# Ordinal methods for a characterization of evolving functional brain networks

Klaus Lehnertz<EMAIL_ADDRESS>Department of Epileptology,
University of Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany
Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn,
Nussallee 14–16, 53115 Bonn, Germany  Interdisciplinary Center for Complex
Systems, University of Bonn, Brühler Straße 7, 53175 Bonn, Germany

###### Abstract

Ordinal time series analysis is based on the idea to map time series to
ordinal patterns, i.e., order relations between the values of a time series
and not the values themselves, as introduced in 2002 by C. Bandt and B. Pompe.
Despite a resulting loss of information, this approach captures meaningful
information about the temporal structure of the underlying system dynamics as
well as about properties of interactions between coupled systems. This –
together with its conceptual simplicity and robustness against measurement
noise – makes ordinal time series analysis well suited to improve
characterization of the still poorly understood spatial–temporal dynamics of
the human brain. This minireview briefly summarizes the state-of-the-art of
uni- and bivariate ordinal time-series-analysis techniques together with
applications in the neurosciences. It will highlight current limitations to
stimulate further developments which would be necessary to advance
characterization of evolving functional brain networks.

> Deriving evolving functional brain networks from observed, long-lasting,
> multivariate time series to improve characterization of various
> physiological and pathophysiological brain dynamics requires suitable and
> robust time-series-analysis techniques, that are capable of deciphering the
> multifaceted nature of the brain’s complex endogenous and exogenous
> interactions. I will recapitulate concepts of ordinal time series analysis,
> showcase its applications in the neurosciences, and will discuss limitations
> and necessary developments to improve characterization of the complex
> networked dynamics system human brain.

##  I Introduction

Ordinal time series analysis is a special type of symbolic analysis [1, 2]
which makes use of symbols that are ordinal patterns (also referred to as
order patterns or permutation patterns) of length of at least 2. C. Bandt and
B. Pompe introduced this concept in 2002 in their seminal paper [3] together
with permutation entropy as a natural complexity measure of time series. Let
$x_{i}=x(i),i=1,\ldots,N$, denote a sequence of observations (or time series)
from some system $X$. For a given, but otherwise arbitrary $i$, $m$ amplitude
values $X_{i}=\\{x(i),x(i+l),\ldots,x(i+(m-1)l)\\}$ are arranged in an
ascending order $\\{x(i+(k_{i1}-1)l)\leq x(i+(k_{i2}-1)l)\leq\ldots\leq
x(i+(k_{im}-1)l)\\}$, where $l$ and $m$ denote the appropriately chosen [4, 5,
6] time delay and embedding dimension (cf. Takens’ embedding theorem [7, 8]).
In case of equal amplitude values, one can e.g. carry out the rearrangement
according to the associated index $k$, i.e., for
$x(i+(k_{i1}-1)l)=x(i+(k_{i2}-1)l)$ one can write $x(i+(k_{i1}-1)l)\leq
x(i+(k_{i2}-1)l)$ if $k_{i1}<k_{i2}$ thereby ensuring that every $X_{i}$ is
uniquely mapped onto one of the $m!$ possible permutations. A permutation
symbol – or ordinal pattern – is then defined as
$s_{i}\equiv(k_{i1},k_{i2},\ldots,k_{im})$ and captures qualitative
information about the temporal structure of the underlying time series (see
Fig 1).

Figure 1: Possible outcomes for ordinal patterns (permutation symbols) $s$
using embedding dimensions $m=2$ and $m=3$ and fixed embedding delay $l=1$.

Ordinal time series analysis is conceptually simple, computationally fast and
comparably robust against measurement noise. Compared to other symbolization
techniques [2], the derivation of ordinal patterns does not require a priori
knowledge about the data range, which rendered ordinal time series analysis
beneficial for investigations of empirical data from various scientific
domains [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. A large proportion of
studies was concerned with problems such as distinguishing chaos from noise,
improving the detection of determinism [20, 21, 22] or of dynamical changes
[23], system identification [24], or quantifying time reversibility [25],
thereby employing ordinal-pattern-derived quantifier for entropy [16, 17],
complexity [12], or combinations thereof [26].

In this minireview, we will concentrate on ordinal time-series-analysis
techniques that aim at characterizing properties of interactions – strength,
direction, and coupling function –, since these currently form the basis of
complex-network-based studies in diverse scientific fields including
geophysics, meteorology, and the neurosciences [27, 28, 29, 30, 31, 32, 33,
34]. With this ansatz, one assumes that a spatially extended complex system
can be represented by a complex network which, however, requires
identification of vertices and edges. In many cases, such an identification is
straightforward, but it remains a challenging issue when investigating the
system’s dynamics [35, 36, 37, 38, 39, 40, 41, 42]. Network vertices are
usually assumed to represent distinct subsystems and edges represent
interactions between them, and these vertices and edges constitute a
functional (or interaction) network. In an evolving functional network,
properties of edges (and/or vertices) are time-dependent [43]. In case that a
direct access to interactions and their time-dependencies is not possible
(e.g. via probing), one usually resorts to linear and non-linear time-series-
analysis techniques to quantify interaction properties from pairs of time
series of appropriate system observables. These techniques originate from
diverse fields such as statistics, synchronization theory, non-linear
dynamics, information theory, statistical physics, and from the theory of
stochastic processes [44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56],
given that interactions can manifest themselves in various aspects of the
dynamics. While the majority of studies on (evolving) functional networks is
based on binary (an edge exists or not) or weighted networks (the weight of an
edge is given by the strength of interaction), further improvements can be
expected by considering weighted and directed networks, thereby including
knowledge about coupling functions that contain detailed information about the
functional mechanisms underlying an interaction and that prescribe the
physical rule specifying how an interactions occurs [57].

##  II Ordinal Methods for a characterization of interactions

Current bivariate ordinal time-series-analysis techniques allow
characterization of strength and direction of interactions; whether some of
these techniques also allow for a characterization of coupling functions needs
further investigations.

Estimators for the strength of interactions center around the phenomenon of
synchronization and its various forms of appearance – from complete via phase
and lag synchronization to generalized synchronization [44, 58]). In case of
generalized synchronization, the relationship between two empirical time
series can be characterized by an order parameter that is based on the
consistent changing tendency of their permutation entropies [59] (see Fig. 2).
The value of the order parameter can be used to assess the strength of an
interaction. In case of phase synchronization, the latter can be assessed with
a metric (so-called ordinal synchronization) that is based on the dot product
between two ordinal vectors [60]. Analysis techniques based on transcripts
[61] as well as those based on conceptual extensions (so-called coupling
complexity) [62, 63] allow one to detect and characterize various forms of
synchronization and the strength of an interaction can be assessed with
different derived estimators.

Figure 2:  Schematic of time-resolved ordinal-pattern-based estimation of
direction of interactions. Sequences of permutation symbols (here: $A,B,C$,
and $D$) derived from successive segments of time series (moving-window
technique). Direction of interactions can be quantified by estimating the
differential flow of information between symbol sequences (from sequence 1 to
sequence 2 and vice versa) via transition probabilities $p$ between current
and past states (black arrows).

The majority of estimators for the direction of interactions are based on the
information-theoretic functional conditional mutual information [64, 65] and
need to be defined asymmetrically to allow detection of a directed flow of
information (see Fig. 2). Among these estimators are directionality indices
based on the so-called PI approach [66], on symbolic transfer entropy [67, 68,
69, 70, 71], on momentary information transfer [72], on transcripts [73, 74],
on closeness mapping [75], on the ratio of the number of missing joint ordinal
patterns [76], or on joined symbolic recurrences [77]. Some of these
estimators take into account coupling delays [72, 70], which is of importance
as it allows for improved physical interpretations. There are also extensions
that enable the time-resolved investigation of directional relationships
between coupled dynamical systems from short and transient noisy time series
[71]. With respect to complex-network-based studies, an estimator for the
direction of interactions should allow for distinguishing direct from indirect
directional couplings, as this is a key to avoid severe misinterpretations of
possible causal relationships. So far, only one of the aforementioned
approaches (symbolic transfer entropy) has been extended using partialization
analysis to allow for such a distinction [78, 79]. It remains to be
investigated, however, whether these extensions suffer from limitations
similar to the ones identified for other partialized estimators for the
direction of interactions when investigating interactions in larger networks
(number of nodes $\gg 10$) [80].

I briefly mention complementary approach from the field of time series
networks [81], i.e., a transformation of a time series into the complex
network domain, namely transition networks derived from ordinal patterns as
well as cross and joint ordinal transition networks. These networks can then
be investigated further to characterize strength and direction of interactions
[82, 83, 81, 84, 85]. There are, however, some issues that would need to be
fully resolved to allow judging general applicability of these approaches.

##  III Towards characterizing evolving functional brain networks with ordinal
methods

Ordinal methods appear to be ideally suited to improve characterization of the
complex time-varying dynamics of the human brain [86] given its contrasting
and, at times, complicated forms of appearance: oscillations at a variety of
frequencies [87] coexisting with scale-free dynamics ($1/f^{\alpha}$-like
power spectrum (with $\alpha\in\left[1,3\right]$) at many spatial-temporal
scales [88, 89, 90, 91, 92, 93]). Indeed, a number of studies provided
evidence for univariate ordinal time-series-analysis techniques to detect
spatial-temporal changes of EEG data related to epileptic activity in a
quantitative and efficient way that may provide helpful information for
diagnostic and therapeutic purposes [94, 23, 95, 96, 97, 98, 99, 100, 101,
102, 103, 104]. Similar observations were made for other neurological and
neurodevelopmental disorders such as traumatic brain injury [105], mild
cognitive impairment and Alzheimer’s disease [106], Parkinson’s disease [107]
and attention-deficit/hyperactivity disorder [108], and all these disorders
impose a high individual, medical, psychosocial and socioeconomic burden for
those affected. The techniques were also shown to allow differentiating sleep
states [109, 101, 110, 111, 112] and various awake states (including different
mental activities) [113, 114, 115], brain dynamics during resting states
related to different age groups [116], as well as brain states related to
anesthesia [117, 118, 119, 120].

Regardless of the success of univariate ordinal time-series-analysis
techniques, one should keep in mind that the majority of physiological and
pathophysiological changes in brain dynamics investigated so far are
accompanied by distinct modifications of the respective brain dynamics’
frequency content which – in many cases – is visible to the naked eye. Future
studies should thus demonstrate advantages of univariate ordinal time-series-
analysis techniques over classical techniques [121, 122], also to increase
their acceptance for clinical applications.

In contrast to the aforementioned rather large number of possible applications
of univariate ordinal time-series-analysis techniques, there are so far only a
few studies that employed bivariate ordinal techniques to characterize
interactions between various pairs of brain regions. Although mainly
restricted to multichannel recordings from subjects with epilepsy, techniques
allowed a comparably thorough characterization of strength [59] and direction
[67] of short- to long-ranged interactions between structurally identical and
nonidentical coupled but not yet fully synchronized brain regions – covering
lobes and hemispheres – from time-resolved analyses of multiday recordings
that captured a large variety of physiological and pathophysiological brain
states [123, 4, 67, 68, 70, 124]. These investigations also provided evidence
for bivariate ordinal time-series-analysis techniques to allow an improved
characterization of brain interaction dynamics that can be regarded predictive
of an impeding epileptic seizure [124], a prerequisite for the development of
refined seizure prevention or control techniques [125]. Nevertheless, these
studies also identified potential limitations if confounding variables such as
delayed interactions, asymmetric signal-to-noise ratios, number of interacting
systems or connection densities are not taken into account. In general, these
studies suggested to estimate both strength and direction of interactions in
order to effectively distinguish various coupling regimes (uncoupled, weak to
strong couplings) and to avoid misinterpretations when investigating
directional interactions between complex dynamical systems [124].

Bivariate ordinal time-series-analysis techniques were also shown to reliably
detect and characterize changes of transient (in the order of a few 100
milliseconds) directional interactions between brain regions associated with
cognitive control [71] as well as to identify topographical reorganizations of
interactions between brain regions related to coma [126], stroke [127], or
anesthesia-induced unconsciousness [128].

By now, investigations of evolving functional brain networks employing
bivariate ordinal time-series-analysis techniques are rare and exclusively
related to epilepsy [129, 130]. A highly time-resolved investigation of
importance of vertices in evolving functional brain networks during a large
number of seizures identified “hub”-like brain regions that appear to be of
relevance for the complex spatial-temporal spreading dynamics [129].
Interestingly, these brain regions only rarely coincided with the clinically
defined seizure onset zone, which calls for revisiting the role of the latter
in seizure generation [131, 130, 132, 133]. Eventually, a comparative study
employed bivariate ordinal [59, 67] and phase-synchronization-based [134, 135]
time-series-analysis techniques to characterize weighted and directed
interactions in functional brain networks – that evolve over several days –
from a large group of subjects with epilepsy [130]. On a population-sample
level and despite the heterogeneity of investigated cases, both approaches
appeared to provide comparable information about the network characteristics.
On the level of individual subjects, however, the approaches provided largely
independent, non-redundant information but with a varying contrast. This can
probably be related to the various concepts (different synchronization
phenomena, information flow) from which time-series-analysis techniques were
derived.

##  IV The next steps

Summarizing the state-of-the-art of bivariate ordinal time-series-analysis
techniques in use to characterize complex interactions, it can be noted that
there are more estimators for the direction of an interaction than for the
strength, notwithstanding the coupling function. While estimators for the
strength of interactions were mainly designed to capture the various forms of
synchronization, the majority of estimators for the direction are based on the
information-theoretic functional conditional mutual information, for which
many approximations are available. Given this imbalance, many studies in the
neurosciences (as well as in other disciplines) employed ordinal time-series-
analysis techniques to either estimate the direction or the strength of
interactions, or estimated both but by employing ordinal techniques that were
derived from different concepts. In the latter case, it is important to note
that the techniques’ efficiency may be influenced differently by a number of
confounding factors: volume conduction effects [136, 137], propagation delays
and delayed couplings [138, 70, 139], asymmetric signal-to-noise ratios [140,
141, 142] or eigenfrequency ratios [141, 143] (in case of oscillating
(sub)systems), peculiarities of the recording [144, 145, 146, 147, 148, 149,
150], pre-processing steps such as filtering [151, 152], the techniques’
capability to distinguish between (apparent) interdependencies due to common
sources and (true) interdependencies due to interacting (sub)systems [141,
147], the techniques’ capability to distinguish between direct and indirect
interactions [153, 154, 80, 155], or the techniques’ different sensitivities
for the various types of synchronization phenomena, to name just a few. Many
confounding factors can be identified by investigating e.g. coupled
paradigmatic model systems with well-known properties. The impact of such
factors can be minimized by improving the techniques’ robustness and by using
surrogate concepts [156, 157, 158], but these are rarely used in the context
of ordinal time-series-analysis techniques. More importantly, we still lack
reliable surrogate concepts to probe for the direction of interactions.

An exception to the aforementioned conceptual mixture are transcripts-based
estimators for strength and direction of interactions [61, 73], which share a
common conceptual base. So far, their general applicability and particularly
for the neurosciences has been demonstrated only by way of example, and their
susceptibility to the aforementioned (and other) confounding factors remains
to be elucidated.

Future extensions and/or improvements of estimators for strength and direction
of pairwise and higher-order[159] interactions, of coupling functions as well
as of concepts underlying ordinal methods can be expected to further increase
capability of ordinal time-series-analysis techniques to investigate time-
evolving networks – in the neurosciences as well as in other scientific
domains concerned with networked dynamical systems.

##  V Conclusions

In this paper, I have briefly reviewed the state-of-the-art of univariate and
bivariate ordinal time-series-analysis techniques thereby focusing on
applications in the neurosciences. I also discussed current limitations to
stimulate further developments which would be necessary to advance
characterization of evolving functional brain networks during both
physiological and pathophysiological conditions. Ordinal time series analysis
carries the potential to improve characterization of the still poorly
understood spatial–temporal dynamics of the human brain.

## Acknowledgements

In memory of our esteemed colleague Karsten Keller, who unexpectedly passed
away in April 2022.

## Data availability

Data sharing is not applicable to this article as no new data were created or
analyzed in this study.

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# On the numerical solution of stochastic oscillators driven by time-varying
and random forces

R. D’Ambrosio1 and C. Scalone1

###### Abstract.

In this work, we provide a specific trigonometric stochastic numerical method
for linear oscillators with high constant frequencies, driven by a nonlinear
time-varying force and a random force. We present some theoretical
considerations and numerical experiments on popular related physical models.

Keywords: Stochastic differential equations; stochastic oscillators; periodic
time varying force; Filon’s quadrature.

AMS Subject Classification: 65L07, 60H10, 60H35.

1 Department of Engineering and Computer Science and Mathematics, University
of L’Aquila
Via Vetoio, Loc. Coppito - 67100 L’Aquila, Italy,
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Manuscript received xx

##  1\. Introduction

Providing accurate numerical methods for discretizing mathematical models of
oscillating phenomena without imposing severe stepsize restrictions is a very
thorough topic. Scientific literature is extremely rich in models describing
the dynamics of different types of oscillators, both in deterministic and
stochastic setting. Because of the variety of models, an appropriate numerical
treatment for any particular kind of oscillator is required. More
specifically, many interesting examples of stochastic oscillator are obtained
by the introduction in the equation of a deterministic oscillator of a noisy
ingredient, which can be, for example, an additive and/ or a multiplicative
noise, a random frequency, a random damping, and so on (see [12] and
references therein for a survey).

In this paper, we focus on stochastic harmonic undamped oscillators driven by
both a deterministic time-dependent force and a random Gaussian forcing. This
is modeled by a second order stochastic differential equation of the type:

$\ddot{x}=-\omega^{2}x+g(t)+\varepsilon\xi(t)$ (1)

where $\xi(t)$ is a white noise process satisfying
$\mathbb{E}|\xi(t)\xi(t^{\prime})|=\delta(t-t^{\prime})$ and $\omega$ is a
positive real constant.

This kind of stochastic oscillators is very popular in physics literature, see
[12, 13, 18], therefore a specific numerical treatment may be very useful for
approximating their solutions. In the existing literature, there are several
works devoted to investigate different aspects of stochastic oscillatory
problems. An analysis of long-term features of the linear oscillator

$\ddot{x}=-x+\varepsilon\xi(t)$

(i.e. equation (1) with $\omega=1$ and $g\equiv 0$) is provided in [20]. In
[5, 6], the authors introduced a family of stochastic trigonometric numerical
methods based on a variation of constant formula for nonlinear equations of
the form

$\ddot{x}=-\omega^{2}x+g(x)+\varepsilon\xi(t)$ (2)

where $g$ is a positional force and it is supposed to be potential. The
investigation of long-term properties of the equation, of the numerical
approximation and their comparison, is typical in the study of stochastic
oscillatory problems (see, for instance, [2, 3, 4, 5, 6, 8, 10, 19, 21] and
references therein). Our aim is to join two ingredients, i.e. a variation of
constant formula and specific quadrature rules to define a numerical treatment
specific for problem (1). The paper is organized as follows: in Section 2, we
start from a semi-discretization based on a variation of constants formula; in
Section 3 we remember the essential aspects of Filon’s quadrature formulas.
Section 4 is dedicated to theoretical convergence results and, finally, in
Section 5 we test our method on famous physical models.

##  2\. Semidiscretization based on a variation of constant.

Equation (1) is equivalent to the following first order system of two
equations in the variables $X_{t}$ (the position of the particle) and $V_{t}$
(its velocity):

$\begin{cases}dX_{t}=V_{t}dt\\\ dV_{t}=-\omega^{2}X_{t}dt+g(t)dt+\varepsilon
dW_{t}\end{cases}$ (3)

Let us consider the matrix

$R(t\omega)=\exp\left(\ t\begin{bmatrix}0&I\\\
-\omega^{2}&0\end{bmatrix}\right)=\begin{bmatrix}\cos(t\omega)&\omega^{-1}\sin(t\omega)\\\
-\omega\sin(t\omega)&\cos(t\omega)\end{bmatrix}.$

Then, the variation of constant formula applied to the system (3) is given by

$\displaystyle\begin{pmatrix}X_{t}\\\
V_{t}\end{pmatrix}=R(t\omega)\begin{pmatrix}X_{0}\\\ V_{0}\end{pmatrix}$
$\displaystyle+\int_{0}^{t}\begin{pmatrix}\omega^{-1}\sin((t-s)\omega)\\\
\cos((t-s)\omega)\end{pmatrix}g(s)ds$ (4)
$\displaystyle+\varepsilon\int_{0}^{t}\begin{pmatrix}\omega^{-1}\sin((t-s)\omega)\\\
\cos((t-s)\omega)\end{pmatrix}dW_{s}.$

The discretization of the stochastic part leads to

$\displaystyle\begin{pmatrix}X_{n+1}\\\
V_{n+1}\end{pmatrix}=R(h\omega)\begin{pmatrix}X_{n}\\\ V_{n}\end{pmatrix}$
$\displaystyle+\int_{t_{n}}^{t_{n+1}}\begin{pmatrix}\omega^{-1}\sin((t_{n+1}-s)\omega)\\\
\cos((t_{n+1}-s)\omega)\end{pmatrix}g(s)ds$ (5)
$\displaystyle+\varepsilon\begin{pmatrix}\omega^{-1}\sin(h\omega)\\\
\cos(h\omega)\end{pmatrix}\Delta W_{n},$

where $\Delta W_{n}=W(t_{n+1})-W(t_{n})$ denotes the Wiener increment. As
nicely discussed in [6], when $\omega\gg 1$, the problem is stiff, therefore
numerical methods able to get good accuracy with step sizes $h$, whose product
with $\omega$ do not need to be small. This means that the error bounds in the
position of the methods should be independent of the product $h\omega$. The
use of the variation of constant formula (4) to design numerical schemes
suited for such problems, is very used in the deterministic setting, see [14,
16]. The methods derived in [5] and [6] are the stochastic counterpart of the
trigonometric methods. Our problem (1) differs from (2) because we have the
deterministic force $g=g(t)$ acting on the system and we aim to construct
accurate numerical procedures combining the variation of constant formula (4)
and appropriare quadrature rules.

##  3\. Filon’s quadrature formula

In this section, we aim to remind few relevant issues on Filon quadrature
formulas, see [7, 11], which are perfectly suitable for our case. Filon
formulas are specialized for integrals of the type

$\int_{a}^{b}\sin(kx)\psi(x)dx,$ (6)

where $k$ is a real constant. When $k$ is large, the rapid oscillation of
$\sin(kx)$, the ordinary quadrature formulae require the division of the range
of integration into a wide number subinterval. We consider even number of
nodes $2n$ in the interval $[a,b]$, so that

$b=a+2nh$

and we set

$\theta=kh,\quad x_{r}=a+rh,\quad\psi_{r}=\psi(x_{r}).$

Filon formula is then given by

$\int_{a}^{b}\sin(kx)\psi(x)dx=h[\alpha(\psi(a)\cos(ka)-\psi(b)\cos(kb)+\beta
S_{2r}+\gamma S_{2r-1}],$ (7)

where

$\displaystyle\alpha$
$\displaystyle=1/\theta+\cos\theta\sin\theta/\theta^{2}-2\sin^{2}(\theta)/\theta^{3},$
$\displaystyle\beta$
$\displaystyle=2[(1+cos^{2}\theta)/\theta^{2}-2\cos\theta\sin\theta/\theta^{3},$
$\displaystyle\gamma$
$\displaystyle=4(\sin\theta/\theta^{3}-\cos\theta/\theta^{2}),$ $\displaystyle
S_{2r}$
$\displaystyle=2\sum_{r=0}^{n}\psi(x_{2r})\sin(x_{2r})-\psi(a)\sin(ka)-\psi(b)\sin(kb),$
$\displaystyle S_{2r+1}$
$\displaystyle=\sum_{r=1}^{n}\psi(x_{2r-1})\sin(x_{2r-1}).$

For our purposes, it is appropriate to recall the following associated formula

$\int_{a}^{b}\sin(kx+z)\psi(x)dx=h[\alpha(\psi(a)\cos(ka+z)-\psi(b)\cos(kb+z)+\beta
S_{2r}+\gamma S_{2r-1}]$ (8)

setting $z=\frac{\pi}{2}$, we get the formula for
$\int_{a}^{b}\cos(kx)\psi(x)dx$. Let $E_{s},E_{c}$ designate respectively the
error in Filon’s sine and cosine formulas. Of we require to choose $h$ in a
way that $\theta<1$ this means that for large values of $k$, the choice of a
small $h$ is mandatory. However, if $f(x)$ can be well approximated by a
piecewise quadratic function $p(x)$ using a coarse mesh, then we do not need
for $\theta$ to be small. More precisely, if

$\max_{x\in[a,b]}|f(x)-p(x)|\leq\delta,$

then, a uniform bound on the error in Filon’s formula is given by
$(b-a)\delta$. When $\theta<1$, set

$H(\theta)=\left\lvert\frac{\sin\theta}{3\theta^{2}}+\frac{\sin\theta}{\theta^{3}}-\frac{\cos\theta}{\theta^{4}}\right\rvert$

and

$M=\max_{a\leq x\leq b}|f^{(3)}(x)|,$

then it has been shown in [7] that

$E_{s},E_{c}\leq H(\theta)M(b-a)h^{3}+\mathcal{O}(h^{4}).$ (9)

##  4\. Convergence

This section is dedicated to study the convergence of the introduced scheme.
We start recalling the following result form [6] concerning the the global
mean-square error of the stochastic trigonometric integrator for the linear
case.

######  Proposition 4.0.1.

Consider the numerical solution of the linear problem

$\ddot{x}=-\omega^{2}x+\varepsilon\dot{W}_{t}$

given by

$\begin{pmatrix}X_{n+1}\\\
V_{n+1}\end{pmatrix}=R(h\omega)\begin{pmatrix}X_{n}\\\
V_{n}\end{pmatrix}+\varepsilon\begin{pmatrix}\omega^{-1}\sin(h\omega)\\\
\cos(h\omega)\end{pmatrix}\Delta W_{n}.$ (10)

The mean-square errors of the above scheme, with a step size $h\leq h_{0}$
(with a sufficiently small $h_{0}$ independent of $\varepsilon$) for which
$h/\varepsilon\geq c_{0}>0$ holds, satisfy

$\left(\mathbb{E}|X_{t_{n}}-X_{n}|^{2}\right)^{1/2}\leq C\varepsilon\leq Ch,$
$\left(\mathbb{E}|V_{t_{n}}-V_{n}|^{2}\right)^{1/2}\leq CT^{1/2},$

for $nh\leq T$. The constant $C$ is independent of $\varepsilon$, $h$ and $n$.

Following the approach of [6] we prove the following result.

######  Theorem 4.0.2.

Let us suppose the assumptions of Proposition 4.0.1 satisfied. Let us suppose
that on each subinterval $[t_{n-j-1},t_{n-j}]$ of the mesh, the function $g$
can be well approximated by a piecewise quadratic functions $p_{j}(x)$, i.e.

$\max_{t\in[t_{n-j-1},t_{n-j}]}|g(t)-p_{j}(t)|\leq\delta_{j}.$ (11)

If $\delta:=\max_{j}\delta_{j}$, we have that the global mean-square error in
the position is given by

$\mathbb{E}|X_{n}-X_{t_{n}}|^{2}\leq Ch^{2}+D\omega^{-2}h^{4},$ (12)

where the constant $D$ depends only by $T$ and $\delta$. Instead, the global
mean-square error in the velocity is estimated as

$\mathbb{E}|V_{n}-V_{t_{n}}|^{2}\leq(C++\delta^{2}h^{2})T^{2}.$ (13)

###### Proof.

Let us write the numerical approximation as

$X_{n}=\cos(nh\omega)X_{0}+\omega^{-1}\sin(nh\omega)V_{0}+d_{1}+s_{1},$
$V_{n}=-\omega\sin(nh\omega)X_{0}+\cos(nh\omega)V_{0}+d_{2}+s_{2},$

with

$\displaystyle d_{1}$
$\displaystyle=\omega^{-1}\varepsilon\sum_{j=0}^{n-1}\Bigg{(}\cos(jh\omega)\mathcal{Q}^{S}_{F}(t_{n-j-1},t_{n-j},\omega)+\sin(jh\omega)\mathcal{Q}^{C}_{F}(t_{n-j-1},t_{n-j},\omega)\Bigg{)},$
$\displaystyle s_{1}$
$\displaystyle=\omega^{-1}\varepsilon\sum_{j=0}^{n-1}\left(\cos(jh\omega)\sin(jh\omega)+\sin(h\omega)\cos(jh\omega)\right)\Delta
W_{n-j-1},$ $\displaystyle d_{2}$
$\displaystyle=\varepsilon\sum_{j=0}^{n-1}\Bigg{(}\sin(jh\omega)\mathcal{Q}^{S}_{F}(t_{n-j-1},t_{n-j},\omega)-\cos(jh\omega)\mathcal{Q}^{C}_{F}(t_{n-j-1},t_{n-j},\omega)\Bigg{)},$
$\displaystyle s_{2}$
$\displaystyle=\varepsilon\sum_{j=0}^{n-1}\left(\sin(jh\omega)\sin(h\omega)-\cos(h\omega)\cos(jh\omega)\right)\Delta
W_{n-j-1}.$

Let us also write the exact solution as

$X_{t_{n}}=\cos(nh\omega)X_{0}+\omega^{-1}\sin(nh\omega)V_{0}+\hat{d}_{1}+\hat{s}_{1},$
$V_{t_{n}}=-\omega\sin(nh\omega)X_{0}+\cos(nh\omega)V_{0}+\hat{d}_{2}+\hat{s}_{2},$

where

$\displaystyle\hat{d}_{1}$
$\displaystyle=\omega^{-1}\varepsilon\sum_{j=0}^{n-1}\Bigg{(}\cos(jh\omega)\int_{t_{n-j-1}}^{t_{n-j}}\sin((t_{n-j}-s)\omega)g(s)ds$
$\displaystyle\qquad\qquad\qquad+\sin(jh\omega)\int_{t_{n-j-1}}^{t_{n-j}}\cos((t_{n-j}-s)\omega)g(s)ds\Bigg{)},$
$\displaystyle\hat{s}_{1}$
$\displaystyle=\omega^{-1}\varepsilon\sum_{j=0}^{n-1}\Bigg{(}\cos(jh\omega)\int_{t_{n-j-1}}^{t_{n-j}}\sin((t_{n-j}-s)\omega)dW_{s}$
$\displaystyle\qquad\qquad\qquad+\sin(jh\omega)\int_{t_{n-j-1}}^{t_{n-j}}\cos((t_{n-j}-s)\omega)dW_{s}\Bigg{)},$
$\displaystyle\hat{d}_{2}$
$\displaystyle=\varepsilon\sum_{j=0}^{n-1}\Bigg{(}\sin(jh\omega)\int_{t_{n-j-1}}^{t_{n-j}}\sin((t_{n-j}-s)\omega)g(s)ds$
$\displaystyle\qquad\qquad\qquad-\cos(jh\omega)\int_{t_{n-j-1}}^{t_{n-j}}\cos((t_{n-j}-s)\omega)g(s)ds\Bigg{)},$
$\displaystyle\hat{s}_{2}$
$\displaystyle=\varepsilon\sum_{j=0}^{n-1}\Bigg{(}\sin(jh\omega)\int_{t_{n-j-1}}^{t_{n-j}}\sin((t_{n-j}-s)\omega)dW_{s}$
$\displaystyle\qquad\qquad\qquad+\cos(jh\omega)\int_{t_{n-j-1}}^{t_{n-j}}\cos((t_{n-j}-s)\omega)dW_{s}\Bigg{)}.$

The values $\mathcal{Q}^{S}_{F}(t_{n-j-1},t_{n-j},\omega)$ and
$\mathcal{Q}^{C}_{F}(t_{n-j-1},t_{n-j},\omega)$ respectively denote the
Filon’s approximation of the integrals
$\int_{t_{n-j-1}}^{t_{n-j}}\sin((t_{n-j}-s)\omega)g(s)ds$ and
$\int_{t_{n-j-1}}^{t_{n-j}}\cos((t_{n-j}-s)\omega)g(s)ds$.

The global mean-square error in the position is given by

$\mathbb{E}|X_{n}-X_{t_{n}}|^{2}=\mathbb{E}|d_{1}+s_{1}-\hat{d}_{1}-\hat{s}_{1}|^{2}\leq
2\mathbb{E}|d_{1}-\hat{d}_{1}|^{2}+2\mathbb{E}|s_{1}-\hat{s}_{1}|^{2}.$

The term $\mathbb{E}|d_{1}-\hat{d}_{1}|^{2}$ is the global mean-square error
in the linear case (see Prop. 4.0.1 and [6]). Since

$\displaystyle|d_{1}-\hat{d}_{1}|$
$\displaystyle\leq\omega^{-1}\varepsilon\sum_{j=0}^{n-1}\lvert\mathcal{E}^{S}_{F}(t_{n-j-1},t_{n-j},\omega)+\mathcal{E}^{C}_{F}(t_{n-j-1},t_{n-j},\omega)\rvert$
$\displaystyle\leq\omega^{-1}\varepsilon\sum_{j=0}^{n-1}2\delta_{j}h\leq
2\omega^{-1}\varepsilon\delta nh=2\omega^{-1}\varepsilon\delta T,$

then

$\mathbb{E}|X_{n}-X_{t_{n}}|^{2}\leq
4C\varepsilon^{2}+8\omega^{-2}\delta^{2}T^{2}\varepsilon^{2}.$

The global mean-square error in the velocity is given by

$\mathbb{E}|V_{n}-V_{t_{n}}|^{2}=\mathbb{E}|d_{2}+s_{2}-\hat{d}_{2}-\hat{s}_{2}|^{2}\leq
2\mathbb{E}|d_{2}-\hat{d}_{2}|^{2}+2\mathbb{E}|s_{2}-\hat{s}_{2}|^{2}.$

The term $\mathbb{E}|s_{2}-\hat{s}_{2}|^{2}$ is the global mean-square error
in the linear case (see Prop. 4.0.1 and [6]) in the velocity.

Since, as before,

$|\hat{d}_{2}-\hat{s}_{2}|\leq T\delta\varepsilon,$

finally we get

$\mathbb{E}|V_{n}-V_{t_{n}}|^{2}\leq CT^{2}\delta^{2}+\varepsilon^{2}T^{2}.$

∎∎

######  Remark 4.0.3.

In the case in which the “piecewise quadratic” hypothesis on $g$ is not
satisfied, so we have to choose $h$ in a way that $\theta<1$, we can use the
bounds (9) to give an estimate of the mean square-errors. Since

$\displaystyle|d_{1}-\hat{d}_{1}|\leq$
$\displaystyle\omega^{-1}\varepsilon\sum_{j=0}^{n-1}\lvert\mathcal{E}^{S}_{F}(t_{n-j-1},t_{n-j},\omega)+\mathcal{E}^{C}_{F}(t_{n-j-1},t_{n-j},\omega)\rvert$
$\displaystyle\leq\omega^{-1}\varepsilon\sum_{j=0}^{n-1}2H(\theta)Mh(h/r)^{3}\leq
2\omega^{-1}\varepsilon nhH(\theta)M(h/r)^{3},$

we have

$\mathbb{E}|X_{n}-X_{t_{n}}|^{2}\leq
4C\varepsilon^{2}+8\omega^{-2}T^{2}H^{2}(\theta)M^{2}h_{1}^{6}\varepsilon^{2},$

where $h_{1}=h/r$ in the quadrature formula.

##  5\. Numerical Experiments

We propose an example very popular in physics literature (see [1, 12, 13])

$\ddot{x}=-\omega^{2}x-Q\cos(\Omega t)+\varepsilon\xi(t).$ (14)

We integrate (14) over the interval $[0,1]$, with initial conditions
$X_{0}=0.8$ and $V_{0}=1$. We employ the Filon’s formula choosing five nodes.
The values of the constants are $Q=5$, $\Omega=20$ and $\varepsilon=0.3$.
Following the approach in [15], Figure 1 provides the plot of the strong
errors of the numerical solutions obtained by (5), discretizing the integrals
by Filon, Lobatto (also with five points) and the trapezoidal quadrature
formulas, at the endpoint $T=1$. More precisely, in Figure 1, we have set
$\omega=10$ (top) and $\omega=50$ (bottom). In Figure 2, we plot the results
for $\omega=100$. These figures show that Filon method outperforms the other
ones.

---
Figure 1. Strong errors at $T=1$ of the numerical solutions. In the top figure
we set $\omega=10$, while in the bottom figure on the left $\omega=50$. Figure
2. Strong errors of the numerical solutions at $T=1$ for $\omega=100$.

As second example, we propose the problem

$\ddot{x}=-\omega^{2}x-Q\cos(\Omega t)-R\sin(\Theta t)+\varepsilon\xi(t),$
(15)

with $\varepsilon=0.3$, $Q=3$, $\Omega=30$, $R=2$ and $\Theta=25$. We
integrate (14) over the interval $[0,1]$, with initial conditions $X_{0}=0.8$
and $V_{0}=1$. Figure 3 shows the strong errors in $T=1$, setting $\omega=50$
(top) and $\omega=150$ (bottom), of the three methods, corresponding to the
three different quadrature to fully discretize (5). Also in this case Filon
method outperforms the other ones.

---
Figure 3. Strong errors in the position for $\omega=50$ (top) and in for
$\omega=150$ (bottom), related to problem 15.

##  6\. Conclusions

This work is dedicated to present a specific numerical strategy for stochastic
oscillators of type (1). Numerical experiments show that our approach give the
best generalized performance against a full discretization of (5) via Lobatto
or Trapezoidal rule. In particular, the Trapezoidal option may be seen as the
analogue of the method of [6] for the case in which $g=g(t)$. Moreover, in
many examples of oscillators we can find the co-presence of a forcing term
$g=g(t)$ and a non linearity with respect to $x$. In these cases, it is
possible to join the approaches to accurately simulate the solutions. Our
general idea is to correctly adapt the numerical strategy to the problem; in
particular, models like (1) with time-varying frequencies and/or presence of
damping may be objects of future works.

Acknowledgments
This work is supported by GNCS-INDAM project and by PRIN2017-MIUR project. The
authors are member of the INDAM Research group GNCS. The authors are thankful
to Dr. Monia Capanna for their helpful discussions.

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