![Mean Square Convergent Numerical Methods
for Nonlinear Random Differential Equations
J.-C. Cortés1,
, L. Jódar1
, R.-J. Villanueva1
, and L. Villafuerte2
1
Instituto Universitario de Matemática Multidisciplinar
Universidad Politécnica de Valencia
Edificio 8G, 2a
, P.O. Box 22012, Valencia, Spain
{jccortes,ljodar,rjvillan}@imm.upv.es
2
Facultad de Ingenierı́a
Universidad Autónoma de Chiapas
Calle 4a
Ote. Nte. 1428, Tuxtla Gutiérrez, Chiapas, México
laura.villafuerte@unach.mx
Abstract. This paper deals with the construction of numerical solution
of nonlinear random matrix initial value problems by means of a random
Euler scheme. Conditions for the mean square convergence of the method
are established avoiding the use of pathwise information. Finally, one in-
cludes several illustrative examples where the main statistics properties
of the stochastic approximation processes are given.
Keywords: Random differential equation, mean square calculus, nu-
merical solution.
1 Introduction
Random differential equations are useful to model problems involving rates of
changes of quantities representing variables under uncertainties or randomness,
being in fact stochastic processes instead of deterministic functions [1], [2], [3],
[10], [11] and [15]. The model takes the form
Ẋ(t) = F (X(t), t) , t0 ≤ t ≤ te,
X(t0) = X0,
⎫
⎬
⎭
(1.1)
where X0 is a second order random variable and, the unknown X(t) as well as
the second member F(X(t), t) are second order stochastic processes.
The initial value problem (1.1) has been treated from the theoretical point
of view by many authors in the random framework [14,13] as well as in the
deterministic one [4]. We may distinguish two main approaches. One of them
deals with the treatment of problem (1.1) as an initial value problem in an
abstract Banach space, [14], [4], and the other is the so-called sample approach,
where throughout the realizations of problem (1.1) one gets information about
Corresponding author.
M.L. Gavrilova and C.J.K. Tan (Eds.): Trans. on Comput. Sci. VII, LNCS 5890, pp. 1–21, 2010.
c
Springer-Verlag Berlin Heidelberg 2010](https://image.slidesharecdn.com/2071871-250609201503-fb504f7d/85/Transactions-On-Computational-Science-Vii-1st-Edition-J-C-Corts-16-320.jpg)
![2 J.-C. Cortés et al.
the stochastic process solution of (1.1), [13, Appendix A], [9] and [7]. Here we will
present an approach that is independent of the pathwise behavior of the involved
processes. Finally, it is worthwhile to point out that in the random framework
is of major interest to know the probability density function of the stochastic
process solution of problem (1.1). Although in this paper we are not concerned
in this important aspect, we emphasize that in [5] one provides a general method
to tackle that problem taking advantage of the powerful deterministic tools.
As in the deterministic framework, given a right-hand side term F (X(t), t)
into (1.1), often it is not possible to derive an exact process solution, then one
must resort to numerical techniques. It is important to remark that unlike to the
deterministic case, dealing with numerical methods, it is not enough to construct
discrete approximating stochastic processes that converge to the exact theoreti-
cal solution, but it is also necessary to compute at least the expectation and the
variance functions of the approximating process, in order to have a representable
statistic idea of the solution process.
This paper is organized as follows. Section 2 deals with some preliminary
definitions, results, notations and examples that clarify the presentation of the
paper as well the extension of the results to the matrix framework, necessary to
afford problem (1.1) for the case of a system of random differential equations.
Section 3 is addressed to the presentation and the proof of the convergence for
the random Euler method in the mean square sense, for the scalar case. The case
of systems of random differential equations is treated in section 4. Illustrative
examples are included in section 5. Last section 6 is devoted to conclusions.
2 Preliminaries
This section deals with some preliminary notations, results and examples that
will clarify the presentation of the main results of the paper related to the random
Euler method for solving numerically initial value problems associated to random
differential equations.
Let (Ω, F, P) be a probability space. In the following we are interested in
second order real random variables (2-r.v.’s), Y : Ω → R having a density
probability function, fY (y), such that
E
Y 2
=
∞
−∞
y2
fY (y)dy +∞,
where E [·] denotes the expectation operator. The space of all 2-r.v.’s defined on
(Ω, F, P) and endowed with the norm
Y =
E
Y 2
1/2
, (2.1)
has a Banach space structure, denoted by L2. If Xij
: 1 ≤ i ≤ r, 1 ≤ j ≤ s is
the set of r ×s 2-r.v.’s, then the second order random matrix (2-r.m.) associated
to this family is defined as](https://image.slidesharecdn.com/2071871-250609201503-fb504f7d/85/Transactions-On-Computational-Science-Vii-1st-Edition-J-C-Corts-17-320.jpg)
![Mean Square Convergent Numerical Methods 3
X =
⎡
⎢
⎣
X11
· · · X1s
.
.
.
...
.
.
.
Xr1
· · · Xrs
⎤
⎥
⎦ . (2.2)
The set of all 2-r.m.’s X of size r × s endowed with the norm
Xr×s = max
1≤i≤r
s
j=1
Xij
, (2.3)
has a Banach space structure denoted by Lr×s
2 . Given an interval T ⊆ R, a
stochastic process {X(t), t ∈ T } defined on (Ω, F, P) is called a second order
stochastic process (2-s.p.), if for each t ∈ T , X(t) is a 2-r.v. In an analogous way
if for each t ∈ T , X(t) is a 2-r.m. of size r × s, then {X(t), t ∈ T } is a second
order matrix stochastic process (2-m.s.p.). In what follows, we shall assume that
each r.v., r.m., s.p. and m.s.p. are of second order unless the contrary is stated.
We say that a sequence of 2-r.m.’s {Xn}n≥0 is mean square (m.s.) convergent
to X ∈ Lr×s
2 , and will be denoted by Xn
m.s.
−
−
−
−
→
n→∞
X, if
lim
n→∞
Xn − Xr×s = 0. (2.4)
From the corresponding properties for its components, see [13, p.88], if {Xn}n≥0
is a sequence of random matrices in Lr×s
2 m.s. convergent to X, then
E [Xn] −
−
−
−
→
n→∞
E [X] , (2.5)
where using the notation introduced at (2.2), E [X] =
E
Xij
r×s
. In the par-
ticular case where the 2-m.s.p. X(t) is a vector, say X(t), its expectation function
is the deterministic vector function E [X(t)] =
E
Xi
(t)
r×1
and note that by
definition one gets E
XT
(t)
= (E [X(t)])
T
where XT
(t) denotes the transposed
vector of X(t). The covariance matrix function of {X(t), t ∈ T } is defined by
ΛX(t) = E
(X(t) − E [X(t)]) (X(t) − E [X(t)])
T
=
vij
(t) r×r
, (2.6)
where
vij
(t) = E
Xi
(t) − E
Xi
(t)
Xj
(t) − E
Xj
(t)
= E
Xi
(t)Xj
(t)
− E
Xi
(t)
E
Xj
(t)
, 1 ≤ i, j ≤ r, t ∈ T.
(2.7)
Note that vii
(t) denoted by V
Xi
(t)
= E
Xi
(t)
2
−
E
Xi
(t)
2
is the vari-
ance of the r.v. Xi
(t), 1 ≤ i ≤ r. If E [X(t)] = 0, then ΛX(t) is called the
correlation matrix function of {X(t), t ∈ T }. Also, for the covariance matrix
one gets the following property (see, [13, p.88])
Xn
m.s.
−
−
−
−
→
n→∞
X ⇒ ΛXn
−
−
−
−
→
n→∞
ΛX . (2.8)](https://image.slidesharecdn.com/2071871-250609201503-fb504f7d/85/Transactions-On-Computational-Science-Vii-1st-Edition-J-C-Corts-18-320.jpg)
![4 J.-C. Cortés et al.
Given a matrix A =
aij
in Rr×r
, we denote by A∞ the norm defined as [6,
p.57]
A∞ = max
1≤i≤r
r
j=1
aij
. (2.9)
The following lemma deals with the norm of the product of a deterministic matrix
function by a random matrix of compatible sizes and it will play an important
role in the following:
Lemma 2.1. Let A, B be matrices in Rr×r
, Rs×s
respectively, and X, Y ∈
Lr×s
2 , then
AXr×s ≤ A∞ Xr×s , YBr×s ≤ B∞ Yr×s . (2.10)
Proof. Since the procedure for establishing these two inequalities is analogous,
we only prove the first one. By (2.3) and (2.9) it follows that
AXr×s = max
1≤i≤r
s
j=1
r
k=1
aik
Xkj
2
≤ max
1≤i≤r
s
j=1
r
k=1
aik
Xkj
2
≤ max
1≤i≤r
r
k=1
s
j=1
aik
Xkj
2
= max
1≤i≤r
r
k=1
aik
s
j=1
Xkj
2
≤ max
1≤i≤r
r
k=1
aik
max
1≤k≤r
s
j=1
Xkj
2
= max
1≤i≤r
r
k=1
aik
Xr×s
= A∞ Xr×s .
We say that a 2-m.s.p. {X(t) : t ∈ T } in Lr×s
2 is m.s. continuous at t ∈ T , T an
interval of the real line, if
lim
τ→0
X(t + τ) − X(t)r×s = 0, t, t + τ ∈ T,
and it is m.s. differentiable at t ∈ T , if there exists a 2-m.s.p. denoted by
Ẋ(t) : t ∈ T
such that
lim
τ→0
X(t + τ) − X(t)
τ
− Ẋ(t)
r×s
= 0, t, t + τ ∈ T. (2.11)
Example 2.2. Let Y be a 2-r.v. and let us consider the 2-s.p. Y (t) = Y · t for
t lying in the interval T . Note that Y (t) is m.s. differentiable at t because for τ
a real number such that t + τ ∈ T , one gets](https://image.slidesharecdn.com/2071871-250609201503-fb504f7d/85/Transactions-On-Computational-Science-Vii-1st-Edition-J-C-Corts-19-320.jpg)
![Mean Square Convergent Numerical Methods 5
Y (t + τ) − Y (t)
τ
− Y
2
= E
Y · (t + τ) − Y · t
τ
− Y
2
= E
Y 2
t + τ − t
τ
− 1
2
−
−
−
→
τ→0
0.
Definition 2.3. Let S be a bounded set in Lr×s
2 , an interval T ⊆ R and h 0,
we say that F : S × T → Lr×s
2 is randomly bounded time uniformly continuous
in S if
lim
h→0
ω(S, h)=0, (2.12)
where
ω(S, h) = sup
X∈S⊂Lr×s
2
sup
|t−t|≤h
F(X, t) − F(X, t )r×s . (2.13)
Example 2.4. Let us consider the second vector stochastic process F(X, t) =
A(t)X + G(t), 0 ≤ t ≤ te, where
X =X=
X1
X2
, A(t) = A=
0 1
−ω2
0 −2ω0ξ
, G(t)=G(t)=
0
−Y (t)
(2.14)
(for convenience we have identified vectorial notation with matrix notation) being
Y (t) =
m
j=1
taje−αj t
cos (ωjt + θj) , t ≥ 0, (2.15)
and aj, αj, ωj and ξ are positive real numbers, and θj are pairwise independent
2-r.v.’s uniformly distributed on [0, 2π]. Note that
(A(t)X + G(t))
T
=
X2
, −ω2
0X1
− 2ω0ξX2
− Y (t)
, (2.16)
and
(F(X, t) − F(X, t ))
T
= [0 , Y (t ) − Y (t)] . (2.17)
Now by considering the following relationship (see [13, p.47])
E [cos (ωjt + θj) cos (ωkt + θk)] =
0 if j = k
1
2
cos (ωj(t − t )) if j = k
, (2.18)
one gets
E
(Y (t ) − Y (t))
2
=
m
j=1
1
2
a2
j t2
e−2αj t
+ (t )2
e−2αj t
!
−
m
j=1
a2
j tt e−αj (t+t
)
cos (ωj(t − t ))
!
.
(2.19)](https://image.slidesharecdn.com/2071871-250609201503-fb504f7d/85/Transactions-On-Computational-Science-Vii-1st-Edition-J-C-Corts-20-320.jpg)
![6 J.-C. Cortés et al.
Hence, using the fact that the exponential and cosines deterministic functions
involved in (2.19) are continuous with respect to variable t, from definition 2.3
it follows that F(X, t) is randomly bounded time uniformly continuous.
For the sake of clarity in the presentation, we include in this section by enunci-
ating without proof some results that may be found in [13, chap.4].
Theorem 2.5 (Integration by parts). Let {X(t), t ∈ T} be a 2-s.p. m.s. dif-
ferentiable on T = [t0, t] and let the deterministic function h(t, u) be continuous
on T × T whose partial derivative ∂h(t, u)/∂u exists. If one denotes by
Y (t) =
t
t0
h(t, u)Ẋ(u)du, (2.20)
(where the above integral is considered in the Riemann mean square sense) then,
Y (t) = [h(t, u)X(u)]u=t
u=t0
−
t
t0
∂h(t, u)
∂u
X(u)du. (2.21)
Taking h(t, u) ≡ 1 in (2.20)-(2.21) one deduces the following
Theorem 2.6 (Fundamental theorem of the m.s. calculus). Let
{X(t), t ∈ T } be a 2-s.p. m.s. differentiable on T = [t0, t], such that Ẋ(t) is
m.s. Riemann integrable on T , then one gets
t
t0
Ẋ(u)du = X(t) − X(t0). (2.22)
Example 2.7. In the framework of the example 2.2, note that taking the m.s.
differentiable process X(t) = Y · t and applying the formula of integration by
parts for h(t, u) ≡ 1 one gets
t
t0
Y du = Y (t − t0) .
Proposition 2.8. If {X(t), t ∈ T } is a 2-s.p. m.s. continuous on T = [t0, t],
then
t
t0
X(u)du
≤
t
t0
X(u) du ≤ MX(t − t0) , MX = max
t0≤u≤t
X(u) . (2.23)
3 Convergence of the Scalar Random Euler Method
Let us consider the random initial value problem (1.1) under the following hy-
potheses on F : S × T → L2, with S ⊂ L2
– H1: F(X, t) is m.s. randomly bounded time uniformly continuous.
– H2: F(X, t) satisfies the m.s. Lipschitz condition](https://image.slidesharecdn.com/2071871-250609201503-fb504f7d/85/Transactions-On-Computational-Science-Vii-1st-Edition-J-C-Corts-21-320.jpg)
![Mean Square Convergent Numerical Methods 7
F(X, t) − F(Y, t) ≤ k(t) X − Y ,
te
t0
k(t)dt +∞. (3.1)
Note that condition H2 guarantees the m.s. continuity of F(X, t) with respect
to the first variable while H1 guarantees the continuity of F(X, t) with respect
to the second variable. Hence and from the the inequality
F(X, t) − F(Y, t ) ≤ F(X, t) − F(Y, t) + F(Y, t) − F(Y, t ) ,
one gets the m.s. continuity of F(X, t) with respect to both variables.
Let us introduce the random Euler method for problem (1.1) defined by
Xn+1 = Xn + hF(Xn, tn) , n ≥ 0
X0 = X (t0)
(3.2)
where Xn, F(Xn, tn) are 2-r.v.’s, h = tn − tn−1, with tn = t0 + nh, for n =
0, 1, 2, . . .. We wish to prove that under hypotheses H1 and H2, the Euler method
(3.2) is m.s. convergent in the fixed station sense, i.e., fixed t ∈ [t0, te] and taking
n so that t = tn = t0 + nh, the m.s. error
en = Xn − X(t) = Xn − X (tn) , (3.3)
tends to zero in L2, as h → 0, n → ∞ with t − t0 = nh.
Note that under hypotheses H1 and H2, theorem 5.1.2. of [13, p. 118] guaran-
tees the existence and uniqueness of a m.s. solution X(t) in [tn, tn+1] ⊂ [t0, te],
and by the m.s. fundamental theorem of calculus, i.e., theorem 2.6 it follows that
X (tn+1) = X (tn) +
tn+1
tn
Ẋ(u)du, n ≥ 0. (3.4)
From (3.2)-(3.4) it follows that
en+1 − en = (Xn+1 − Xn) − (X (tn+1) − X (tn))
= hF (Xn, tn) −
tn+1
tn
Ẋ(u)du.
(3.5)
Note also that as F(Xn, tn) ∈ L2, the first term appearing in the right-hand side
of (3.5) can be written as follows, see example 2.7,
hF (Xn, tn) = F (Xn, tn) (tn+1 − tn) =
tn+1
tn
F (Xn, tn) du. (3.6)
By (3.5), (3.6) and using that Ẋ(u) = F (X(u), u), one gets
en+1 = en +
tn+1
tn
F (Xn, tn) − Ẋ(u)
du
= en +
tn+1
tn
[F (Xn, tn) − F (X(u), u)] du.
(3.7)](https://image.slidesharecdn.com/2071871-250609201503-fb504f7d/85/Transactions-On-Computational-Science-Vii-1st-Edition-J-C-Corts-22-320.jpg)
Transactions On Computational Science Vii 1st Edition J C Corts
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- 6. Lecture Notes in Computer Science 5890 Commenced Publication in 1973 Founding and Former Series Editors: Gerhard Goos, Juris Hartmanis, and Jan van Leeuwen Editorial Board David Hutchison Lancaster University, UK Takeo Kanade Carnegie Mellon University, Pittsburgh, PA, USA Josef Kittler University of Surrey, Guildford, UK Jon M. Kleinberg Cornell University, Ithaca, NY, USA Friedemann Mattern ETH Zurich, Switzerland John C. Mitchell Stanford University, CA, USA Moni Naor Weizmann Institute of Science, Rehovot, Israel Oscar Nierstrasz University of Bern, Switzerland C. Pandu Rangan Indian Institute of Technology, Madras, India Bernhard Steffen TU Dortmund University, Germany Madhu Sudan Microsoft Research, Cambridge, MA, USA Demetri Terzopoulos University of California, Los Angeles, CA, USA Doug Tygar University of California, Berkeley, CA, USA Moshe Y. Vardi Rice University, Houston, TX, USA Gerhard Weikum Max-Planck Institute of Computer Science, Saarbruecken, Germany
- 7. Marina L. Gavrilova C.J. Kenneth Tan (Eds.) Transactions on Computational Science VII 1 3
- 8. Editors-in-Chief Marina L. Gavrilova University of Calgary, Department of Computer Science 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada E-mail: mgavrilo@ucalgary.ca C.J. Kenneth Tan Exascala Ltd. Unit 9, 97 Rickman Drive, Birmingham B15 2AL, UK E-mail: cjtan@exascala.com Library of Congress Control Number: 2009943910 CR Subject Classification (1998): F.2.1, G.1, G.3, G.1.8, G.4, J.2 ISSN 0302-9743 (Lecture Notes in Computer Science) ISSN 1866-4733 (Transaction on Computational Science) ISBN-10 3-642-11388-5 Springer Berlin Heidelberg New York ISBN-13 978-3-642-11388-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. springer.com © Springer-Verlag Berlin Heidelberg 2010 Printed in Germany Typesetting: Camera-ready by author, data conversion by Scientific Publishing Services, Chennai, India Printed on acid-free paper SPIN: 12832507 06/3180 5 4 3 2 1 0
- 9. LNCS Transactions on Computational Science Computational science, an emerging and increasingly vital field, is now widely recognized as an integral part of scientific and technical investigations, affecting researchers and practitioners in areas ranging from aerospace and automotive research to biochemistry, electronics, geosciences, mathematics, and physics. Computer systems research and the exploitation of applied research naturally complement each other. The increased complexity of many challenges in computational science demands the use of supercomputing, parallel processing, sophisticated algorithms, and advanced system software and architecture. It is therefore invaluable to have input by systems research experts in applied computational science research. Transactions on Computational Science focuses on original high-quality research in the realm of computational science in parallel and distributed environments, also encompassing the underlying theoretical foundations and the applications of large- scale computation. The journal offers practitioners and researchers the opportunity to share computational techniques and solutions in this area, to identify new issues, and to shape future directions for research, and it enables industrial users to apply leading- edge, large-scale, high-performance computational methods. In addition to addressing various research and application issues, the journal aims to present material that is validated – crucial to the application and advancement of the research conducted in academic and industrial settings. In this spirit, the journal focuses on publications that present results and computational techniques that are verifiable. Scope The scope of the journal includes, but is not limited to, the following computational methods and applications: • Aeronautics and Aerospace • Astrophysics • Bioinformatics • Climate and Weather Modeling • Communication and Data Networks • Compilers and Operating Systems • Computer Graphics • Computational Biology • Computational Chemistry • Computational Finance and Econometrics • Computational Fluid Dynamics • Computational Geometry
- 10. VI LNCS Transactions on Computational Science • Computational Number Theory • Computational Physics • Data Storage and Information Retrieval • Data Mining and Data Warehousing • Grid Computing • Hardware/Software Co-design • High-Energy Physics • High-Performance Computing • Numerical and Scientific Computing • Parallel and Distributed Computing • Reconfigurable Hardware • Scientific Visualization • Supercomputing • System-on-Chip Design and Engineering
- 11. Editorial The Transactions on Computational Science journal is part of the Springer series Lecture Notes in Computer Science, and is devoted to the gamut of computational science issues, from theoretical aspects to application-dependent studies and the vali- dation of emerging technologies. The journal focuses on original high-quality research in the realm of computational science in parallel and distributed environments, encompassing the facilitating theo- retical foundations and the applications of large-scale computations and massive data processing. Practitioners and researchers share computational techniques and solutions in the area, identify new issues, and shape future directions for research, as well as enable industrial users to apply the techniques presented. The current issue is devoted to core computational science techniques: grid com- puting, advanced numerical methods and stochastic systems, as well as the application of such research, which naturally complement each other. The issue is comprised of two parts. Part 1 focuses on computations of stochastic systems and Part 2 on compu- tational methods for complex systems. Part 1, edited by Prof. Magdy A. El-Tawil, Cairo University, Egypt, is devoted to state-of-the-art research on computations using stochastic methods. Systems with stochastic uncertainties have received significant attention in applied and engineering domains, where algorithms to compute approximate or exact solutions can be con- structed. The five papers in Part 1 of this issue are devoted to the description, applica- tion and presentation of such algorithms for applied stochastic nonlinear systems. Each paper addresses a specific computational problem and provides a stochastic solution to it. The five manuscripts are devoted to: the wave equation on a torus, the discrete-time retrial queue in communication networks, the nonlinear diffusion equa- tion with a stochastic nonhomogeneity, the comprehensive presentation of the homo- topy WHEP technique and the Pickard algorithm, and the construction of a numerical mean square solution of a nonlinear random initial value problem. Part 2 continues the topic with an in-depth look at computational science research in the areas of numerical methods and complex systems applications. The four papers comprising Part 2 are devoted to the following topics: protecting the integrity and confidentiality of data and code stored in systems and ensuring integrity of computa- tions through comprehensive analysis of secure multiprocessor architectures; address- ing the issues of non-physical deviations in the numerical solution through compensa- tion of the scheme dispersion and dissipation by non-linear methods; presenting new computational algorithms for the study of quantum 3D disordered spin systems; and finally developing a new grid-based approach for chemical reaction computations. Each paper provides a detailed experimentation or an applied example to complement the methodology presented. In conclusion, we would like to extend our sincere appreciation to Prof. Magdy A. El-Tawil, to all authors for submitting their manuscripts, and to all associate editors and external referees for their meticulous and valuable reviews. We would also like to
- 12. Editorial VIII express our gratitude to the LNCS editorial staff of Springer, in particular Alfred Hofmann, Ursula Barth and Anna Kramer, who supported us at every stage of the project. It is our hope that the fine collection of papers presented in this issue will be a valuable resource for Transactions on Computational Science readers and will stimu- late further research into the vibrant area of computational science applications. November 2009 Marina L. Gavrilova C.J. Kenneth Tan
- 13. Computations of Stochastic Systems Special Issue Guest Editor Preface The modern probability theory has been evolved from extensive works of highly ap- preciated scientists over the past decades. However, the probabilistic models are not introduced for mere curiosity, they are the true models. The statement that "the world is nonlinear and stochastic" is a true one. The problem arises from the complications of the analysis of the probabilistic model, which makes necessary the adaptation of the well known deterministic analysis to probabilistic analysis. In this special issue, the probabilistic analysis is illustrated through introducing an Ito-Stratonovich formula for the wave equation on a torus. The stochastic numerical analysis is expressed through the construction of a numerical mean square solution of a nonlinear random initial value problem by means of a random Euler scheme. The random approximation is illustrated through the approximate solution of a nonlinear diffusion equation with a stochastic nonhomogeneity using the homotopy WHEP technique. Finally, the numerical methods for the analysis of systems in computer and communication networks are discussed through a discrete-time retrial queue with impatient customers and a server subject to starting failures. The methods presented in this issue are now increasing in popularity and are widely applied in analysis, modeling, engineering and physics. It is expected they will grow to cover all the scientific areas in which the deterministic systems previously prevailed. November 2009 Magdy A. El-Tawil Professor of Mathematics Cairo University, Faculty of Engineering, Engineering Mathematics Department Giza, Egypt
- 14. LNCS Transactions on Computational Science – Editorial Board Marina L. Gavrilova, Editor-in-chief University of Calgary, Canada Chih Jeng Kenneth Tan, Editor-in-chief OptimaNumerics, UK Tetsuo Asano JAIST, Japan Brian A. Barsky University of California at Berkeley, USA Alexander V. Bogdanov Institute for High Performance Computing and Data Bases, Russia Martin Buecker Aachen University, Germany Rajkumar Buyya University of Melbourne, Australia Hyungseong Choo Sungkyunkwan University, Korea Danny Crookes Queen's University Belfast, UK Tamal Dey Ohio State University, USA Ivan Dimov Bulgarian Academy of Sciences, Bulgaria Magdy El-Tawil Cairo University, Egypt Osvaldo Gervasi Università degli Studi di Perugia, Italy Christopher Gold University of Glamorgan, UK Rodolfo Haber Council for Scientific Research, Spain Andres Iglesias University of Cantabria, Spain Deok-Soo Kim Hanyang University, Korea Ivana Kolingerova University of West Bohemia, Czech Republic Vipin Kumar Army High Performance Computing Research Center, USA Antonio Lagana Università degli Studi di Perugia, Italy D.T. Lee Institute of Information Science, Academia Sinica, Taiwan Laurence Liew Platform Computing, Singapore Nikolai Medvedev Novosibirsk Russian Academy of Sciences, Russia Graham M Megson University of Reading, UK Edward D. Moreno UEA – University of Amazonas state, Brazil Youngsong Mun Soongsil University, Korea Dimitri Plemenos Université de Limoges, France Viktor K. Prasanna University of Southern California, USA Muhammad Sarfraz KFUPM, Saudi Arabia Dale Shires Army Research Lab, USA Masha Sosonkina Ames Laboratory, USA Alexei Sourin Nanyang Technological University, Singapore David Taniar Monash University, Australia Athanasios Vasilakos University of Western Macedonia, Greece Chee Yap New York University, USA Igor Zacharov SGI Europe, Switzerland Zahari Zlatev National Environmental Research Institute, Denmark
- 15. Table of Contents Part 1: Computations of Stochastic Systems Guest Editor: Magdy A. El-Tawil Mean Square Convergent Numerical Methods for Nonlinear Random Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 J.-C. Cortés, L. Jódar, R.-J. Villanueva, and L. Villafuerte A Discrete-Time Geo/G/1 Retrial Queue with Starting Failures and Impatient Customers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Abdel-Karim Aboul-Hassan, Sherif I. Rabia, and Ahmed A. Al-Mujahid Using Homotopy-WHEP Technique in Solving Nonlinear Diffusion Equation with Stochastic No Homogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Magdy A. El-Tawil and Noha A. Al-Mulla Itô-Stratonovitch Formula for the Wave Equation on a Torus . . . . . . . . . . 68 Rémi Léandre The Solution of Non-linear Diffusion Equation under Stochastic Nonhomogeneity Using Symbolic WHEP and Pickard Algorithms . . . . . . 75 Magdy A. El-Tawil and Noha A. Al-Mulla Part 2: Computational Methods for Complex Systems An Analysis of Secure Processor Architectures . . . . . . . . . . . . . . . . . . . . . . . 101 Siddhartha Chhabra, Yan Solihin, Reshma Lal, and Matthew Hoekstra Compensation of the Scheme Dispersion and Dissipation by Artificial Non-linear Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 A.V. Porubov, D. Bouche, and G. Bonnaud of Quantum 3D Disordered Spin System under the Influence of External Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Ashot S. Gevorkyan, Chin-Kun Hu, and Sergei Flach Grid-Technology for Chemical Reactions Calculation . . . . . . . . . . . . . . . . . 154 G.G. Balint-Kurti, A.V. Bogdanov, A.S. Gevorkyan, Yu.E. Gorbachev, T. Hakobyan, G. Nyman, and I. Shoshmina Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 New Mathematical Conception and Computation Algorithm for Study of Quantum 3D Disordered Spin System under the Influence of External Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ashot S. Gevorkyan, Chin-Kun Hu, and Sergei Flach Erratum E1 Retracted: New Mathematical Conception and Computation Algorithm for Study
- 16. Mean Square Convergent Numerical Methods for Nonlinear Random Differential Equations J.-C. Cortés1, , L. Jódar1 , R.-J. Villanueva1 , and L. Villafuerte2 1 Instituto Universitario de Matemática Multidisciplinar Universidad Politécnica de Valencia Edificio 8G, 2a , P.O. Box 22012, Valencia, Spain {jccortes,ljodar,rjvillan}@imm.upv.es 2 Facultad de Ingenierı́a Universidad Autónoma de Chiapas Calle 4a Ote. Nte. 1428, Tuxtla Gutiérrez, Chiapas, México laura.villafuerte@unach.mx Abstract. This paper deals with the construction of numerical solution of nonlinear random matrix initial value problems by means of a random Euler scheme. Conditions for the mean square convergence of the method are established avoiding the use of pathwise information. Finally, one in- cludes several illustrative examples where the main statistics properties of the stochastic approximation processes are given. Keywords: Random differential equation, mean square calculus, nu- merical solution. 1 Introduction Random differential equations are useful to model problems involving rates of changes of quantities representing variables under uncertainties or randomness, being in fact stochastic processes instead of deterministic functions [1], [2], [3], [10], [11] and [15]. The model takes the form Ẋ(t) = F (X(t), t) , t0 ≤ t ≤ te, X(t0) = X0, ⎫ ⎬ ⎭ (1.1) where X0 is a second order random variable and, the unknown X(t) as well as the second member F(X(t), t) are second order stochastic processes. The initial value problem (1.1) has been treated from the theoretical point of view by many authors in the random framework [14,13] as well as in the deterministic one [4]. We may distinguish two main approaches. One of them deals with the treatment of problem (1.1) as an initial value problem in an abstract Banach space, [14], [4], and the other is the so-called sample approach, where throughout the realizations of problem (1.1) one gets information about Corresponding author. M.L. Gavrilova and C.J.K. Tan (Eds.): Trans. on Comput. Sci. VII, LNCS 5890, pp. 1–21, 2010. c Springer-Verlag Berlin Heidelberg 2010
- 17. 2 J.-C. Cortés et al. the stochastic process solution of (1.1), [13, Appendix A], [9] and [7]. Here we will present an approach that is independent of the pathwise behavior of the involved processes. Finally, it is worthwhile to point out that in the random framework is of major interest to know the probability density function of the stochastic process solution of problem (1.1). Although in this paper we are not concerned in this important aspect, we emphasize that in [5] one provides a general method to tackle that problem taking advantage of the powerful deterministic tools. As in the deterministic framework, given a right-hand side term F (X(t), t) into (1.1), often it is not possible to derive an exact process solution, then one must resort to numerical techniques. It is important to remark that unlike to the deterministic case, dealing with numerical methods, it is not enough to construct discrete approximating stochastic processes that converge to the exact theoreti- cal solution, but it is also necessary to compute at least the expectation and the variance functions of the approximating process, in order to have a representable statistic idea of the solution process. This paper is organized as follows. Section 2 deals with some preliminary definitions, results, notations and examples that clarify the presentation of the paper as well the extension of the results to the matrix framework, necessary to afford problem (1.1) for the case of a system of random differential equations. Section 3 is addressed to the presentation and the proof of the convergence for the random Euler method in the mean square sense, for the scalar case. The case of systems of random differential equations is treated in section 4. Illustrative examples are included in section 5. Last section 6 is devoted to conclusions. 2 Preliminaries This section deals with some preliminary notations, results and examples that will clarify the presentation of the main results of the paper related to the random Euler method for solving numerically initial value problems associated to random differential equations. Let (Ω, F, P) be a probability space. In the following we are interested in second order real random variables (2-r.v.’s), Y : Ω → R having a density probability function, fY (y), such that E Y 2 = ∞ −∞ y2 fY (y)dy +∞, where E [·] denotes the expectation operator. The space of all 2-r.v.’s defined on (Ω, F, P) and endowed with the norm Y = E Y 2 1/2 , (2.1) has a Banach space structure, denoted by L2. If Xij : 1 ≤ i ≤ r, 1 ≤ j ≤ s is the set of r ×s 2-r.v.’s, then the second order random matrix (2-r.m.) associated to this family is defined as
- 18. Mean Square Convergent Numerical Methods 3 X = ⎡ ⎢ ⎣ X11 · · · X1s . . . ... . . . Xr1 · · · Xrs ⎤ ⎥ ⎦ . (2.2) The set of all 2-r.m.’s X of size r × s endowed with the norm Xr×s = max 1≤i≤r s j=1 Xij , (2.3) has a Banach space structure denoted by Lr×s 2 . Given an interval T ⊆ R, a stochastic process {X(t), t ∈ T } defined on (Ω, F, P) is called a second order stochastic process (2-s.p.), if for each t ∈ T , X(t) is a 2-r.v. In an analogous way if for each t ∈ T , X(t) is a 2-r.m. of size r × s, then {X(t), t ∈ T } is a second order matrix stochastic process (2-m.s.p.). In what follows, we shall assume that each r.v., r.m., s.p. and m.s.p. are of second order unless the contrary is stated. We say that a sequence of 2-r.m.’s {Xn}n≥0 is mean square (m.s.) convergent to X ∈ Lr×s 2 , and will be denoted by Xn m.s. − − − − → n→∞ X, if lim n→∞ Xn − Xr×s = 0. (2.4) From the corresponding properties for its components, see [13, p.88], if {Xn}n≥0 is a sequence of random matrices in Lr×s 2 m.s. convergent to X, then E [Xn] − − − − → n→∞ E [X] , (2.5) where using the notation introduced at (2.2), E [X] = E Xij r×s . In the par- ticular case where the 2-m.s.p. X(t) is a vector, say X(t), its expectation function is the deterministic vector function E [X(t)] = E Xi (t) r×1 and note that by definition one gets E XT (t) = (E [X(t)]) T where XT (t) denotes the transposed vector of X(t). The covariance matrix function of {X(t), t ∈ T } is defined by ΛX(t) = E (X(t) − E [X(t)]) (X(t) − E [X(t)]) T = vij (t) r×r , (2.6) where vij (t) = E Xi (t) − E Xi (t) Xj (t) − E Xj (t) = E Xi (t)Xj (t) − E Xi (t) E Xj (t) , 1 ≤ i, j ≤ r, t ∈ T. (2.7) Note that vii (t) denoted by V Xi (t) = E Xi (t) 2 − E Xi (t) 2 is the vari- ance of the r.v. Xi (t), 1 ≤ i ≤ r. If E [X(t)] = 0, then ΛX(t) is called the correlation matrix function of {X(t), t ∈ T }. Also, for the covariance matrix one gets the following property (see, [13, p.88]) Xn m.s. − − − − → n→∞ X ⇒ ΛXn − − − − → n→∞ ΛX . (2.8)
- 19. 4 J.-C. Cortés et al. Given a matrix A = aij in Rr×r , we denote by A∞ the norm defined as [6, p.57] A∞ = max 1≤i≤r r j=1 aij . (2.9) The following lemma deals with the norm of the product of a deterministic matrix function by a random matrix of compatible sizes and it will play an important role in the following: Lemma 2.1. Let A, B be matrices in Rr×r , Rs×s respectively, and X, Y ∈ Lr×s 2 , then AXr×s ≤ A∞ Xr×s , YBr×s ≤ B∞ Yr×s . (2.10) Proof. Since the procedure for establishing these two inequalities is analogous, we only prove the first one. By (2.3) and (2.9) it follows that AXr×s = max 1≤i≤r s j=1 r k=1 aik Xkj 2 ≤ max 1≤i≤r s j=1 r k=1 aik Xkj 2 ≤ max 1≤i≤r r k=1 s j=1 aik Xkj 2 = max 1≤i≤r r k=1 aik s j=1 Xkj 2 ≤ max 1≤i≤r r k=1 aik max 1≤k≤r s j=1 Xkj 2 = max 1≤i≤r r k=1 aik Xr×s = A∞ Xr×s . We say that a 2-m.s.p. {X(t) : t ∈ T } in Lr×s 2 is m.s. continuous at t ∈ T , T an interval of the real line, if lim τ→0 X(t + τ) − X(t)r×s = 0, t, t + τ ∈ T, and it is m.s. differentiable at t ∈ T , if there exists a 2-m.s.p. denoted by Ẋ(t) : t ∈ T such that lim τ→0 X(t + τ) − X(t) τ − Ẋ(t) r×s = 0, t, t + τ ∈ T. (2.11) Example 2.2. Let Y be a 2-r.v. and let us consider the 2-s.p. Y (t) = Y · t for t lying in the interval T . Note that Y (t) is m.s. differentiable at t because for τ a real number such that t + τ ∈ T , one gets
- 20. Mean Square Convergent Numerical Methods 5 Y (t + τ) − Y (t) τ − Y 2 = E Y · (t + τ) − Y · t τ − Y 2 = E Y 2 t + τ − t τ − 1 2 − − − → τ→0 0. Definition 2.3. Let S be a bounded set in Lr×s 2 , an interval T ⊆ R and h 0, we say that F : S × T → Lr×s 2 is randomly bounded time uniformly continuous in S if lim h→0 ω(S, h)=0, (2.12) where ω(S, h) = sup X∈S⊂Lr×s 2 sup |t−t|≤h F(X, t) − F(X, t )r×s . (2.13) Example 2.4. Let us consider the second vector stochastic process F(X, t) = A(t)X + G(t), 0 ≤ t ≤ te, where X =X= X1 X2 , A(t) = A= 0 1 −ω2 0 −2ω0ξ , G(t)=G(t)= 0 −Y (t) (2.14) (for convenience we have identified vectorial notation with matrix notation) being Y (t) = m j=1 taje−αj t cos (ωjt + θj) , t ≥ 0, (2.15) and aj, αj, ωj and ξ are positive real numbers, and θj are pairwise independent 2-r.v.’s uniformly distributed on [0, 2π]. Note that (A(t)X + G(t)) T = X2 , −ω2 0X1 − 2ω0ξX2 − Y (t) , (2.16) and (F(X, t) − F(X, t )) T = [0 , Y (t ) − Y (t)] . (2.17) Now by considering the following relationship (see [13, p.47]) E [cos (ωjt + θj) cos (ωkt + θk)] = 0 if j = k 1 2 cos (ωj(t − t )) if j = k , (2.18) one gets E (Y (t ) − Y (t)) 2 = m j=1 1 2 a2 j t2 e−2αj t + (t )2 e−2αj t ! − m j=1 a2 j tt e−αj (t+t ) cos (ωj(t − t )) ! . (2.19)
- 21. 6 J.-C. Cortés et al. Hence, using the fact that the exponential and cosines deterministic functions involved in (2.19) are continuous with respect to variable t, from definition 2.3 it follows that F(X, t) is randomly bounded time uniformly continuous. For the sake of clarity in the presentation, we include in this section by enunci- ating without proof some results that may be found in [13, chap.4]. Theorem 2.5 (Integration by parts). Let {X(t), t ∈ T} be a 2-s.p. m.s. dif- ferentiable on T = [t0, t] and let the deterministic function h(t, u) be continuous on T × T whose partial derivative ∂h(t, u)/∂u exists. If one denotes by Y (t) = t t0 h(t, u)Ẋ(u)du, (2.20) (where the above integral is considered in the Riemann mean square sense) then, Y (t) = [h(t, u)X(u)]u=t u=t0 − t t0 ∂h(t, u) ∂u X(u)du. (2.21) Taking h(t, u) ≡ 1 in (2.20)-(2.21) one deduces the following Theorem 2.6 (Fundamental theorem of the m.s. calculus). Let {X(t), t ∈ T } be a 2-s.p. m.s. differentiable on T = [t0, t], such that Ẋ(t) is m.s. Riemann integrable on T , then one gets t t0 Ẋ(u)du = X(t) − X(t0). (2.22) Example 2.7. In the framework of the example 2.2, note that taking the m.s. differentiable process X(t) = Y · t and applying the formula of integration by parts for h(t, u) ≡ 1 one gets t t0 Y du = Y (t − t0) . Proposition 2.8. If {X(t), t ∈ T } is a 2-s.p. m.s. continuous on T = [t0, t], then t t0 X(u)du ≤ t t0 X(u) du ≤ MX(t − t0) , MX = max t0≤u≤t X(u) . (2.23) 3 Convergence of the Scalar Random Euler Method Let us consider the random initial value problem (1.1) under the following hy- potheses on F : S × T → L2, with S ⊂ L2 – H1: F(X, t) is m.s. randomly bounded time uniformly continuous. – H2: F(X, t) satisfies the m.s. Lipschitz condition
- 22. Mean Square Convergent Numerical Methods 7 F(X, t) − F(Y, t) ≤ k(t) X − Y , te t0 k(t)dt +∞. (3.1) Note that condition H2 guarantees the m.s. continuity of F(X, t) with respect to the first variable while H1 guarantees the continuity of F(X, t) with respect to the second variable. Hence and from the the inequality F(X, t) − F(Y, t ) ≤ F(X, t) − F(Y, t) + F(Y, t) − F(Y, t ) , one gets the m.s. continuity of F(X, t) with respect to both variables. Let us introduce the random Euler method for problem (1.1) defined by Xn+1 = Xn + hF(Xn, tn) , n ≥ 0 X0 = X (t0) (3.2) where Xn, F(Xn, tn) are 2-r.v.’s, h = tn − tn−1, with tn = t0 + nh, for n = 0, 1, 2, . . .. We wish to prove that under hypotheses H1 and H2, the Euler method (3.2) is m.s. convergent in the fixed station sense, i.e., fixed t ∈ [t0, te] and taking n so that t = tn = t0 + nh, the m.s. error en = Xn − X(t) = Xn − X (tn) , (3.3) tends to zero in L2, as h → 0, n → ∞ with t − t0 = nh. Note that under hypotheses H1 and H2, theorem 5.1.2. of [13, p. 118] guaran- tees the existence and uniqueness of a m.s. solution X(t) in [tn, tn+1] ⊂ [t0, te], and by the m.s. fundamental theorem of calculus, i.e., theorem 2.6 it follows that X (tn+1) = X (tn) + tn+1 tn Ẋ(u)du, n ≥ 0. (3.4) From (3.2)-(3.4) it follows that en+1 − en = (Xn+1 − Xn) − (X (tn+1) − X (tn)) = hF (Xn, tn) − tn+1 tn Ẋ(u)du. (3.5) Note also that as F(Xn, tn) ∈ L2, the first term appearing in the right-hand side of (3.5) can be written as follows, see example 2.7, hF (Xn, tn) = F (Xn, tn) (tn+1 − tn) = tn+1 tn F (Xn, tn) du. (3.6) By (3.5), (3.6) and using that Ẋ(u) = F (X(u), u), one gets en+1 = en + tn+1 tn F (Xn, tn) − Ẋ(u) du = en + tn+1 tn [F (Xn, tn) − F (X(u), u)] du. (3.7)
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- 24. spread for them on his right. The camels came next, and knelt down gently, whereupon the houris sprang from their palanquins with a lightness and grace which astonished Roland more than all. Their feet hardly left an imprint in the sand. Like the favourites of the harem, they also approached Roland, and, kissing the ground before him, ranged themselves on his left. Then like a flood advanced the troop of celestial dancers, tripping along to the sound of castanets, flutes, theorbos, timbrels, guitars, and mandolins, amid loud singing, accompanied by the most lively strains of music. The animation of their movements increased or diminished according to the rhythm, which they marked by accurate beats of the foot and clapping of hands, in slow or quick time. Their eyes were now filled with soft languor—now darted glances of fire. Balancing themselves from the hips, they swung their bodies and waved their arms with ease and grace. At times a comb, unable to imprison such a wealth of tresses, fell out, and freed locks that were as dark as the night. But now the Prophet gave the signal: the dances ceased, and the houris flew, like a flock of frightened birds, to take their position opposite Roland, and under shelter of the elephants. Mahomet, in his turn, drew nearer to the nephew of Charlemagne, who immediately dismounted—an act of courtesy to age he invariably observed. “May Allah, who has made all things of earth and heaven, of day and night, extend his blessing to you in this world and in the one you inhabit! You are welcome,” said the Prophet! “I must ask your pardon for the poverty of this reception, as our meeting has been arranged at such short notice that I have only had time to bring as my suite a few of my immediate followers, and the troops which happen to be my guard of honour for the day. Besides, I feared that in surrounding myself with too great pomp, I might seem to be offering a defiance to a late enemy, whom I only desire to make a friend of. If I have not treated you with more ceremony, it is because I wish to treat you like a brother.”
- 25. Roland made a wry face, which the Prophet thought it convenient to attribute to the glare of the sun in his eyes, and therefore made a sign to four angels, who immediately flew off and spread a rosy cloud before the luminary. . “I accept your explanation,” said Roland, coolly, half doubting whether the Prophet were not making fun of him. “I have equal need of pardon; but if I have come without a fitting retinue, you must attribute it to my desire to answer your invitation promptly.” After this exchange of courtesies, Roland commenced the conversation by saying, “You will forgive me if I beg you at once to inform me what it is that has obtained me the honour of this interview, as I am in a hurry to return to earth. I mount guard to- night in the Emperor’s tent, and I never like to fail in the performance of duty.” “Never fear,” said Mahomet; “I’ll have the sun put back. We have all time for our interview.” “I am all attention.” “There is not a more valiant knight than you living. Your single arm is worth an army. Your judgment is sound, your decision speedy ——” “How much do you expect for this panegyric? I warn you, before you go any further, not to set too high a price on it, as I have a clear estimate of my modest worth.” “I am in the habit of giving far more than I get, so fear not, but suffer me to proceed. In my youth I was called El Amin—‘the Safe Man.’ I know that I possess a generous soul, and that none can be more loyal than you.” “This eulogy is evidently the prologue of some treason you are going to ask of me.” “If it be treason to leave a bad cause for a good one, to renounce attempts which are futile, and to accept good fortune when it is offered, I have, in effect, treason to propose to you.”
- 26. “By the Trinity! but you are putting a high price on compliments for which nobody asked you!” “I swear by the holy mountain—by the temple of pilgrimage—by the vault of heaven and the depths of ocean—that the divine vengeance is about to fall! nothing can delay it. The convulsed skies shall totter! the uprooted mountains shall move! I swear by the resting-place of the star—” “Of a truth, here is plenty of fine words!” said Roland, shrugging his shoulders. “When we gallant Christian knights make a statement, they believe us without our having to call in the aid of the sky, and sea, and stars.” “As surely as I overthrew the three idols of Mecca, Lata, Aloza, and Menât, the Christians shall be driven from Spain, and their lands invaded. Their army shall be dispersed, and shall fly shamefully. Their hour is come, and it will be bitter and terrible.” “I have read all that in the Koran,” answered Roland, who felt his patience failing him. “But that does not say what you want of me, or why you are thus wasting my time. Since the future is revealed to you, and you are so certain of our approaching overthrow, there can be no obstacle to my returning to my post.” “Yes, the future is ours. You alone delay the coming of the day of glory. We shall conquer, but while you live it will be only at the price of terrible sacrifices that we can purchase victory. Why persist in returning to a world in which death awaits you? I offer you the sovereignty of this realm, its wealth, its women, its warriors. The inhabitants of air, earth, and water, the stars which move in the firmament—all that is gifted with reason or instinct, essence and matter—in one word, everything shall belong to you and owe to you unreserved obedience. If the sun annoys you, the moon shall take its place. Give but the sign, and rivers shall dry up to let you pass. A population more vast than all the nations of earth put together shall live only to serve you. These warriors are brave.” “Of what use is their bravery if they have no enemies to contend with?”
- 27. “These horses are more swift than the wind.” “Of what service is their speed, since there is here no goal that I desire to reach?” “These women are lovely.” “Their beauty is sheer waste, for I do not love them!” “Durandal is famous on earth, and yet the humblest of these soldiers could cut it in two with the edge of his poniard.” “Enough!” interposed Roland. “I have already told you I am in a hurry. You have not, I imagine, the impudence to suppose you are rich enough in wonders to induce me to commit a base action—your Allah himself would be ashamed of such a thing. You have told me I am the bravest of living knights: should I be so if I feared the death you threaten me with? ‘My single arm is worth a whole army,’ you add. Have I any right, then, to deprive my comrades of its aid at that moment, of all others, when you profess that they are in danger? ‘My judgment is sound:’ allow me to offer you a further proof of it by laughing at your menaces, and predicting your complete overthrow. Mahomet and Jupiter will soon meet and shake hands, and the crescent will be sent where the old moons go.” “You will not listen?” “I have heard too much already!” “Behold these lovely creatures, who stretch out their arms towards you! “They but make me see how far lovelier my Aude is.” “See the lands I offer you!” “What is a region of wonders compared with the spot where a man was born?” “Roland, by the faith of Mahomet! you shall never again behold the land of France!” “I am a Christian, besides being a Frenchman. The native land to which I aspire is Heaven, and that birthplace you cannot prevent me from beholding once more.”
- 28. “Infidel hound!” said the Prophet, “I——” But the words were such as Roland could not listen to patiently. Mahomet did not finish his sentence, for the gauntlet of the knight smote him on the mouth. Original Size -- Medium-Size Original Size -- Medium-Size
- 29. I CHAPTER X. WHEN ROLAND REMEMBERS HIS LATIN, AND THE DEVIL FORGETS HIS. AM unable to tell you what followed. Even Roland had no clear recollection. When he recovered his senses, he rose and cast his eyes round him, to find himself in the midst of a vast sandy plain, stretching on all sides to the horizon. The sun poured its hostile rays upon him so fiercely, that in a few minutes his armour became insupportably hot. The atmosphere was so charged with electricity, that the plume of his helmet crackled, and gave out sparks. In vain he searched the horizon for a place of shelter—there was nothing to be seen but level plain and blue sky. Gigantic red ants came and went busily—they were the only occupants of this desert. All of a sudden he beheld before him in the distance white mosques, knots of palms, and a sea-port with some vessels at anchor, and others sailing out of the harbour. He saw, too, long caravans, which journeyed to the city gates. Roland felt his courage revive, and set out in the direction of the city. But he did not appear to come any the closer to it; he took to running until he fell down with fatigue on the burning sand. Then the city seemed to turn of a yellow hue, the blue of the sea grew paler, and was lost in that of the sky; the trees vanished, and the Count of Mans found himself once more alone in the desert. “Why come to a halt?” said he to himself. “Better move forward in any direction at hap-hazard. I can only gain by the change.” He rose, determined to struggle on as long as his limbs would sustain him. What was his surprise to see, in an opposite direction to that he had just been pursuing, a mountain covered with verdure, on the summit of which stood a castle! Three walls of
- 30. circumvallation surrounded it. At the foot of each flowed a river covered with vessels of war. Three hanging ladders of marvellous workmanship united the three platforms of the fortress, and four bastions guarded the approach to each ladder. Original Size -- Medium-Size Roland once more pushed on; but as he advanced, the fortress rose into the skies, until, after about an hour’s walking, he found himself with nothing before him save the blank horizon of the desert. Then despair seized him. He sank on his knees, crossed himself, and shed four tears, the first he had ever wept. They fell on the sand, and there formed four springs for a stream of cool and clear water. Roland received from this new vigour, and having rendered thanks to Providence, he was preparing to move forward, when he remarked with surprise a great stirring of the sand. Little
- 31. clouds of dust began to rise in all directions, although there was not a breath stirring. Then the sand began to whirl round incessantly, marking a great circle at a short distance from our hero. As it began to whirl, it heaped itself up, drawn towards the centre by some strange force of attraction. You would have said that some gigantic polypus was sucking up all the sand of the desert. After a few minutes there mounted, still eddying round, a huge column, which grew as Roland watched it, until the summit was lost to sight in the sky. A hot wind, like the harmattan of the Guinea coast, rose and drove the sand before it in clouds. The sun turned red as molten iron. The pillar of sand at last lost its equilibrium, and fell with a horrible rushing sound. Roland closed his eyes, but he did not recoil. Hearing a great roar of laughter, he instinctively clutched his sword by the hilt. What he saw next induced him to draw it from its sheath. The sand, in falling, had reared a mound, the base of which formed an enormous circle, in the centre of which Roland perceived, with surprise, a huge monster buried in sand to his waist. It was Eblis, the Devil of the East. Original Size -- Medium-Size His Majesty was a hundred feet in height, which is a respectable size, even for a demon of the highest rank. His black skin, striped with red, was covered with small scales, which made it glisten like armour. His hair was so long and curly, a snake might have lost its way in it. His flat nose was pierced with a ring of admirable workmanship, as you see done to the wild bulls of the Roman Campagna. His white teeth, set with precious stones, gave to his
- 32. smile a very variegated appearance. His small eyes assumed, one after the other, all the prismatic colours, which made it impossible to sustain his gaze. His ears, which exactly resembled those of an elephant, flapped on his shoulders; but he had, to make up for it, a tail sixty feet long, terminating in a hooked claw, which could have wielded the Monument easily as a toothpick. Eblis had no other covering than his wings, which were large, soft, and marvellously pliable, and in which he delighted to wrap himself. Conceive, further, that a phosphorescent gleam played incessantly over the monster’s skin, and you will easily understand why Roland unsheathed Durandal. Eblis was writhing with laughter. “I haven’t roared so through all eternity, upon my honour! Here, I say, my little man, do you know you have just done a master- stroke?” This familiar tone displeased Roland. “I have just met Mahomet,” continued Eblis, “and you have broken five of his front teeth. I have seen a good many prophets in my time, but I vow, on the faith of the accursed, I never saw one in such a rage. I have, in honour of the blow, given three days’ holiday in the infernal regions. There will be concerts, balls, hunts, and theatres. I have had written, by one of our best authors, a little comedy in the style of Apollodorus, in the last scene of which Mahomet receives a hundred strokes of the bastinado. I have given orders to an army of cooks; you can hear even here a rattle of stew- pans altogether refreshing. I will undertake to let you see we are not so backward in this respect as people pretend. You will meet with many old friends among the guests; we have quite a crowd of visitors just now. My wife, who is a lively one, will be delighted to make your acquaintance. Come, let me present you to her as the best of my friends.” “Babbler!” exclaimed Roland, but little flattered at these marks of friendship. “What right have you to address me in this style?”
- 33. Eblis, who was not accustomed to be treated so cavalierly, was dumb with surprise for a moment. “By my father’s horns!” said he, at last, “I must have misunderstood you. Give me your hand, Roland, to disabuse me of the error.” He stretched out his tail to the knight, who, however, only drew back a few steps. “What, puny wretch!” shrieked Eblis, turning as white with rage as it was possible for one so black to do. “I shall send you back to earth. Do you think I am of the same stuff as Mahomet?” “But here Roland flung his second gauntlet in the demon’s face. “That makes the pair!” said the nephew of Charlemagne, placing himself in an attitude of defence. “Zacoum Zimzim Galarabak!” shouted Eblis, mad with fury. (You must know that is the most terrible oath that can be uttered in the Saracen tongue.) The earth shook and gaped at Roland’s feet. He felt himself launched into space. His armour suddenly became icy cold. “If I get back without an attack of rheumatism I shall be lucky,” said the knight. He heard around him the flapping of wings; it was a troop of afreets and djins. “Reflect, Roland. There is yet time. Mahomet is prepared to forgive you.” All the answer Roland vouchsafed was the intoning of the canticle — “Sub tuum Fræsidium confugimus.” “In a few moments your body will be dashed to pieces on earth. Remember the wondrous things the Prophet offered to share with you.” “Sancta Dei genitrix; nostras deprecationes ne despicias,” continued Roland. And now it seemed to him that, instead of falling
- 34. at hazard, he was being gently carried. The chorus of afreets and djins was left far behind, but he still heard the sound of pinions. Original Size -- Medium-Size “Set your mind at rest,” said a voice so exquisitely musical that Roland trembled to hear it. “I am the Archangel Michael. Our Blessed Lady has sent me to preserve you. She had been touched by your constancy and courage. Repose in safety on my wings, and we shall soon reach earth.” And, in truth, in a few minutes’ time the Count of Mans, to his astonishment, found himself before Saragossa. He was at prayer in his tent when he heard the voice of Miton. “My dear Roland, where are you?” cried the Count of Rennes, anxiously. “Here I am,” said the knight, hurrying to his friend.
- 35. Original Size -- Medium-Size “Charlemagne, who knows how punctual you are, seeing you were ten minutes behind your time to take on your guard, has sent to look for you in every direction. You are pale, my dear Count; what has happened to you?” “I will tell you all about it,” said Roland, as he hastened to his post near the Emperor. END OF THE SECOND BOOK.
- 36. BOOK THE THIRD — THE FORTRESS OF FEAR. Original Size -- Medium-Size
- 37. C CHAPTER I. THE FOUR FOES OF CROQUEMITAINE. HARLEMAGNE had an excellent memory. He never omitted to ponder over the dangers to which Mitaine was exposed at every turn. He had the scene of the late ambush carefully searched by his spies in the first place, and afterwards by his soldiers. All, on their return, made the same report. They said the forest was inhabited, and there was a good deal of talk about a castle called “The Fortress of Fear,” which was to be found somewhere in the neighbourhood, although nobody they met with had seen it. None, however, doubted its existence. If a child disappeared, or any cattle were carried off, the trembling peasants said, “The Lord of Fear-fortress had taken them.” If a fire broke out anywhere, it was the Lord of Fear-fortress who must have lit it. The origin of all accidents, mishaps, catastrophes, or disasters was traced to the mysterious owner of this invisible castle. “I should like to have the mystery cleared up,” said Charlemagne to himself. “I can hardly resign myself to the belief that it is Ganelon, my old brother-in-arms.” He called his knights together. “My faithful champions, I need four of you for a perilous adventure, I know not where I am sending you—I know not whether you will return. Who will risk death for my good favour?” All the knights at once flung themselves at his feet, each entreating the Emperor to honour him with his choice. “You place me in a difficult position,” said the Emperor, greatly moved; “I see that chance must point out the four champions. I can without fear trust to it, for you are all equally brave.”
- 38. The names of all the knights present were put into a helmet, and Mitaine played the part of Destiny to the best of her power, little thinking she was choosing her own champions and avengers. The first name she called out was that of Allegrignac of Cognac, Count of Salençon and Saintonge. “The lot suits me admirably,” said the Emperor, giving a friendly wave of his hand to the knight. “You know the language of the country, and will be a safe guide for your companions.” Mitaine next named the Baron of Mont-Rognon, Lord of Bourglastic, Tortebesse, and elsewhere. “This is indeed a capital choice! There is no stouter arm in the Arvennes than yours; and if there be a postern to be burst open by a powerful shoulder, you will be there, Mont-Rognon.” “Porc-en-Truie, Lord of Machavoine,” cried Mitaine. “I am in luck to-day, by St. James! You are known to be experienced, Porc-en-Truie, and you will conduct the adventure, I entrust to you, to a prosperous end, I feel sure. But I am curious to know who is my fourth champion.” “Maragougnia, Count of Rioin,” said Mitaine. “Now we have wisdom, strength, and cunning. Maragougnia can give the serpent points at wisdom, and beat him. If I do not succeed with such knights I shall despair altogether.” Charlemagne withdrew with his four champions, told them of the perils to which his god-child had been exposed, the investigation he had instituted, the suspicions he had entertained; and finally, he spoke of the Fortress of Fear, winding up in these terms:— “I am anxious to square accounts with this Croquemitaine. You will pass through the forest till you arrive at Alagon, a little hamlet on the banks of the Ebro. There you will inquire for the Fonda del Caïman, or, if you prefer it, the sign of the Crocodile. You will there rest yourselves for a short time, and then set out on your quests. You, Allegrignac, striking off from the river, will pursue your course towards Pampeluna. You, Mont-Rognon, will proceed in the direction of Catalyud; and look out for the Saracens, my friend, who on that
- 39. side are disgusted enough with the trouble we have given them. You, Porc-en-Truie, will make for Fuentes. If you are guided by me, you will travel by night only, and conceal yourself carefully by day. You will appreciate my counsel when once you are on the road. You, finally, my gallant Maragougnia, will have to direct your steps towards Lerida, but you will not go beyond the river Alcander. I have reserved this expedition for you because it is the most hazardous— there, you need not thank me. I understand you! Quarter the country in every direction, and find out for me this Fortress of Fear. He who brings me the head of its dreaded lord shall be created a baron and peer of my realm.” The Emperor replenished the purses of his champions, and took leave of them with an embrace. When they’ found themselves alone they interchanged looks of bewilderment. “What do you think of that?” said Porc-en-Truie, with a grimace. “That I shall be a duke,” said Allegrignac, cutting a caper. “This adventure won’t take me a minute!” “To think that we must set out to-night!” said Mont-Rognon, in tones of regret; “and to think that I have ordered a splendid supper for to-night, which my fellows will get the benefit of!” “To think that we shall none of us ever come back again!” said Maragougnia, in a melancholy voice, as he wiped away a tear with the sleeve of his chain-mail. “Pshaw! who knows?” broke in Porc-en-Truie, with a smile. “Let us set out, and then we can see!” They appointed to meet on the borders of the forest, and within an hour afterwards they’ were all on the spot, equipped for war or for travel. Porc-en-Truie, Lord of Machavoine, was a great fellow of thirty years of age, more skilled in avoiding blows than in dealing them. He invariably shirked all his military duties, not because he was a coward, but because he was incorrigibly idle. He had been known to tramp three hours afoot to save himself the trouble of saddling his horse, and he had killed his dearest friend in a tournament, in order
- 40. to terminate a long and fatiguing tilting match. He arrived at the rendezvous on horseback, with no weapon but his sword. “How imprudent!” cried Allegrignac, the moment he saw him coming. “Are we going to a wedding only, or are you desirous of emulating Miton’s great feat at the Tourney of Fronsac?” “I hate a load of weapons, and I don’t mean to kill myself for this Mitaine—for whom, between you and me, I don’t care a grain of mustard-seed!” Original Size -- Medium-Size Allegrignac of Cognac, Count of Salençon, was twenty-five years of age, and six feet six high. He had an open countenance, a stout heart, an untiring tongue, limbs of steel, a stomach of leather, and a very slender patrimony. His hair was curly, his teeth were white. He was as proud as a Spaniard, as brave as a Frenchman, as simple- minded as a goose. He was possessed of a pleasant contralto voice, a cheerful spirit, and a grey horse called Serenade. Picture to yourself a figure clad in complete steel, and with weapons of vast weight, like one of those armed and bandy-legged giants you see in a procession of trades, capable of lifting enormous weights, not to mention cattle, and any other unconsidered trifles he could lay hands on, and you have a portrait of the Baron of Mont- Rognon, Lord of Bourglastic, Tortebesse, and elsewhere. This huge
- 41. mass of muscle existed only to eat and drink. He was a descendant of Esau on his father’s side, and of Gargantua on his mother’s. He once performed a gigantic feat—he killed six hundred Saracens who happened to get in his way as he was going to dinner. He had an elastic stomach, and a mouth armed with four rows of teeth. Having described his stomach and his mouth, I need not go on with the likeness, for all that remained were mere incidental appurtenances. He arrived third at the place of meeting, leading by the halter a mule laden with provisions and bottles. “What’s this?” said Allegrignac, laughingly. “That!” said Mont-Rognon, offended at his bluntness. “That’s supper.” “What’s the use of that?” said Porc-en-Truie. Mont-Rognon the Monstrous. Mont-Rognon in a hurry for his dinner “Charlemagne has ordered us to perish for him,” broke in the Lord of Bourglastic, “but he did not stipulate that we should perish of hunger.”
- 42. Original Size -- Medium-Size Maragougnia, Count of Riom, was the last to arrive. He was equipped in the most gloomy style. His armour was of browned steel, sprinkled with silver tears. From the coronet that surmounted his helmet sprang a few mangy black feathers, which drooped over his shoulders like the branches of a weeping willow, and all the rest of his accoutrements were to match. He had one extraordinary quality, which was his strong point— instead of making him lose his head, fear only gave him increased presence of mind. They related deeds of prowess of his which were, in reality, only prodigies of cowardice. He did everything with a profound air of melancholy. His first wife, they say, died of yawning; the second perished of sheer weariness in three weeks. Behind him came a page, who might be considered to have originated the sombre livery worn nine hundred years later by the page of the Duchess of Marlborough.* * Vide “Malbrouck:”— “Elle voit venir son page De unir tout habillé.”
- 43. This lugubrious squire bore the count’s change of arms—to wit: two daggers of mercy; three swords, various; one lance; one helmet; one morion; two daggers, poisoned; one battle-axe; one flail, iron; one shield; one breastplate; one shirt of mail; two pairs of gauntlets; three pairs of spurs. “Good heavens!” said Allegrignac; “are we going to equip all the nation for war? Look, Porc-en-Truie! the Count of Riom has stripped the armouries of his ten castles.” “I wouldn’t stir an inch,” said Porc-en-Truie, in the interval of a couple of yawns, “to assure myself that Maragougnia has done something silly. If you assured me to the contrary, I might perhaps be surprised into getting up to see. And yet no! I couldn’t believe it; so I should stay where I was.” Porc-en-Truie, I must observe, sat himself down on the grass the moment he arrived. “You’re quite welcome to laugh at my prudence,” said Maragougnia, “but I don’t forget we are going to certain death.” “Certain death! Fiddlesticks! I mean yet to rival the Methusalems of the period,” said Porc-en-Truie, rising. “And now let’s be off, if we are to reach Alagon to-night.” “To prepare for death,” said Maragougnia, dashing away a tear with his gauntlet. “To go to sleep,” said Porc-en-Truie, with a yawn. “To try a throw with the dice,” said Allegrignac, jingling the money in his purse. “To make a good supper,” said Mont-Rognon, with a hollow voice, gnashing his teeth like castanets. In ten minutes the four knights had entered the wood. At sunset Alleericmac was hammering with his fist at the door of the Fonda del Caïman.
- 44. Original Size -- Medium-Size Original Size -- Medium-Size
- 45. T CHAPTER II. THE SIGN OF THE CROCODILE. HE innkeeper was a man of middle size, half Spaniard and half Moor, with a big body and thin leys, a brown skin and grey eyes. He had acquired considerable reputation in the district for his mode of dressing calves’ feet with saffron, and his handiness in stabbing people in the right place. He made everything a matter of trade, and used to regret that he had inherited no religious opinions which he could have abjured at a fixed price to be got either from the Saracens or the Christians. For the rest, he was a most obliging host, provided your purse was well supplied; and I believe I shall put the finishing stroke to the likeness when I say he was the biggest robber in all Spain, from Pontevedra to Girone. Ali Pépé opened the door. One is always forgetting something, and I forgot to tell you his name was Ali Pépé. “Where’s the landlady?” asked Allegrignac, twisting his moustache. “I want a bed,” yawned Porc-en-Truie. “Some supper!” growled Mont-Rognon. Maragougnia said nothing. He was absorbed in studying the inn, and the estimate he formed seemed far from satisfactory. Ali Pépé stood on the defensive, blocking the entrance of the inn. “Your lordships appear of too exalted a station for me to omit to inform you that you will find the accommodation here very unsuited to you.” “Here’s frankness and disinterestedness! But where can we find better accommodation?”
- 46. Original Size -- Medium-Size “My inn is the only one in the district.” “Then make way for us,” said Mont-Rognon, catching up Ali Pope by the girdle, and carrying him in at arm’s length into the kitchen. “We shall be able to converse better here!” Maragougnia entered last. He tried all the locks, in order to see whether the doors closed securely. He examined all the outlets, sounded the panels, and ordered his squire to bring him his arms. “We want four beds,” said Porc-en-Truie. “In the same room,” said Maragougnia, who had a horror of being solitary. “First of all we want supper,” bellowed Mont-Rognon; “don’t let us forget the most important of our wants.” “A modest supper,” suggested Maragougnia, who was afraid of the expense. “A modest supper!” bellowed the Lord of Bourglastic. “Don’t you do anything of the kind, landlord, or I’ll burn the place about your ears. Empty your poultry-yard, drag your fish-ponds, uncork your bottles; set to work—kill, pluck, draw, and broach,—in short, make ready, to the best of your power, a feast for an emperor or a sultan!” “You will lay for me separately,” said the Count of Riom, tearfully, “a few radishes and some wine of first-rate——”
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