
z n−1
..
rj z j = (0, . . . , 1)R
σR (z) =
.
1
j=0
!
(see [13] for example).
L EMMA C.1. Let T1 and T2 = toeplitz(β0 , . . . , βn−1 ) be two n × n lower
triangular Toeplitz matrices. Then
σT2 T1 (z) = σT2 (z)σT1 (z) + O (z n ) .
Pn−1
for |z| < 1. Moreover, if β0 6= 0 and if j=0 |βj | can be bounded independently
of n, then
(C.1)
(C.2)
σT −1 T1 (z) =
2
σT1 (z)
+ O (z n ) .
σT2 (z)
Proof. The product T2 T1 is again a lower triangular Toeplitz matrix. The
relation (C.1) is a discrete convolution formula which is easy to verify by multiplying out the symbols of the two matrices and comparing it with the symbol of
SPECTRAL APPROXIMATION OF TIME WINDOWS
27
the product of the two matrices. The second relation (C.2) can be derived from
(C.1) as
σT1 (z) = σT2 (T −1 T1 ) (z) = σT2 (z)σT −1 T1 (z) + O (z n )
2
2
Pn−1
and the fact that |σT2 (z)| ≤ j=0 |βj | = O(1) for |z| < 1 independently of n.
Pn−1
Note that the condition in Lemma C.1 that j=0 |βj | be bounded independently of the dimension n of T2 is in particular satis£ed if T 2 has a £xed bandwidth.
REFERENCES
[1] K. B URRAGE , Z. JACKIEWICZ , S. P. N ØRSETT AND R. R ENAUT, Preconditioning waveform
relaxation iterations for differential systems, BIT, 36: 54–76, 1996.
[2] K. B URRAGE , Z. JACKIEWICZ AND R. R ENAUT, Waveform relaxation techniques for pseudospectral methods, Numer. Methods Partial Differential Equations, 12: 245–263, 1996.
[3] K. B URRAGE , G. H ERTONO , Z. JACKIEWICZ AND B. D. W ELFERT, Acceleration of convergence of static and dynamic iterations, BIT, 41: 645–655, 2001.
[4] C. C ANUTO , M.Y. H USSAINI , A. Q UARTERONI AND T.A. Z ANG, Spectral Methods in Fluid
Mechanics, Springer-Verlag, New York, Berlin, Heidelberg, 1988.
[5] K. D EKKER AND J. G. V ERWER, Stability of Runge-Kutta methods for stiff nonlinear differential equations, North-Holland, Amsterdam, 1984.
[6] D. G OTTLIEB AND J.S. H ESTHAVEN, Spectral methods for hyperbolic problems, J. Comput.
Appl. Math., 128: 83–131 2001.
[7] E. H AIRER , S. P. N ØRSETT AND G. WANNER, Solving Ordinary Differential Equations I,
Nonstiff Problems, Springer Series in Computational Math., 2nd rev. ed., Springer-Verlag,
New York, 1993.
[8] E. H AIRER AND G. WANNER, Solving Ordinary Differential Equations II, Stiff and
Differential–Algebraic Problems, Springer Series in Computational Math., 2nd rev ed.,
Springer-Verlag, New York, 1996.
[9] P. H ENRICI, Discrete Variable Methods in Ordinary Differential Equations, John Wiley & Sons,
Inc., New York, 1962.
[10] Z. JACKIEWICZ , B. OWREN AND B. W ELFERT, Pseudospectra of waveform relaxation operators, Comp. & Math. with Appls, 36: 67–85, 1998.
[11] B. L EIMKUHLER, Estimating waveform relaxation convergence, SIAM J. Sci. Comput., 14:
872–889, 1993.
[12] E. L ELARASMEE, The waveform relaxation methods for the time domain analysis of large scale
nonlinear dynamical systems, Ph.D. Thesis, University of California, Berkeley, California,
1982.
[13] L. R EICHEL AND L.N. T REFETHEN, Eigenvalues and pseudo-eigenvalues of Toeplitz matrices, Linear Algebra Appl., 162: 153–185, 1992.
[14] L.N. T REFETHEN, Pseudospectra of linear operators, SIAM Rev., 39: 383–406, 1997.