text: A PRIORI ESTIMATES OF A FINITE ELEMENT METHOD FOR FRACTIONAL DIFFUSION PROBLEMS BY ENERGY ARGUMENTS † † ‡ SAMIR KARAA AND KASSEM MUSTAPHA AND AMIYA K. PANI Abstract. In this article, the Galerkin piecewise-linear ﬁnite element (FE) method is applied to approximate the solution of time-fractional diﬀusion equations with variable diﬀusivity on bounded convex domains. Standard energy arguments do not provide satisfactory results for such a problem due to the low regularity of its exact solution. Using a delicate energy analysis, a priori optimal ∞ error bounds in L2(Ω)-, H1(Ω)-, and quasi-optimal in L (Ω)-norms are derived for the semidiscrete FE scheme for cases with smooth and nonsmooth initial data. The main tool of our analysis is based on a repeated use of an integral operator and use of a tm type of weights to take care of the singular behavior at t = 0. The generalized Leibniz formula for fractional derivatives is found to play a key role in our analysis. Numerical experiments are presented to illustrate some of the theoretical results. Key words. Fractional diﬀusion equation, ﬁnite element approximation, energy argument, error analysis, nonsmooth data, generalized Leibniz formula. AMS subject classiﬁcations. 65M60, 65M12, 65M15 In this paper, we aim to investigate the error analysis via energy arguments of a semidiscrete Galerkin ﬁnite element (FE) method for time-fractional diﬀusion problems of the form: ﬁnd u = u(x, t) such that (cid:48) − L × u (x, t) + ∂1 α u(x, t) = f(x, t) in Ω (0, T], (1.1a) t × u(x, t) = 0 on ∂Ω (0, T], (1.1b) u(x, 0) = u (x) in Ω, (1.1c) 0 L − ∇ R where u = div(a(x) u), Ω is a bounded, convex polygonal domain in 2 with boundary ∂Ω, f, a and u are given functions deﬁned on their respective domains. (cid:48) 0 − − Here, u is the partial derivative of u with respect to time and ∂1 α := RD1 α is the t Riemann–Liouville time-fractiona(cid:90)l derivative deﬁned by: for 0 < α < 1, − − ∂ I ∂ t − tα 1 ∂1 αϕ(t) := αϕ(t) := ω (t s)ϕ(s) ds with ω (t) := , (1.2) t ∂t ∂t α α Γ(α) 0 I → − − L ( α is the Riemann–Liouville time-fractional integral). As α 1 , ∂1 α ( u) L t converges to u, and thus, problem (1.1) reduces to the classical diﬀusion equation. Throughout the paper, we assume that the source term f and the diﬀusivity coeﬃcient function a are suﬃciently regular and ≤ ≤ ∞ 0 < a a(x) a < on Ω. (1.3) min max † Department of Mathematics and Statistics, Sultan Qaboos University, P. O. Box 36, Al-Khod 123, Muscat, Oman. Email: skaraa@squ.edu.om † Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia. Email:kassem@kfupm.edu.sa ‡ Department of Mathematics, Industrial Mathematics Group, Indian Institute of Technology Bombay, Powai, Mumbai-400076. Email: akp@math.iitb.ac.in 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: A

label: paragraph

text: PRIORI

text: ESTIMATES

text: OF

text: A

text: FINITE

text: ELEMENT

text: METHOD

text: FOR

text: FRACTIONAL

text: DIFFUSION

text: PROBLEMS

text: BY

text: ENERGY

text: ARGUMENTS

text: SAMIR

text: KARAA

text: AND

text: KASSEM

text: MUSTAPHA

text: AND

text: AMIYA

text: K.

text: PANI

text: Abstract.

text: In

label: abstract

text: this

text: article,

text: the

text: Galerkin

text: piecewise-linear

text: ﬁnite

text: element

text: (FE)

text: method

text: is

text: applied

text: to

text: approximate

text: the

text: solution

text: of

text: time-fractional

text: diﬀusion

text: equations

text: with

text: variable

text: diﬀusivity

text: on

text: bounded

text: convex

text: domains.

text: Standard

text: energy

text: arguments

text: do

text: not

text: provide

text: satisfactory

text: results

text: for

text: such

text: a

text: problem

text: due

text: to

text: the

text: low

text: regularity

text: of

text: its

text: exact

text: solution.

text: Using

text: a

text: delicate

text: energy

text: analysis,

text: a

text: priori

text: optimal

text: error

text: bounds

text: in

text: and

text: quasi-optimal

text: in

text: L

text: (Ω)-norms

text: are

text: derived

text: for

text: the

text: semidiscrete

text: FE

text: scheme

text: for

text: cases

text: with

text: smooth

text: and

text: nonsmooth

text: initial

text: data.

text: The

text: main

text: tool

text: of

text: our

text: analysis

text: is

text: based

text: on

text: a

text: repeated

text: use

text: of

text: an

text: integral

text: operator

text: and

text: use

text: of

text: a

text: tm

text: type

text: of

text: weights

text: to

text: take

text: care

text: of

text: the

text: singular

text: behavior

text: at

text: t

text: The

text: generalized

text: Leibniz

text: formula

text: for

text: fractional

text: derivatives

text: is

text: found

text: to

text: play

text: a

text: key

text: role

text: in

text: our

text: analysis.

text: Numerical

text: experiments

text: are

text: presented

text: to

text: illustrate

text: some

text: of

text: the

text: theoretical

text: results.

text: Key

text: words.

text: Fractional

text: diﬀusion

text: equation,

text: ﬁnite

text: element

text: approximation,

text: energy

text: argument,

text: error

text: analysis,

text: nonsmooth

text: data,

text: generalized

text: Leibniz

text: formula.

text: AMS

text: subject

text: classiﬁcations.

text: In

text: this

text: paper,

text: we

text: aim

text: to

text: investigate

text: the

text: error

text: analysis

text: via

text: energy

text: arguments

text: of

text: a

text: semidiscrete

text: Galerkin

text: ﬁnite

text: element

text: (FE)

text: method

text: for

text: time-fractional

text: diﬀusion

text: problems

text: of

text: the

text: form:

text: ﬁnd

text: u

text: u(x,

text: t)

text: such

text: that

text: L

text: u

text: (x,

text: t)

text: ∂1

text: α

text: u(x,

text: t)

text: f(x,

text: t)

text: in

text: Ω

text: T],

text: t

text: u(x,

text: t)

text: on

text: ∂Ω

text: T],

text: u(x,

text: u

text: (x)

text: in

text: Ω,

text: L

text: R

text: where

text: u

text: div(a(x)

text: u),

text: Ω

text: is

text: a

text: bounded,

text: convex

text: polygonal

text: domain

text: in

text: with

text: boundary

text: ∂Ω,

text: f,

text: a

text: and

text: u

text: are

text: given

text: functions

text: deﬁned

text: on

text: their

text: respective

text: domains.

text: Here,

text: u

text: is

text: the

text: partial

text: derivative

text: of

text: u

text: with

text: respect

text: to

text: time

text: and

text: ∂1

text: α

text: α

text: is

text: the

text: t

text: Riemann–Liouville

text: derivative

text: deﬁned

text: by:

text: for

text: α

text: I

text: t

text: tα

text: ∂1

text: αϕ(t)

text: αϕ(t)

text: ω

text: (t

text: s)ϕ(s)

text: ds

text: with

text: ω

text: (t)

text: t

text: ∂t

text: ∂t

text: α

text: α

text: Γ(α)

text: I

text: L

text: α

text: is

text: the

text: Riemann–Liouville

text: time-fractional

text: integral).

text: As

text: α

text: ∂1

text: α

text: u)

text: L

text: t

text: converges

text: to

text: u,

text: and

text: thus,

text: problem

text: reduces

text: to

text: the

text: classical

text: diﬀusion

text: equation.

text: Throughout

text: the

text: paper,

text: we

text: assume

text: that

text: the

text: source

text: term

text: f

text: and

text: the

text: diﬀusivity

text: coeﬃcient

text: function

text: a

text: are

text: suﬃciently

text: regular

text: and

text: a

text: a(x)

text: a

text: on

text: Ω.

text: min

text: max

text: Department

text: of

text: Mathematics

text: and

text: Statistics,

text: Sultan

text: Qaboos

text: University,

text: P.

text: O.

text: Box

text: Al-Khod

text: Muscat,

text: Oman.

text: Email:

text: skaraa@squ.edu.om

text: Department

text: of

text: Mathematics

text: and

text: Statistics,

text: King

text: Fahd

text: University

text: of

text: Petroleum

text: and

text: Minerals,

text: Dhahran,

text: Saudi

text: Arabia.

text: Email:kassem@kfupm.edu.sa

text: Department

text: of

text: Mathematics,

text: Industrial

text: Mathematics

text: Group,

text: Indian

text: Institute

text: of

text: Technology

text: Bombay,

text: Powai,

text: Email:

text: akp@math.iitb.ac.in

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: FEM FOR FRACTIONAL DIFFUSION MODELS 19 Table 7.3 Behavior of the weighted error Eµ, as the number of spatial mesh elements increases, for diﬀerent choices of the power weight exponent µ. In each case, we use 1300 time subintervals. M E CR E CR E CR E CR 0 0.5 0.75 1 4 1.0160e-00 3.453e-02 1.531e-02 9.898e-03 8 9.7501e-01 0.0594 8.545e-03 2.015 2.245e-03 2.769 1.525e-03 2.699 16 7.0054e-01 0.4769 3.852e-03 1.150 6.809e-04 1.721 4.783e-04 1.672 32 3.2311e-01 1.1164 1.776e-03 1.117 2.000e-04 1.767 1.442e-04 1.730 64 1.5301e-01 1.0783 8.409e-04 1.078 6.234e-05 1.682 4.945e-05 1.544 [12] W. Mclean, Regularity of solutions to a time-fractional diﬀusion equation, ANZIAM J., 52 (2010), 123–138. [13] W. McLean, Fast summation by interval clustering for an evolution equation with memory, SIAM J. Sci. Comput., 34 (2012), 3039–3056. [14] W. McLean and K. Mustapha, Time-stepping error bounds for fractional diﬀusion problems with non-smooth initial data, J. Comput. Phys., 293 (2015), 201–217. [15] W. McLean and V. Thom´ee, Numerical solution via Laplace transforms of a fractional order evolution equation, J. Integral Equations Appl., 22 (2010), 57-–94. [16] W. McLean and V. Thom´ee, Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional order evolution equation, IMA J. Numer. Anal., 30 (2010), 208–230. [17] K. Mustapha, An implicit ﬁnite diﬀerence time-stepping method for a sub-diﬀusion equation, with spatial discretization by ﬁnite elements, IMA J. Numer. Anal., 31 (2011), 719–739. [18] K. Mustapha and W. McLean, Piecewise-linear, discontinuous Galerkin method for a fractional diﬀusion equation, Numer. Algor., 56 (2011), 159–184. − [19] K. Mustapha and D. Scho¨tzau, Well-posedness of hp version discontinuous Galerkin methods for fractional diﬀusion wave equations, IMA J. Numer. Anal., 34 (2014), 1226–1246. [20] J. Ren and Z. Z. Sun, Numerical algorithm with high spatial accuracy for the fractional diﬀusion-wave equation with Neumann boundary conditions, J. Sci. Comput., 56 (2013), 381–408. [21] V. Thom´ee. Galerkin ﬁnite element methods for parabolic problems, Springer, 1997 [22] Y. N. Zhang and Z. Z. Sun, Alternating direction implicit schemes for the two-dimensional fractional sub-diﬀusion equation, J. Comput. Phys., 230 (2011), 8713–8728. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FEM

text: FOR

text: FRACTIONAL

text: DIFFUSION

text: MODELS

text: Table

text: Behavior

label: caption

text: of

text: the

text: weighted

text: error

text: Eµ,

text: as

text: the

text: number

text: of

text: spatial

text: mesh

text: elements

text: increases,

text: for

text: diﬀerent

text: choices

text: of

text: the

text: power

text: weight

text: exponent

text: µ.

text: In

text: each

text: case,

text: we

text: use

text: time

text: subintervals.

text: M

label: table

text: E

text: CR

text: E

text: CR

text: E

text: CR

text: E

text: CR

label: reference

text: W.

text: Mclean,

text: Regularity

text: of

text: solutions

text: to

text: a

text: time-fractional

text: diﬀusion

text: equation,

text: ANZIAM

text: J.,

text: W.

text: McLean,

text: Fast

text: summation

text: by

text: interval

text: clustering

text: for

text: an

text: evolution

text: equation

text: with

text: memory,

text: SIAM

text: J.

text: Sci.

text: Comput.,

text: W.

text: McLean

text: and

text: K.

text: Mustapha,

text: Time-stepping

text: error

text: bounds

text: for

text: fractional

text: diﬀusion

text: problems

text: with

text: non-smooth

text: initial

text: data,

text: J.

text: Comput.

text: Phys.,

text: W.

text: McLean

text: and

text: V.

text: Thom´ee,

text: Numerical

text: solution

text: via

text: Laplace

text: transforms

text: of

text: a

text: fractional

text: order

text: evolution

text: equation,

text: J.

text: Integral

text: Equations

text: Appl.,

text: W.

text: McLean

text: and

text: V.

text: Thom´ee,

text: Maximum-norm

text: error

text: analysis

text: of

text: a

text: numerical

text: solution

text: via

text: Laplace

text: transformation

text: and

text: quadrature

text: of

text: a

text: fractional

text: order

text: evolution

text: equation,

text: IMA

text: J.

text: Numer.

text: Anal.,

text: K.

text: Mustapha,

text: An

text: implicit

text: ﬁnite

text: diﬀerence

text: time-stepping

text: method

text: for

text: a

text: sub-diﬀusion

text: equation,

text: with

text: spatial

text: discretization

text: by

text: ﬁnite

text: elements,

text: IMA

text: J.

text: Numer.

text: Anal.,

text: K.

text: Mustapha

text: and

text: W.

text: McLean,

text: Piecewise-linear,

text: discontinuous

text: Galerkin

text: method

text: for

text: a

text: fractional

text: diﬀusion

text: equation,

text: Numer.

text: Algor.,

text: K.

text: Mustapha

text: and

text: D.

text: Scho¨tzau,

text: Well-posedness

text: of

text: hp

text: version

text: discontinuous

text: Galerkin

text: methods

text: for

text: fractional

text: diﬀusion

text: wave

text: equations,

text: IMA

text: J.

text: Numer.

text: Anal.,

text: J.

text: Ren

text: and

text: Z.

text: Z.

text: Sun,

text: Numerical

text: algorithm

text: with

text: high

text: spatial

text: accuracy

text: for

text: the

text: fractional

text: diﬀusion-wave

text: equation

text: with

text: Neumann

text: boundary

text: conditions,

text: J.

text: Sci.

text: Comput.,

text: V.

text: Thom´ee.

text: Galerkin

text: ﬁnite

text: element

text: methods

text: for

text: parabolic

text: problems,

text: Springer,

text: Y.

text: N.

text: Zhang

text: and

text: Z.

text: Z.

text: Sun,

text: Alternating

text: direction

text: implicit

text: schemes

text: for

text: the

text: two-dimensional

text: fractional

text: sub-diﬀusion

text: equation,

text: J.

text: Comput.

text: Phys.,

text: ##LTLine##

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text: FEM FOR FRACTIONAL DIFFUSION MODELS 3 − − | | uniform, a quasi-optimal error estimate of order t α(3 δ)/2 ln h 5/2 h2 in the ∞ stronger L (Ω)-norm is proved, see Theorem 5.5. The proposed technique has several attractive features. Some of these are: (1) allowing variable coeﬃcients, and smooth and nonsmooth initial data in the error analysis, (2) the quasi-uniform FE mesh assumption is not required to show the convergence results in Hm-norm for m = 0, 1, and (3) the proposed technique can be applied to other fractional model problems with smooth and nonsmooth initial data. For instance, we discussed in Section 5 the extension of the achieved error bounds in Theorems 4.4, 5.3 and 5.5 to the time-fractional diﬀusion equation: for 0 < α < 1, L C∂αu(x, t) + u(x, t) = f(x, t), (1.4) t I − (cid:48) where C∂αv(t) := 1 αv (t) is the time-fractional Caputo derivative. The error t estimate in Theorem 4.4 provides an improvement of the result obtained by Jin et al. [8, Theorem 3.7], where the error analysis of the lumped mass FE method was also considered. For constant diﬀusivity and under the assumption that the mesh is quasi-uniform, the FE error bound in [8] involves a logarithmic factor which was derived using a semigroup approach. The error analysis approach in this work can also be used to investigate the error estimates for the FE method applied to the time-fractional Rayleigh-Stokes problem (described by the time-fractional diﬀerential equation) presented in the recent work [1], which is close to the one in [15]. Outline of the paper. In Section 2, we recall some smoothness properties of the solution u, we also state and derive some technical results. In Section 3, we introduce our semidiscrete FE scheme and recall some FE error results. We claim that a direct application of energy arguments to problem (1.1) does not lead to optimal convergence ∈ rates even when the initial data u H˙ 2(Ω). In Section 4, for both smooth and 0 nonsmooth initial data, we derive error estimates for the FE problem in the L2(Ω)- norm, see Theorem 4.4. The generalized Leibniz formula is an essential ingredient in our error analysis. In Section 5, a superconvergence result in the H1-norm is obtained, see Theorem 5.2, and as a consequence, an optimal gradient FE error estimate in the L2(Ω)-norm is derived in Theorem 5.3, and a quasi-optimal FE error bound in ∞ ⊂ R the L (Ω)-norm is achieved for Ω 2, see Theorem 5.5. Particularly relevant to this a priori error analysis is the appropriate use of several properties of the time- fractional integral and derivative operators. In Section 5, we show that the achieved error estimates for problem (1.1) are valid for the FE discretization of the fractional diﬀusion model (1.4). Numerical tests are presented in Section 5 to conﬁrm some of our theoretical ﬁndings. Throughout the paper, C is a generic positive constant that may depend on α and T, but is independent of the spatial mesh element size h. 2. Regularity and technical results. Smoothness properties of the solution u of the fractional diﬀusion problem (1.1) play a key role in the error analysis of the Galerkin FE method, particularly, since u has singularity near t = 0, even for smooth given data. Below, we state the required regularity results for problem (1.1) in terms of the initial data u and the source term f. Over the convex domain Ω, by combining 0 ≤ ≤ the results of Theorems 4.1, 4.2 and 5.6 in [12], for 0 r, µ 2, we obtain

text: FEM

text: FOR

text: FRACTIONAL

text: DIFFUSION

text: MODELS

label: list

text: uniform,

text: a

text: quasi-optimal

text: error

text: estimate

text: of

text: order

text: t

text: ln

text: h

text: in

text: the

text: stronger

text: L

text: (Ω)-norm

text: is

text: proved,

text: see

text: Theorem

text: The

text: proposed

text: technique

text: has

text: several

text: attractive

text: features.

text: Some

text: of

text: these

text: are:

text: allowing

text: variable

text: coeﬃcients,

text: and

text: smooth

text: and

text: nonsmooth

text: initial

text: data

text: in

text: the

text: error

text: analysis,

text: the

text: quasi-uniform

text: FE

text: mesh

text: assumption

text: is

text: not

text: required

text: to

text: show

text: the

text: convergence

text: results

text: in

text: Hm-norm

text: for

text: m

text: and

text: the

text: proposed

text: technique

text: can

text: be

text: applied

text: to

text: other

text: fractional

text: model

text: problems

text: with

text: smooth

text: and

text: nonsmooth

text: initial

text: data.

text: For

text: instance,

text: we

text: discussed

text: in

text: Section

text: the

text: extension

text: of

text: the

text: achieved

text: error

text: bounds

text: in

text: Theorems

text: and

text: to

text: the

text: time-fractional

text: diﬀusion

text: equation:

text: for

text: α

text: L

text: C∂αu(x,

text: t)

text: u(x,

text: t)

text: f(x,

text: t),

text: t

text: I

text: where

text: C∂αv(t)

text: αv

text: (t)

text: is

text: the

text: time-fractional

text: Caputo

text: derivative.

text: The

text: error

text: t

text: estimate

text: in

text: Theorem

text: provides

text: an

text: improvement

text: of

text: the

text: result

text: obtained

text: by

text: Jin

text: et

text: al.

text: Theorem

text: where

text: the

text: error

text: analysis

text: of

text: the

text: lumped

text: mass

text: FE

text: method

text: was

text: also

text: considered.

text: For

text: constant

text: diﬀusivity

text: and

text: under

text: the

text: assumption

text: that

text: the

text: mesh

text: is

text: quasi-uniform,

text: the

text: FE

text: error

text: bound

text: in

text: involves

text: a

text: logarithmic

text: factor

text: which

text: was

text: derived

text: using

text: a

text: semigroup

text: approach.

text: The

text: error

text: analysis

text: approach

text: in

text: this

text: work

text: can

text: also

text: be

text: used

text: to

text: investigate

text: the

text: error

text: estimates

text: for

text: the

text: FE

text: method

text: applied

text: to

text: the

text: time-fractional

text: Rayleigh-Stokes

text: problem

text: (described

text: by

text: the

text: time-fractional

text: diﬀerential

text: equation)

text: presented

text: in

text: the

text: recent

text: work

text: which

text: is

text: close

text: to

text: the

text: one

text: in

text: Outline

text: of

text: the

text: paper.

text: In

text: Section

text: we

text: recall

text: some

text: smoothness

text: properties

text: of

text: the

text: solution

text: u,

text: we

text: also

text: state

text: and

text: derive

text: some

text: technical

text: results.

text: In

text: Section

text: we

text: introduce

text: our

text: semidiscrete

text: FE

text: scheme

text: and

text: recall

text: some

text: FE

text: error

text: results.

text: We

text: claim

text: that

text: a

text: direct

text: application

text: of

text: energy

text: arguments

text: to

text: problem

text: does

text: not

text: lead

text: to

text: optimal

text: convergence

text: rates

text: even

text: when

text: the

text: initial

text: data

text: u

text: H˙

text: In

text: Section

text: for

text: both

text: smooth

text: and

text: nonsmooth

text: initial

text: data,

text: we

text: derive

text: error

text: estimates

text: for

text: the

text: FE

text: problem

text: in

text: the

text: norm,

text: see

text: Theorem

text: The

text: generalized

text: Leibniz

text: formula

text: is

text: an

text: essential

text: ingredient

text: in

text: our

text: error

text: analysis.

text: In

text: Section

text: a

text: superconvergence

text: result

text: in

text: the

text: is

text: obtained,

text: see

text: Theorem

text: and

text: as

text: a

text: consequence,

text: an

text: optimal

text: gradient

text: FE

text: error

text: estimate

text: in

text: the

text: is

text: derived

text: in

text: Theorem

text: and

text: a

text: quasi-optimal

text: FE

text: error

text: bound

text: in

text: R

text: the

text: L

text: (Ω)-norm

text: is

text: achieved

text: for

text: Ω

text: see

text: Theorem

text: Particularly

text: relevant

text: to

text: this

text: a

text: priori

text: error

text: analysis

text: is

text: the

text: appropriate

text: use

text: of

text: several

text: properties

text: of

text: the

text: time-

text: fractional

text: integral

text: and

text: derivative

text: operators.

text: In

text: Section

text: we

text: show

text: that

text: the

text: achieved

text: error

text: estimates

text: for

text: problem

text: are

text: valid

text: for

text: the

text: FE

text: discretization

text: of

text: the

text: fractional

text: diﬀusion

text: model

text: Numerical

text: tests

text: are

text: presented

text: in

text: Section

text: to

text: conﬁrm

text: some

text: of

text: our

text: theoretical

text: ﬁndings.

text: Throughout

text: the

text: paper,

text: C

text: is

text: a

text: generic

text: positive

text: constant

text: that

text: may

text: depend

text: on

text: α

text: and

text: T,

text: but

text: is

text: independent

text: of

text: the

text: spatial

text: mesh

text: element

text: size

text: h.

text: Regularity

text: and

text: technical

text: results.

text: Smoothness

text: properties

text: of

text: the

text: solution

text: u

text: of

text: the

text: fractional

text: diﬀusion

text: problem

text: play

text: a

text: key

text: role

text: in

text: the

text: error

text: analysis

text: of

text: the

text: Galerkin

text: FE

text: method,

text: particularly,

text: since

text: u

text: has

text: singularity

text: near

text: t

text: even

text: for

text: smooth

text: given

text: data.

text: Below,

text: we

text: state

text: the

text: required

text: regularity

text: results

text: for

text: problem

text: in

text: terms

text: of

text: the

text: initial

text: data

text: u

text: and

text: the

text: source

text: term

text: f.

text: Over

text: the

text: convex

text: domain

text: Ω,

text: by

text: combining

text: the

text: results

text: of

text: Theorems

text: and

text: in

text: for

text: r,

text: µ

text: we

text: obtain


text: 4 S. Karaa, K. Mustapha and A. K. Pani (cid:104) (cid:88) (cid:90) (cid:105) (cid:107) (cid:107) ≤ − (cid:107) (cid:107) (cid:96)+1 t (cid:107) (cid:107) t(cid:96) u((cid:96))(t) Ct αµ/2 u + (1 + tαµ/2) sm f(m)(s) ds r+µ 0 r r (2.1) m=0 0 ≤ − (cid:80) (cid:82) C(1 + Tαµ/2)t αµ/2d (u , f), r 0 ∈ { } (cid:107) (cid:107) (cid:107) (cid:107) (cid:107) · (cid:107) for (cid:96) 0, 1 , where d (u , f) = u + 2 T sm f(m)(s) ds. Here, r 0 0 r m=0 0 r r denotes the norm on the Hilbert space H˙ r(Ω)(cid:88)deﬁned by ∞ (cid:107) (cid:107) (cid:107) (cid:107) v 2 = Ar/2v 2 = λr(v, φ )2, r j j j=1 { }∞ where λ (with 0 < λ < λ < . . .) are the eigenvalues of the operator elliptic Lj j=1 1 2 { }∞ operator (subject homogeneous Dirichlet boundary conditions) and φ are the j j≤=1 associated orthonormal eigenfunctions. Noting that, H˙ r(Ω) = Hr(Ω) for 0 r < 1/2, ∩ and for convex polygonal domains, H˙ r(Ω) = Hr(Ω) H1(Ω) for 1/2 < r < 5/2, where 0 Hr(Ω) (with H0(Ω) = L2(Ω)) is the standard Sobolev space. I − Next, we state the positivity properties of the fractional operators α and ∂1 α, t and derive some technical results that will be used in the subsequent sections. By · · L [19, Lemma 3.1(ii)] and since the bilinear form A( , ) associated with the operator ∇ ∇ (that is, A(v, w) = (a v, w)) is symmetric positive deﬁnite on the Sobolev space → H1(Ω(cid:90)), it follows that for piecewis(cid:90)e time continuous functions ϕ : [0, T] H1(Ω), 0 0 √ T I ≥ T (cid:107) ∇I (cid:107) ≥ A( αϕ, ϕ)(cid:112)dt cos(απ/2) a α/2ϕ 2 dt 0 for 0 < α < 1, (2.2) 0 0 (cid:107) (cid:107) where ϕ := (ϕ, ϕ) denotes the L2-norm. · · By [13, Lemma A.1] and again since the bilinear form A( , ) is symmetric positive deﬁnite, th(cid:90)e following holds: for W1,1(0, T; H1(Ω)), (cid:90) 0 √ T − ≥ 1 − T (cid:107) ∇ (cid:107) A(∂1 αϕ(t), ϕ(t)) dt sin(απ/2)Tα 1 a ϕ(t) 2 dt . (2.3) t 2 0 0 (cid:82) (cid:82) The next lemma will be used frequently in our convergence analysis. In the ∈ | | ≤ | | proof, we use the following integral inequality: if for any τ (0, t), φ(τ) 2 φ(0) 2 + | | | | | | ≤ | | | | τ t 2 φ(s) ψ(s) ds, then φ(t) φ(0) + ψ(s) ds. 0 ∈ { } B 0 − B I Lemma 2.1. Let κ 0, 1 and let α = ∂1 α or α = α. Assume that t − (cid:48) B ∀ ∈ κ(v(t), χ) + (1 κ)(v (t), χ) + A( αv(t), χ) = (w(t), χ), χ V , (2.4) h (cid:90) ∈ (cid:16) (cid:90) (cid:17) (cid:90) for t (0, T]. Then t (cid:107) (cid:107) − (cid:107) (cid:107) ≤ − (cid:107) (cid:107) t (cid:107) (cid:107) 2 t (cid:107) (cid:107) κ v 2 ds + (1 κ) v(t) 2 (1 κ) v(0) + w ds + κ w 2 ds. 0 0 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: S.

text: Karaa,

text: K.

text: Mustapha

text: and

text: A.

text: K.

text: Pani

label: equation

text: t

text: Ct

text: αµ/2

text: u

text: sm

text: f(m)(s)

text: ds

text: r+µ

text: r

text: r

text: (u

text: f),

text: r

text: for

text: where

text: d

text: (u

text: f)

text: u

text: T

text: sm

text: f(m)(s)

text: ds.

text: Here,

text: r

text: r

text: r

text: r

text: denotes

text: the

text: norm

text: on

text: the

text: Hilbert

text: space

text: H˙

text: by

text: v

text: λr(v,

text: φ

text: r

text: j

text: j

text: where

text: λ

text: (with

text: λ

text: λ

text: are

text: the

text: eigenvalues

text: of

text: the

text: operator

text: elliptic

text: Lj

text: operator

text: (subject

text: homogeneous

text: Dirichlet

text: boundary

text: conditions)

text: and

text: φ

text: are

text: the

text: j

text: associated

text: orthonormal

text: eigenfunctions.

text: Noting

text: that,

text: H˙

text: r(Ω)

text: Hr(Ω)

text: for

text: r

text: and

text: for

text: convex

text: polygonal

text: domains,

text: H˙

text: r(Ω)

text: Hr(Ω)

text: for

text: r

text: where

text: Hr(Ω)

text: (with

text: is

text: the

text: standard

text: Sobolev

text: space.

text: I

text: Next,

text: we

text: state

text: the

text: positivity

text: properties

text: of

text: the

text: fractional

text: operators

text: α

text: and

text: ∂1

text: α,

text: t

text: and

text: derive

text: some

text: technical

text: results

text: that

text: will

text: be

text: used

text: in

text: the

text: subsequent

text: sections.

text: By

text: L

text: Lemma

text: and

text: since

text: the

text: bilinear

text: form

text: A(

text: associated

text: with

text: the

text: operator

text: (that

text: is,

text: A(v,

text: w)

text: (a

text: v,

text: w))

text: is

text: symmetric

text: positive

text: deﬁnite

text: on

text: the

text: Sobolev

text: space

text: it

text: follows

text: that

text: for

text: time

text: continuous

text: functions

text: ϕ

text: T]

text: T

text: I

text: T

text: ∇I

text: A(

text: αϕ,

text: a

text: α/2ϕ

text: dt

text: for

text: α

text: where

text: ϕ

text: (ϕ,

text: ϕ)

text: denotes

text: the

text: By

text: Lemma

text: and

text: again

text: since

text: the

text: bilinear

text: form

text: A(

text: is

text: symmetric

text: positive

text: deﬁnite,

text: following

text: holds:

text: for

text: T;

text: T

text: T

text: αϕ(t),

text: ϕ(t))

text: dt

text: a

text: ϕ(t)

text: dt

text: t

text: The

text: next

text: lemma

text: will

text: be

text: used

text: frequently

text: in

text: our

text: convergence

text: analysis.

text: In

text: the

text: proof,

text: we

text: use

text: the

text: following

text: integral

text: inequality:

text: if

text: for

text: any

text: τ

text: t),

text: φ(τ)

text: τ

text: t

text: φ(s)

text: ψ(s)

text: ds,

text: then

text: φ(t)

text: ψ(s)

text: ds.

text: B

text: B

text: I

text: Lemma

text: Let

text: κ

text: and

text: let

text: α

text: ∂1

text: α

text: or

text: α

text: α.

text: Assume

text: that

text: t

text: B

text: κ(v(t),

text: χ)

text: κ)(v

text: (t),

text: χ)

text: A(

text: αv(t),

text: χ)

text: (w(t),

text: χ),

text: χ

text: V

text: h

text: for

text: t

text: T].

text: Then

text: t

text: t

text: t

text: κ

text: v

text: ds

text: κ)

text: v(t)

text: κ)

text: w

text: ds

text: κ

text: w

text: ds.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 6 S. Karaa, K. Mustapha and A. K. Pani Next, we show that a direct application of energy arguments to problem (1.1) does not yield satisfactory results due to the low regularity of the continuous solution. − From (3.1) and (3.2), the decomposition e = ρ θ, and the property of the elliptic projection, we obtain the equation in θ as (cid:48) − (cid:48) ∀ ∈ (θ , χ) + A(∂1 αθ, χ) = (ρ , χ) χ V . (3.5) t h Then, the following result holds, whose proof can be found in [15, 18]. ∈ Theorem 3.1. For t (0, T], we have(cid:90) (cid:107) − (cid:107) ≤ (cid:107) (cid:107) t (cid:107) (cid:48) (cid:107) (cid:107) (cid:107) u(t) u (t) θ + ρ (s) ds + ρ(t) . (3.6) h 0 0 Noting that, the solution u of problem (1.1) has singularity near t = 0. For ≡ ∈ instance, if f 0 and u H˙ 2(Ω), then by (3.3) and the regularity property (cid:107) (cid:48) (cid:90) (cid:107) ≤ − −(cid:90) (cid:107) 0 (cid:107) (cid:90) u (t) Ctα(1 m/2) 1 u for m = 1, 2, m 0 2 t (cid:107) (cid:48) (cid:107) ≤ (cid:16) (cid:90)(cid:15) (cid:107) (cid:48) (cid:107) (cid:90) t (cid:107) (cid:48) (cid:17) (cid:107) ρ (s) ds Ch u (s) ds + Ch2 u (s) ds 1 2 0 0 (cid:15) ≤ (cid:16) (cid:15) − t − (cid:107) (cid:107) C h sα/2 1 ds + h2 s 1 ds u 0 2 0 (cid:15) ≤ | | (cid:107) (cid:107) ≤ | |(cid:107) (cid:107) C h(cid:15)α/2 + h2 log (cid:15) ) u Ch2 log h u for (cid:15) = h2/α. 0 2 0 2 | | This leads to a quasi-optimal O(h2 log h ) convergence for the spatial discretization by linear FEs. To achieve an optimal O(h2) convergence, a stronger regularity as- sumption on u is required and that, in turn imposes severe restrictions on initial data. Thus, the upper bound in Theorem 3.1 is not sharp even for the case of smooth initial data, that is, H2-regularity on the initial data u is not suﬃcient to get an optimal 0 O(h2) convergence rate. Furthermore, it is clear that this upper bound is not suitable for the case of nonsmooth initial data. Therefore, we propose in the next section an approach via delicate energy arguments that provides valid error bounds for the problem with both smooth and nonsmooth initial data. 4. L2(Ω)-error estimates. For convenience, we introduce the notations: (cid:48) Θ (t) := tiθ(t) and Θ˙ (t) := tiθ (t) for i = 1, 2. i i In the next lemma, based on the generalized Leibniz formula for fractional derivatives, we state and show some identities for our subsequent use. Lemma 4.1. For 0 < α < 1, the followings hold: − − − − I (a) t∂1 αθ = ∂1 αΘ (1 α) αθ, It I t 1I (b) t αθ = αΘ + α 1+αθ. 1 Proof. The ﬁrst identity follows from the fractional Leibniz formula. To show the − I (cid:48) I I second identity, noting ﬁrst that ∂1 αΘ = αΘ = αθ + αΘ˙ . Hence, by (a), t 1 1 1 − I I t∂1 αθ(t) = αΘ˙ (t) + α αθ(t). (4.1) t 1

text: S.

text: Karaa,

text: K.

text: Mustapha

text: and

text: A.

text: K.

text: Pani

text: Next,

text: we

text: show

text: that

text: a

text: direct

text: application

text: of

text: energy

text: arguments

text: to

text: problem

text: does

text: not

text: yield

text: satisfactory

text: results

text: due

text: to

text: the

text: low

text: regularity

text: of

text: the

text: continuous

text: solution.

text: From

text: and

text: the

text: decomposition

text: e

text: ρ

text: θ,

text: and

text: the

text: property

text: of

text: the

text: elliptic

text: projection,

text: we

text: obtain

text: the

text: equation

text: in

text: θ

text: as

text: (θ

text: χ)

text: αθ,

text: χ)

text: (ρ

text: χ)

text: χ

text: V

text: t

text: h

text: Then,

text: the

text: following

text: result

text: holds,

text: whose

text: proof

text: can

text: be

text: found

text: in

text: Theorem

text: For

text: t

text: T],

text: we

text: t

text: u(t)

text: u

text: (t)

text: θ

text: ρ

text: (s)

text: ds

text: ρ(t)

text: h

text: Noting

text: that,

text: the

text: solution

text: u

text: of

text: problem

text: has

text: singularity

text: near

text: t

text: For

text: instance,

text: if

text: f

text: and

text: u

text: H˙

text: then

text: by

text: and

text: the

text: regularity

text: property

text: u

text: (t)

text: u

text: for

text: m

text: m

text: t

text: t

text: ρ

text: (s)

text: ds

text: Ch

text: u

text: (s)

text: ds

text: u

text: (s)

text: ds

text: t

text: C

text: h

text: ds

text: s

text: ds

text: u

text: C

text: log

text: u

text: log

text: h

text: u

text: for

text: This

text: leads

text: to

text: a

text: quasi-optimal

text: log

text: h

text: convergence

text: for

text: the

text: spatial

text: discretization

text: by

text: linear

text: FEs.

text: To

text: achieve

text: an

text: optimal

text: convergence,

text: a

text: stronger

text: regularity

text: as-

text: sumption

text: on

text: u

text: is

text: required

text: and

text: that,

text: in

text: turn

text: imposes

text: severe

text: restrictions

text: on

text: initial

text: data.

text: Thus,

text: the

text: upper

text: bound

text: in

text: Theorem

text: is

text: not

text: sharp

text: even

text: for

text: the

text: case

text: of

text: smooth

text: initial

text: data,

text: that

text: is,

text: on

text: the

text: initial

text: data

text: u

text: is

text: not

text: suﬃcient

text: to

text: get

text: an

text: optimal

text: convergence

text: rate.

text: Furthermore,

text: it

text: is

text: clear

text: that

text: this

text: upper

text: bound

text: is

text: not

text: suitable

text: for

text: the

text: case

text: of

text: nonsmooth

text: initial

text: data.

text: Therefore,

text: we

text: propose

text: in

text: the

text: next

text: section

text: an

text: approach

text: via

text: delicate

text: energy

text: arguments

text: that

text: provides

text: valid

text: error

text: bounds

text: for

text: the

text: problem

text: with

text: both

text: smooth

text: and

text: nonsmooth

text: initial

text: data.

text: estimates.

text: For

text: convenience,

text: we

text: introduce

text: the

text: notations:

text: Θ

text: (t)

text: tiθ(t)

text: and

text: Θ˙

text: (t)

text: tiθ

text: (t)

text: for

text: i

text: i

text: i

text: In

text: the

text: next

text: lemma,

text: based

text: on

text: the

text: generalized

text: Leibniz

text: formula

text: for

text: fractional

text: derivatives,

text: we

text: state

text: and

text: show

text: some

text: identities

text: for

text: our

text: subsequent

text: use.

text: Lemma

text: For

text: α

text: the

text: followings

text: hold:

text: I

text: (a)

text: αθ

text: ∂1

text: αΘ

text: α)

text: αθ,

text: It

text: I

text: t

text: (b)

text: t

text: αθ

text: αΘ

text: α

text: Proof.

text: The

text: ﬁrst

text: identity

text: follows

text: from

text: the

text: fractional

text: Leibniz

text: formula.

text: To

text: show

text: the

text: I

text: I

text: I

text: second

text: identity,

text: noting

text: ﬁrst

text: that

text: ∂1

text: αΘ

text: αΘ

text: αθ

text: αΘ˙

text: Hence,

text: by

text: (a),

text: t

text: I

text: I

text: αθ(t)

text: αΘ˙

text: (t)

text: α

text: αθ(t).

text: t


text: FEM FOR FRACTIONAL DIFFUSION MODELS 9 − Therefore, by using the identity θ(t) = t 2Θ (t), the inequality in (4.7) and Lemma 2 4.2 will complete the rest of the proof. In the next theorem, we derive optimal convergence results of the FE scheme (3.2) ∈ in the L2-norm for both smooth and nonsmooth initial data u . For u H˙ δ(Ω) with ≤ ≤ − −0 0 ∈ 0 δ 2, we show that the error is bounded by Ch2t α(2 δ)/2 for each t (0, T]. { ∈ } ≤ Recall that, H˙ δ(Ω) = v Hδ(Ω) : v = 0 on ∂Ω for 1/2 < δ 2, while H˙ δ(Ω) = ≤ → − Hδ(Ω) for 0 δ < 1/2. Noting that, in the limiting case α 1 , we recover the (cid:48) − L convergence rates for the parabolic equation u u = f. Theorem 4.4. Let u and u be the solutions of (1.1) and (3.2), respectively, with ∈ h u (0) = P u . Then, for u H˙ δ(Ω), h h 0 0 (cid:107) − (cid:107) ≤ − − ∈ ≤ ≤ (u u )(t) Ch2t α(2 δ)/2d (u , f) for t (0, T] with 0 δ 2 . h δ 0 − − Proof. The desired result follows from the decomposition u u = ρ θ, the h estimate of θ in Lemma 4.3, the bound in (4.9), and the estimate of ρ in (3.4). Remark 4.5. In the proof of the above theorem, we used (4.9) which follows from the projection estimate in (3.4) for m = 2. For m = 1, we follow similar steps − − B ≤ − − where 2 δ will be replaced with 1 δ, to obtain (t) C h2t3 α(1 δ)d2(u , f) for ≤ ≤ ≤ 1 ≤ ≤ δ 0 { } 0 δ 1 . Now, for 1 < δ 2, we notice ﬁrst that for 0 q t with t = max t, h , h (cid:107) (cid:107) ≤ (cid:107) − (cid:107) (cid:107) (cid:107) ≤ I (cid:107) (cid:48) (cid:107) (cid:107) (cid:107) ρ(q) ρ(t ) ρ(q) + ρ(t ) ( ρ (t ) ) + ρ(t ) . h h h h (cid:90) B (cid:16) (cid:17) Substitute this in the deﬁnition of deﬁned in (4.8), we observe 1 B ≤ t (cid:107) (cid:48) (cid:107) I (cid:107) (cid:48) (cid:107) (cid:107) (cid:107) 2 (t) Ct2 s2 ρ (s) 2 ds + Ct3 ( ρ (t ) ) + ρ(t ) . 1 h h 0 To estimate the ﬁrst two terms, we use (3.3) (with (cid:96), m = 1) and the following regu- larity property (which follows from [12, Theorems 4.2 and 5.6]) (cid:107) (cid:48) (cid:107) ≤ − − ∈ ≤ u (t) Ctα(δ 1)/2 1d˜ (u , f), t (0, T], for 1 < δ 2, (4.13) 1 (cid:80)δ 0 (cid:107) (cid:107) (cid:107) (cid:107) where d˜δ(u0, f) = u0 δ + 2m=0 tm f(m) L∞(H˙ 1), we arrive to B ≤ − (cid:107) (cid:107) ≤ (t) Ct3h2tα(δ 1)d˜ (u , f) + Ct3 ρ(t ) 2, for 1 < δ 2 . 1 h δ 0 h − ≤ − − − However, by (3.4) and the inequality tα(δ 1)/2 Ct α(1 δ)/2(t /t)α(δ 1)/2 for 1 < ≤ h h δ 2, we ﬁnd that (cid:107) (cid:107) ≤ − − ≤ − − − ρ(t ) Ch2t α(2 δ)/2d (u , f) Cht α(1 δ)/2(t /t)α(δ 1)/2d (u , f). h h δ 0 h δ 0 Therefore, B ≤ − − ≤ (t) Ch2t3+α(δ 1)(t /t)α(δ 1)d˜2(u , f), for 1 < δ 2 . 1 h δ 0 B Consequently, by using the above bound of in Theorem 4.4, we get the error estimate 1 below that will be used in the forthcoming section to show the convergence of the ∈ gradient FE solution. For t (0, T], ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: FEM

text: FOR

text: FRACTIONAL

text: DIFFUSION

text: MODELS

text: Therefore,

text: by

text: using

text: the

text: identity

text: θ(t)

text: t

text: 2Θ

text: (t),

text: the

text: inequality

text: in

text: and

text: Lemma

text: will

text: complete

text: the

text: rest

text: of

text: the

text: proof.

text: In

text: the

text: next

text: theorem,

text: we

text: derive

text: optimal

text: convergence

text: results

text: of

text: the

text: FE

text: scheme

text: in

text: the

text: for

text: both

text: smooth

text: and

text: nonsmooth

text: initial

text: data

text: u

text: For

text: u

text: H˙

text: δ(Ω)

text: with

text: −0

text: δ

text: we

text: show

text: that

text: the

text: error

text: is

text: bounded

text: by

text: for

text: each

text: t

text: T].

text: Recall

text: that,

text: H˙

text: δ(Ω)

text: v

text: Hδ(Ω)

text: v

text: on

text: ∂Ω

text: for

text: δ

text: while

text: H˙

text: δ(Ω)

text: Hδ(Ω)

text: for

text: δ

text: Noting

text: that,

text: in

text: the

text: limiting

text: case

text: α

text: we

text: recover

text: the

text: L

text: convergence

text: rates

text: for

text: the

text: parabolic

text: equation

text: u

text: u

text: f.

text: Theorem

text: Let

text: u

text: and

text: u

text: be

text: the

text: solutions

text: of

text: and

text: respectively,

text: with

text: h

text: u

text: P

text: u

text: Then,

text: for

text: u

text: H˙

text: δ(Ω),

text: h

text: h

text: (u

text: u

text: )(t)

text: (u

text: f)

text: for

text: t

text: T]

text: with

text: δ

text: h

text: δ

text: Proof.

text: The

text: desired

text: result

text: follows

text: from

text: the

text: decomposition

text: u

text: u

text: ρ

text: θ,

text: the

text: h

text: estimate

text: of

text: θ

text: in

text: Lemma

text: the

text: bound

text: in

text: and

text: the

text: estimate

text: of

text: ρ

text: in

text: Remark

text: In

text: the

text: proof

text: of

text: the

text: above

text: theorem,

text: we

text: used

text: which

text: follows

text: from

text: the

text: projection

text: estimate

text: in

text: for

text: m

text: For

text: m

text: we

text: follow

text: similar

text: steps

text: B

text: where

text: δ

text: will

text: be

text: replaced

text: with

text: δ,

text: to

text: obtain

text: (t)

text: C

text: f)

text: for

text: δ

text: δ

text: Now,

text: for

text: δ

text: we

text: notice

text: ﬁrst

text: that

text: for

text: q

text: t

text: with

text: t

text: max

text: t,

text: h

text: h

text: I

text: ρ(q)

text: ρ(t

text: ρ(q)

text: ρ(t

text: ρ

text: (t

text: ρ(t

text: h

text: h

text: h

text: h

text: B

text: Substitute

text: this

text: in

text: the

text: deﬁnition

text: of

text: deﬁned

text: in

text: we

text: observe

text: B

text: t

text: I

text: (t)

text: ρ

text: (s)

text: ds

text: ρ

text: (t

text: ρ(t

text: h

text: h

text: To

text: estimate

text: the

text: ﬁrst

text: two

text: terms,

text: we

text: use

text: (with

text: m

text: and

text: the

text: following

text: regu-

text: larity

text: property

text: (which

text: follows

text: from

text: Theorems

text: and

text: u

text: (t)

text: Ctα(δ

text: (u

text: f),

text: t

text: T],

text: for

text: δ

text: where

text: f)

text: δ

text: tm

text: f(m)

text: L∞(H˙

text: we

text: arrive

text: to

text: B

text: (t)

text: (u

text: f)

text: ρ(t

text: for

text: δ

text: h

text: δ

text: h

text: However,

text: by

text: and

text: the

text: inequality

text: tα(δ

text: Ct

text: /t)α(δ

text: for

text: h

text: h

text: δ

text: we

text: ﬁnd

text: that

text: ρ(t

text: (u

text: f)

text: Cht

text: /t)α(δ

text: (u

text: f).

text: h

text: h

text: δ

text: h

text: δ

text: Therefore,

text: B

text: (t)

text: /t)α(δ

text: f),

text: for

text: δ

text: h

text: δ

text: B

text: Consequently,

text: by

text: using

text: the

text: above

text: bound

text: of

text: in

text: Theorem

text: we

text: get

text: the

text: error

text: estimate

text: below

text: that

text: will

text: be

text: used

text: in

text: the

text: forthcoming

text: section

text: to

text: show

text: the

text: convergence

text: of

text: the

text: gradient

text: FE

text: solution.

text: For

text: t

text: T],

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Analysis Of Nonstationary Modulated Time Series With Applications to Oceanographic Surface Flow Measurements ∗ Arthur P. Guillaumin 1, Adam M. Sykulski1, Soﬁa C. Olhede1, Jeﬀrey J. Early2, and Jonathan M. Lilly2 1Department of Statistical Science, University College London, UK 2NorthWest Research Associates, Seattle WA, USA Abstract We propose a new class of univariate nonstationary time series models, using the framework of modulated time series, which is appropriate for the analysis of rapidly-evolving time series as well as time series observa- tions with missing data. We extend our techniques to a class of bivariate time series that are isotropic. Exact inference is often not computationally viable for time series analysis, and so we propose an estimation method based on the Whittle-likelihood, a commonly adopted pseudo-likelihood. Our inference procedure is shown to be consistent under standard assumptions, as well as having considerably lower computational cost than exact likelihood in general. We show the utility of this framework for the analysis of drifting instruments, an analysis that is key to characterising global ocean circulation and therefore also for decadal to century-scale climate understanding. Keywords: modulation; nonstationary; periodogram; Whittle likelihood; missing data; surface drifters 1 Introduction This paper introduces a new family of rapidly-evolving time series models, inspired by real data applications, and then develops the appropriate analysis tools for their computationally-eﬃcient and consistent inference. Sta- tistical models for time series observations are usually described by their expectations and covariance structure. Classic families of covariance structure correspond to stationary covariances, governed only by the temporal lags between observed values of the process. The assumption of stationarity greatly simpliﬁes analysis, as it renders the covariance structure homogeneous across time and this motivates averaging for estimation. Unfortunately most often this homogeneous time structure is inadequate as a model for real-world applications, and does not reﬂect the variability of the observed time series. In order to analyse nonstationary time series, using the framework of locally stationary time series is standard (Priestley, 1988, Dahlhaus, 1997). The idea is to allow for a time-varying spectral density. Parametric models for the time-varying spectral density can be ﬁtted via the use of local Fourier transforms, usually requiring a spectral smoothness assumption. The concept of inﬁll asymptotics developed by Dahlhaus (1997) is based on the idea that a growing amount of data is obtained locally in time. Normally, for nonstationary time series analysis, there is a bias-variance trade-oﬀ that occurs when selecting the length of an analysis window. Longer windows will decrease variance, but will simultaneously increase bias due to the variation of the covariance function over the analysis window (Adak, 1998). In our case we shall eliminate the bias, and this will enable us to use longer time window lengths. For this purpose we exploit the notion of a modulated process (Parzen, 1963, Priestley, 1965). A modulated process is a latent stationary process multiplied pointwise by a modulating function. If we observe the modulating function, this framework allows us to deﬁne an averaged autocovariance function, despite the clear nonstationarity of the modulated process. This in turn allows us to introduce the Fourier transform of the averaged autocovariance function of the modulated process, which is equal to its expected periodogram. Through examining the expected periodogram, the properties controlling the latent random process may be inferred even when the modulating function changes very rapidly. The standard class of modulated processes are asymptotically stationary modulated processes (Parzen, 1963, Toloi and Morettin, 1989, Jiang and Hui, 2004). Here the autocovariance of the modulating function converges to a ﬁxed function, which is too restrictive for our real-world application. We introduce a more general class, which we call modulated processes with a signiﬁcant correlation contribution. This more ﬂexible model will still allow ∗ Corresponding author: arthur.guillaumin.14@ucl.ac.uk 1 ##LTLine##

text: Analysis

label: title

text: Of

text: Nonstationary

text: Modulated

text: Time

text: Series

text: With

text: Applications

text: to

text: Oceanographic

text: Surface

text: Flow

text: Measurements

label: author

text: Arthur

text: P.

text: Guillaumin

text: Adam

text: M.

text: Soﬁa

text: C.

text: Jeﬀrey

text: J.

text: and

text: Jonathan

text: M.

text: of

text: Statistical

text: Science,

text: University

text: College

text: London,

text: UK

text: Research

text: Associates,

text: Seattle

text: WA,

text: USA

text: Abstract

text: We

text: propose

text: a

text: new

text: class

text: of

text: univariate

text: nonstationary

text: time

text: series

text: models,

text: using

text: the

text: framework

text: of

text: modulated

text: time

text: series,

text: which

text: is

text: appropriate

text: for

text: the

text: analysis

text: of

text: rapidly-evolving

text: time

text: series

text: as

text: well

text: as

text: time

text: series

text: observa-

text: tions

text: with

text: missing

text: data.

text: We

text: extend

text: our

text: techniques

text: to

text: a

text: class

text: of

text: bivariate

text: time

text: series

text: that

text: are

text: isotropic.

text: Exact

text: inference

text: is

text: often

text: not

text: computationally

text: viable

text: for

text: time

text: series

text: analysis,

text: and

text: so

text: we

text: propose

text: an

text: estimation

text: method

text: based

text: on

text: the

text: Whittle-likelihood,

text: a

text: commonly

text: adopted

text: pseudo-likelihood.

text: Our

text: inference

text: procedure

text: is

text: shown

text: to

text: be

text: consistent

text: under

text: standard

text: assumptions,

text: as

text: well

text: as

text: having

text: considerably

text: lower

text: computational

text: cost

text: than

text: exact

text: likelihood

text: in

text: general.

text: We

text: show

text: the

text: utility

text: of

text: this

text: framework

text: for

text: the

text: analysis

text: of

text: drifting

text: instruments,

text: an

text: analysis

text: that

text: is

text: key

text: to

text: characterising

text: global

text: ocean

text: circulation

text: and

text: therefore

text: also

text: for

text: decadal

text: to

text: century-scale

text: climate

text: understanding.

text: Keywords:

text: modulation;

text: nonstationary;

text: periodogram;

text: Whittle

text: likelihood;

text: missing

text: data;

text: surface

text: drifters

text: Introduction

text: This

text: paper

text: introduces

text: a

text: new

text: family

text: of

text: rapidly-evolving

text: time

text: series

text: models,

text: inspired

text: by

text: real

text: data

text: applications,

text: and

text: then

text: develops

text: the

text: appropriate

text: analysis

text: tools

text: for

text: their

text: computationally-eﬃcient

text: and

text: consistent

text: inference.

text: Sta-

text: tistical

text: models

text: for

text: time

text: series

text: observations

text: are

text: usually

text: described

text: by

text: their

text: expectations

text: and

text: covariance

text: structure.

text: Classic

text: families

text: of

text: covariance

text: structure

text: correspond

text: to

text: stationary

text: covariances,

text: governed

text: only

text: by

text: the

text: temporal

text: lags

text: between

text: observed

text: values

text: of

text: the

text: process.

text: The

text: assumption

text: of

text: stationarity

text: greatly

text: simpliﬁes

text: analysis,

text: as

text: it

text: renders

text: the

text: covariance

text: structure

text: homogeneous

text: across

text: time

text: and

text: this

text: motivates

text: averaging

text: for

text: estimation.

text: Unfortunately

text: most

text: often

text: this

text: homogeneous

text: time

text: structure

text: is

text: inadequate

text: as

text: a

text: model

text: for

text: real-world

text: applications,

text: and

text: does

text: not

text: reﬂect

text: the

text: variability

text: of

text: the

text: observed

text: time

text: series.

text: In

text: order

text: to

text: analyse

text: nonstationary

text: time

text: series,

text: using

text: the

text: framework

text: of

text: locally

text: stationary

text: time

text: series

text: is

text: standard

text: (Priestley,

text: Dahlhaus,

text: The

text: idea

text: is

text: to

text: allow

text: for

text: a

text: time-varying

text: spectral

text: density.

text: Parametric

text: models

text: for

text: the

text: time-varying

text: spectral

text: density

text: can

text: be

text: ﬁtted

text: via

text: the

text: use

text: of

text: local

text: Fourier

text: transforms,

text: usually

text: requiring

text: a

text: spectral

text: smoothness

text: assumption.

text: The

text: concept

text: of

text: inﬁll

text: asymptotics

text: developed

text: by

text: Dahlhaus

text: is

text: based

text: on

text: the

text: idea

text: that

text: a

text: growing

text: amount

text: of

text: data

text: is

text: obtained

text: locally

text: in

text: time.

text: Normally,

text: for

text: nonstationary

text: time

text: series

text: analysis,

text: there

text: is

text: a

text: bias-variance

text: trade-oﬀ

text: that

text: occurs

text: when

text: selecting

text: the

text: length

text: of

text: an

text: analysis

text: window.

text: Longer

text: windows

text: will

text: decrease

text: variance,

text: but

text: will

text: simultaneously

text: increase

text: bias

text: due

text: to

text: the

text: variation

text: of

text: the

text: covariance

text: function

text: over

text: the

text: analysis

text: window

text: (Adak,

text: In

text: our

text: case

text: we

text: shall

text: eliminate

text: the

text: bias,

text: and

text: this

text: will

text: enable

text: us

text: to

text: use

text: longer

text: time

text: window

text: lengths.

text: For

text: this

text: purpose

text: we

text: exploit

text: the

text: notion

text: of

text: a

text: modulated

text: process

text: (Parzen,

text: Priestley,

text: A

text: modulated

text: process

text: is

text: a

text: latent

text: stationary

text: process

text: multiplied

text: pointwise

text: by

text: a

text: modulating

text: function.

text: If

text: we

text: observe

text: the

text: modulating

text: function,

text: this

text: framework

text: allows

text: us

text: to

text: deﬁne

text: an

text: averaged

text: autocovariance

text: function,

text: despite

text: the

text: clear

text: nonstationarity

text: of

text: the

text: modulated

text: process.

text: This

text: in

text: turn

text: allows

text: us

text: to

text: introduce

text: the

text: Fourier

text: transform

text: of

text: the

text: averaged

text: autocovariance

text: function

text: of

text: the

text: modulated

text: process,

text: which

text: is

text: equal

text: to

text: its

text: expected

text: periodogram.

text: Through

text: examining

text: the

text: expected

text: periodogram,

text: the

text: properties

text: controlling

text: the

text: latent

text: random

text: process

text: may

text: be

text: inferred

text: even

text: when

text: the

text: modulating

text: function

text: changes

text: very

text: rapidly.

text: The

text: standard

text: class

text: of

text: modulated

text: processes

text: are

text: asymptotically

text: stationary

text: modulated

text: processes

text: (Parzen,

text: Toloi

text: and

text: Morettin,

text: Jiang

text: and

text: Hui,

text: Here

text: the

text: autocovariance

text: of

text: the

text: modulating

text: function

text: converges

text: to

text: a

text: ﬁxed

text: function,

text: which

text: is

text: too

text: restrictive

text: for

text: our

text: real-world

text: application.

text: We

text: introduce

text: a

text: more

text: general

text: class,

text: which

text: we

text: call

text: modulated

text: processes

text: with

text: a

text: signiﬁcant

text: correlation

text: contribution.

text: This

text: more

text: ﬂexible

text: model

text: will

text: still

text: allow

text: Corresponding

text: author:

text: ##LTLine##


text: The process (22) is a nonstationary version of the complex-valued ﬁrst order autoregressive process (Sykulski et al., 2016a) introduced by Le Breton (1988), and also a discrete-time analogue of the complex-valued Ornstein- Ulhenbeck (OU) process (Arat´o et al., 1962) with time-vary(cid:101)ing oscillation frequency. We now prove in Proposition 5 that the model deﬁned in (22) belongs to our class of bivariate modulated processes. { } P(cid:101)roposition 5 (Modulated process representation). Let Z be the process deﬁned in (22). T(cid:101)here exists a unit- t { } magnitude complex-valued modulating sequence g , and a stationary complex-valued proper process Z such that { } { } t { } t Z is the modulation of Z by the non-random sequence g . More explicitly, we have Z = g Z , for all ∈tN t t t t t t , where, (cid:80) gt = ei tu=1 βu, (23) (cid:48) ≥ Z = rZ − + (cid:15) , t 1, t t 1 t ∼ N − (cid:48) and Z (0, σ2/(1 r2)). The process (cid:15) is a Gaussian white noise process with the same properties 0 C t (zero-mean, variance σ2 and independence of real and imaginary parts) as those of (cid:15) . Deﬁned as such, the latent t complex-valued process Z is stationary and proper. t Proof. See Appendix . The stationary latent process Z deﬁned in (23) is a stationary complex-valued ﬁrst order autoregressive process, t and is Gaussian. Its autocovariance sequence is given by, σ2 ∈ Z c (τ) = − rτ, τ . Z 1 r2 (cid:55)→ It is easy to verify that the mapping (r, σ) (c (0), c (1)) is a one-to-one mapping. In the following proposition, Z Z we stipulate a suﬃcient condition on the frequencies β so that the process deﬁned in (22) satisﬁes our assumption t of signiﬁcant correlation contribution, when represented as a modulated process as deﬁned in Proposition 5. ∈ − ≤ ≤ Proposition 6. Let Z˜ be the process deﬁned by (22). Assume that there exists Ξ [ π, π) and 0 ∆ π such that for all t ∈ N, |β −t Ξ| ≤ ∆. Then Z˜ is a modulated process with signiﬁcant correlation contribution. 2 t t Proof. See Appendix . Hence the complex-valued autoregressive process deﬁned by (22) belongs to the class of processes with a signiﬁcant correlation contribution, and the expected periodogram is a one-to-one mapping from the parameter set × ∞ − [0, 1) [0, ) to the set of non-negative continuous functions on [ π, π], according to Proposition 3—a proposition which is readily extended to the complex-valued processes of this section. 3 Parametric estimation of modulated processes We have explored a class of univariate and bivariate modulat(cid:101)ed processes. The next stage is to describe their { } eﬃcient inference. In this section we describe how the parameters of the latent model for X can be inferred { } t from observing a single realization of the modulated process X . Most authors have focused on the problem t of estimating modulated processes under the assumption of asymptotic stationarity as deﬁned in Deﬁnition 2 (Parzen, 1963, Dunsmuir and Robinson, 1981a,b, Toloi and Morettin, 1989). Although non-parametric estimates have been the key concern in most of the relevant literature, there have been instances where parametric estimation has been considered, see for instance Dunsmuir and Robinson (1981a). Parametric estimation ensures that the estimated autocovariance sequence cˆ (τ) is non-negative deﬁnite, as opposed to using non-parametric estimates X of the form given in (8). Parametric estimation is also preferable when the true model is known, as it uses the observed degrees of freedom more eﬃciently. Herein we consider the problem of parametric estimation for our class of modulated processes with a signiﬁcant correlation contribution, which, we recall, is a generalization of asymptotically stationary modulated processes. We propose an adaptation of the Whittle likelihood (Whittle, 1953), based on the expected (cid:101)period(cid:101)ogram. (cid:101) { } We wish to infer the parameter vector θ of the latent univariate stationary process X within the parameter · · · {t · · · − } set Θ, based on the sample X = (cid:101)X , , X − and the known modulating sequence g : t = 0, , N 1 . (cid:8) (cid:9) 0 N 1 t Because it has been assumed that the latent process is a zero-mean Gaussian process, the same is true for the × mo(cid:101)dulated p(cid:101)rocess. The vector X is multivariate Gaussian with an expected N N autocovari(cid:101)ance matrix · · · − C (θ) = c (t , t ; θ) for t , t = 0, , N 1, where the components of this matrix are given by c (t , t ; θ) = X −X 1 2 1 2 { } X 1 2 g g c (t t ; θ). However, the parameter vector θ of the latent process X can be uniquely determined from t1 t2 X 2 1 t 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: process

text: is

text: a

text: nonstationary

text: version

text: of

text: the

text: complex-valued

text: ﬁrst

text: order

text: autoregressive

text: process

text: (Sykulski

text: et

text: al.,

text: introduced

text: by

text: Le

text: Breton

text: and

text: also

text: a

text: discrete-time

text: analogue

text: of

text: the

text: complex-valued

text: Ornstein-

text: Ulhenbeck

text: (OU)

text: process

text: (Arat´o

text: et

text: al.,

text: with

text: oscillation

text: frequency.

text: We

text: now

text: prove

text: in

text: Proposition

text: that

text: the

text: model

text: deﬁned

text: in

text: belongs

text: to

text: our

text: class

text: of

text: bivariate

text: modulated

text: processes.

text: (Modulated

text: process

text: representation).

text: Let

text: Z

text: be

text: the

text: process

text: deﬁned

text: in

text: exists

text: a

text: unit-

text: t

text: magnitude

text: complex-valued

text: modulating

text: sequence

text: g

text: and

text: a

text: stationary

text: complex-valued

text: proper

text: process

text: Z

text: such

text: that

text: t

text: t

text: Z

text: is

text: the

text: modulation

text: of

text: Z

text: by

text: the

text: non-random

text: sequence

text: g

text: More

text: explicitly,

text: we

text: have

text: Z

text: g

text: Z

text: for

text: all

text: ∈tN

text: t

text: t

text: t

text: t

text: t

text: t

text: where,

text: gt

text: ei

text: βu,

text: Z

text: rZ

text: t

text: t

text: t

text: t

text: N

text: and

text: Z

text: The

text: process

text: is

text: a

text: Gaussian

text: white

text: noise

text: process

text: with

text: the

text: same

text: properties

text: C

text: t

text: (zero-mean,

text: variance

text: σ2

text: and

text: independence

text: of

text: real

text: and

text: imaginary

text: parts)

text: as

text: those

text: of

text: Deﬁned

text: as

text: such,

text: the

text: latent

text: t

text: complex-valued

text: process

text: Z

text: is

text: stationary

text: and

text: proper.

text: t

text: Proof.

text: See

text: Appendix

text: The

text: stationary

text: latent

text: process

text: Z

text: deﬁned

text: in

text: is

text: a

text: stationary

text: complex-valued

text: ﬁrst

text: order

text: autoregressive

text: process,

text: t

text: and

text: is

text: Gaussian.

text: Its

text: autocovariance

text: sequence

text: is

text: given

text: by,

text: σ2

text: Z

text: c

text: (τ)

text: rτ,

text: τ

text: Z

text: It

text: is

text: easy

text: to

text: verify

text: that

text: the

text: mapping

text: (r,

text: σ)

text: (c

text: c

text: is

text: a

text: one-to-one

text: mapping.

text: In

text: the

text: following

text: proposition,

text: Z

text: Z

text: we

text: stipulate

text: a

text: suﬃcient

text: condition

text: on

text: the

text: frequencies

text: β

text: so

text: that

text: the

text: process

text: deﬁned

text: in

text: satisﬁes

text: our

text: assumption

text: t

text: of

text: signiﬁcant

text: correlation

text: contribution,

text: when

text: represented

text: as

text: a

text: modulated

text: process

text: as

text: deﬁned

text: in

text: Proposition

text: Proposition

text: Let

text: Z˜

text: be

text: the

text: process

text: deﬁned

text: by

text: Assume

text: that

text: there

text: exists

text: Ξ

text: π,

text: π)

text: and

text: π

text: such

text: that

text: for

text: all

text: t

text: N,

text: |β

text: −t

text: Ξ|

text: Then

text: Z˜

text: is

text: a

text: modulated

text: process

text: with

text: signiﬁcant

text: correlation

text: contribution.

text: t

text: t

text: Proof.

text: See

text: Appendix

text: Hence

text: the

text: complex-valued

text: autoregressive

text: process

text: deﬁned

text: by

text: belongs

text: to

text: the

text: class

text: of

text: processes

text: with

text: a

text: signiﬁcant

text: correlation

text: contribution,

text: and

text: the

text: expected

text: periodogram

text: is

text: a

text: one-to-one

text: mapping

text: from

text: the

text: parameter

text: set

text: to

text: the

text: set

text: of

text: non-negative

text: continuous

text: functions

text: on

text: π,

text: π],

text: according

text: to

text: Proposition

text: proposition

text: which

text: is

text: readily

text: extended

text: to

text: the

text: complex-valued

text: processes

text: of

text: this

text: section.

text: Parametric

text: estimation

text: of

text: modulated

text: processes

text: We

text: have

text: explored

text: a

text: class

text: of

text: univariate

text: and

text: bivariate

text: processes.

text: The

text: next

text: stage

text: is

text: to

text: describe

text: their

text: eﬃcient

text: inference.

text: In

text: this

text: section

text: we

text: describe

text: how

text: the

text: parameters

text: of

text: the

text: latent

text: model

text: for

text: X

text: can

text: be

text: inferred

text: t

text: from

text: observing

text: a

text: single

text: realization

text: of

text: the

text: modulated

text: process

text: X

text: Most

text: authors

text: have

text: focused

text: on

text: the

text: problem

text: t

text: of

text: estimating

text: modulated

text: processes

text: under

text: the

text: assumption

text: of

text: asymptotic

text: stationarity

text: as

text: deﬁned

text: in

text: Deﬁnition

text: (Parzen,

text: Dunsmuir

text: and

text: Robinson,

text: Toloi

text: and

text: Morettin,

text: Although

text: non-parametric

text: estimates

text: have

text: been

text: the

text: key

text: concern

text: in

text: most

text: of

text: the

text: relevant

text: literature,

text: there

text: have

text: been

text: instances

text: where

text: parametric

text: estimation

text: has

text: been

text: considered,

text: see

text: for

text: instance

text: Dunsmuir

text: and

text: Robinson

text: Parametric

text: estimation

text: ensures

text: that

text: the

text: estimated

text: autocovariance

text: sequence

text: cˆ

text: (τ)

text: is

text: non-negative

text: deﬁnite,

text: as

text: opposed

text: to

text: using

text: non-parametric

text: estimates

text: X

text: of

text: the

text: form

text: given

text: in

text: Parametric

text: estimation

text: is

text: also

text: preferable

text: when

text: the

text: true

text: model

text: is

text: known,

text: as

text: it

text: uses

text: the

text: observed

text: degrees

text: of

text: freedom

text: more

text: eﬃciently.

text: Herein

text: we

text: consider

text: the

text: problem

text: of

text: parametric

text: estimation

text: for

text: our

text: class

text: of

text: modulated

text: processes

text: with

text: a

text: signiﬁcant

text: correlation

text: contribution,

text: which,

text: we

text: recall,

text: is

text: a

text: generalization

text: of

text: asymptotically

text: stationary

text: modulated

text: processes.

text: We

text: propose

text: an

text: adaptation

text: of

text: the

text: Whittle

text: likelihood

text: (Whittle,

text: based

text: on

text: the

text: expected

text: We

text: wish

text: to

text: infer

text: the

text: parameter

text: vector

text: θ

text: of

text: the

text: latent

text: univariate

text: stationary

text: process

text: X

text: within

text: the

text: parameter

text: {t

text: set

text: Θ,

text: based

text: on

text: the

text: sample

text: X

text: X

text: and

text: the

text: known

text: modulating

text: sequence

text: g

text: t

text: N

text: N

text: t

text: Because

text: it

text: has

text: been

text: assumed

text: that

text: the

text: latent

text: process

text: is

text: a

text: zero-mean

text: Gaussian

text: process,

text: the

text: same

text: is

text: true

text: for

text: the

text: The

text: vector

text: X

text: is

text: multivariate

text: Gaussian

text: with

text: an

text: expected

text: N

text: N

text: matrix

text: C

text: (θ)

text: c

text: (t

text: t

text: θ)

text: for

text: t

text: t

text: N

text: where

text: the

text: components

text: of

text: this

text: matrix

text: are

text: given

text: by

text: c

text: (t

text: t

text: θ)

text: X

text: −X

text: X

text: g

text: g

text: c

text: (t

text: t

text: θ).

text: However,

text: the

text: parameter

text: vector

text: θ

text: of

text: the

text: latent

text: process

text: X

text: can

text: be

text: uniquely

text: determined

text: from

text: X

text: t

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: { } by 2πK, where K is one solar day in seconds, so that ω f is in cycles per day. The entire drifter dataset is split into segments of 60 inertial periods in length, accounting for the variation of the inertial period along drifter trajectories, and with 50% overlap between segments. The standard deviation of the inertial frequency along each data segment is taken, and the 200 segments exhibiting the largest ratio of the standard deviation of the inertial frequency, to the magnitude of its mean value along the segment, are identiﬁed for use in this study. These exhibit the largest fractional changes in the inertial frequency, and as shown in Fig. 1(a), are located in the vicinity of the equatorial region where inertial frequency vanishes. (a) (b) 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 (c) (d) 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0 0 -1 -0.5 0 0.5 1 -1.5 -1 -0.5 0 0.5 1 1.5 Figure 2: Fitted variance of the discrete Fourier transform using either the stationary model (in black) or the nonstationary model (in red) to the periodogram (in blue) for segments of data from drifter IDs (a) #79243, (b) #54656, (c) #71845 and (d) #44312. The solid black vertical line is the average inertial frequency, and the dashed vertical black lines are the minimum and maximum observed inertial frequency over the observed time window. The models are ﬁt in the frequency range of 0 to 0.8 cycles per day in (a)–(c), and from 0 to 1.5 cycles per day ◦ in (d) as this drifter is at a higher latitude of 37 S where inertial oscillations occur at a frequency of about 1.2 cycles per day. The ﬁtted models are shown in solid lines within the frequency range, and in dashed lines outside the frequency range. 4.1.2 Stochastic modelling The stochastic modelling of Lagrangian trajectories was investigated in Sykulski et al. (2016c), where the term “Lagrangian” is used because the moving object making the observations (i.e. the drifter) is the frame of reference, as opposed to ﬁxed-point measurements known as Eulerian observations. In that paper, the Lagrangian velocity time series was modelled as a complex-valued time series, with the following 6-parameter power spectral density: A2 B2 S(ω) = (ω − ω{f})2 + λ2 + (ω2 + h2)α , (29) { } ∈ − 1 A > 0, λ > 0, ω f [ π, π], h > 0, B > 0, α > , 2 where ω is giv(cid:101)en in cycles per day. The ﬁrst com(cid:101)ponent of (29) is the spectral density of a complex Ornstein- Uhlenbeck (OU) process (Arat´o et al., 1962), and is used to describe the eﬀect of inertial oscillations at frequency { } ω f . Denoting Z (t) the OU component, where Z (t) is complex-valued, these oscillations are described by OU OU 14 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: by

text: where

text: K

text: is

text: one

text: solar

text: day

text: in

text: seconds,

text: so

text: that

text: ω

text: f

text: is

text: in

text: cycles

text: per

text: day.

text: The

text: entire

text: drifter

text: dataset

text: is

text: split

text: into

text: segments

text: of

text: inertial

text: periods

text: in

text: length,

text: accounting

text: for

text: the

text: variation

text: of

text: the

text: inertial

text: period

text: along

text: drifter

text: trajectories,

text: and

text: with

text: overlap

text: between

text: segments.

text: The

text: standard

text: deviation

text: of

text: the

text: inertial

text: frequency

text: along

text: each

text: data

text: segment

text: is

text: taken,

text: and

text: the

text: segments

text: exhibiting

text: the

text: largest

text: ratio

text: of

text: the

text: standard

text: deviation

text: of

text: the

text: inertial

text: frequency,

text: to

text: the

text: magnitude

text: of

text: its

text: mean

text: value

text: along

text: the

text: segment,

text: are

text: identiﬁed

text: for

text: use

text: in

text: this

text: study.

text: These

text: exhibit

text: the

text: largest

text: fractional

text: changes

text: in

text: the

text: inertial

text: frequency,

text: and

text: as

text: shown

text: in

text: Fig.

text: are

text: located

text: in

text: the

text: vicinity

text: of

text: the

text: equatorial

text: region

text: where

text: inertial

text: frequency

text: vanishes.

text: (a)

text: (b)

text: (c)

text: (d)

text: Figure

text: Fitted

text: variance

text: of

text: the

text: discrete

text: Fourier

text: transform

text: using

text: either

text: the

text: stationary

text: model

text: (in

text: black)

text: or

text: the

text: nonstationary

text: model

text: (in

text: red)

text: to

text: the

text: periodogram

text: (in

text: blue)

text: for

text: segments

text: of

text: data

text: from

text: drifter

text: IDs

text: (a)

text: (b)

text: (c)

text: and

text: (d)

text: The

text: solid

text: black

text: vertical

text: line

text: is

text: the

text: average

text: inertial

text: frequency,

text: and

text: the

text: dashed

text: vertical

text: black

text: lines

text: are

text: the

text: minimum

text: and

text: maximum

text: observed

text: inertial

text: frequency

text: over

text: the

text: observed

text: time

text: window.

text: The

text: models

text: are

text: ﬁt

text: in

text: the

text: frequency

text: range

text: of

text: to

text: cycles

text: per

text: day

text: in

text: (a)–(c),

text: and

text: from

text: to

text: cycles

text: per

text: day

text: in

text: (d)

text: as

text: this

text: drifter

text: is

text: at

text: a

text: higher

text: latitude

text: of

text: S

text: where

text: inertial

text: oscillations

text: occur

text: at

text: a

text: frequency

text: of

text: about

text: cycles

text: per

text: day.

text: The

text: ﬁtted

text: models

text: are

text: shown

text: in

text: solid

text: lines

text: within

text: the

text: frequency

text: range,

text: and

text: in

text: dashed

text: lines

text: outside

text: the

text: frequency

text: range.

text: Stochastic

text: modelling

text: The

text: stochastic

text: modelling

text: of

text: Lagrangian

text: trajectories

text: was

text: investigated

text: in

text: Sykulski

text: et

text: al.

text: where

text: the

text: term

text: “Lagrangian”

text: is

text: used

text: because

text: the

text: moving

text: object

text: making

text: the

text: observations

text: (i.e.

text: the

text: drifter)

text: is

text: the

text: frame

text: of

text: reference,

text: as

text: opposed

text: to

text: ﬁxed-point

text: measurements

text: known

text: as

text: Eulerian

text: observations.

text: In

text: that

text: paper,

text: the

text: Lagrangian

text: velocity

text: time

text: series

text: was

text: modelled

text: as

text: a

text: complex-valued

text: time

text: series,

text: with

text: the

text: following

text: power

text: spectral

text: density:

text: S(ω)

text: (ω

text: λ2

text: A

text: λ

text: ω

text: f

text: π,

text: π],

text: h

text: B

text: α

text: where

text: ω

text: is

text: in

text: cycles

text: per

text: day.

text: The

text: ﬁrst

text: of

text: is

text: the

text: spectral

text: density

text: of

text: a

text: complex

text: Ornstein-

text: Uhlenbeck

text: (OU)

text: process

text: (Arat´o

text: et

text: al.,

text: and

text: is

text: used

text: to

text: describe

text: the

text: eﬀect

text: of

text: inertial

text: oscillations

text: at

text: frequency

text: ω

text: f

text: Denoting

text: Z

text: (t)

text: the

text: OU

text: component,

text: where

text: Z

text: (t)

text: is

text: complex-valued,

text: these

text: oscillations

text: are

text: described

text: by

text: OU

text: OU

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text: Figure 1: (a) The trajectories of the 200 drifters from the Global Drifter Program, analysed in Section , that exhibit the greatest change in Coriolis frequency (f) across 60 inertial cycles, as described in that section; (b) a segment of data of the meridional (latitudinal) positions over time from Drifter ID#43594; and (c) a segment of data of the meridional velocities from this drifter in cm/s. This ﬁgure is produced using the jLab toolbox (Lilly, 2016). · { } (cid:88) transform S ( ), also known as the spectral density of X , X t ∞ 1 − ∈ − S (ω) = c (τ)e iωτ, ω [ π, π]. X 2π −∞ X τ= (cid:90) The spectral density S (ω) is then a continuous function of ω. The corresponding inversion formulae is given for X all integer value τ by π c (τ) = S (ω)eiωτdω. X − X π A consequence of stationarity is that the quantities in question can be stably estimated by averaging in time (Brockwell and Davis, 1991). If c (t, t + τ) is not stationary, but is slowly varying in time, then it can be X estimated by dividing the observed data into multiple segments and performing inference on each segment (Adak, 1998). This does not hold in settings where the time variation is too rapid. Our goal is to estimate c (t, t + τ) in X such settings, in particular when a parametric speciﬁcation is made for the function. 2.1 Classes of modulated processes Modulation is a natural and simple method of producing a nonstationary process (Parzen, 1963). A univariate modulated process is deﬁned as follows. { ∈ N} Deﬁnition 1 (Modulated process). Let X : t be a Gaussian, real-valued, zero-mean stationary process. { ∈ N} t Let g : t be a given bounded real-valued deterministic sequence. Then a modulated process is deﬁned as t one taking the form (cid:101) X = g X (1) t t t ∈ N at all time points t . { } { } Herein we treat g as a known deterministic signal. In our setting the process X , which is referred to as t ∈ ⊂ R t the latent process, is modelled through a ﬁnite set of parameters θ Θ d, where d is a positive integer and Θ is the parameter space. Usually our object of interest is θ, the particular values of parameters that generated the observed realization. For example, if the latent process is an autoregressive process of order p, we then have d = p + 1 if the mean is known (p regressive parameters and the variance of the innovations). We denote the { } autocovariance function of the stationary zero-mean process X by c (τ), or c (τ; θ) when we want to make the t X X dependence on θ explicit. Its Fourier transform, the spectral density, is denoted S (ω) or S (ω; θ), respectively. X X The modulation of the latent process X is a convenient mechanism to account for a wide range of nonstationary t processes. In particular this mechanism has been widely used as a modelling tool for missing data problems,(cid:101)where g is assigned values 0 or 1 when respectively missing or obser(cid:101)ving a data point in time (Jones, 1962). t To understand when we can recover the parameters controlling the latent process X from observing X , (cid:101)we t t need to put further conditions in place on g . The tim(cid:101)e series X /g cannot always be formed as g may be zero for t t t t some time indices, corresponding to missing observations. Another reason is that we may not directly observe X , t but instead we may observe an aggregated process X + Z , where Z is a stationary process (or more generally t t t 3 ##LTFigure## ##LTFigure## ##LTLine##

text: Figure

text: (a)

text: The

text: trajectories

text: of

text: the

text: drifters

text: from

text: the

text: Global

text: Drifter

text: Program,

text: analysed

text: in

text: Section

text: that

text: exhibit

text: the

text: greatest

text: change

text: in

text: Coriolis

text: frequency

text: (f)

text: across

text: inertial

text: cycles,

text: as

text: described

text: in

text: that

text: section;

text: (b)

text: a

text: segment

text: of

text: data

text: of

text: the

text: meridional

text: (latitudinal)

text: positions

text: over

text: time

text: from

text: Drifter

text: and

text: (c)

text: a

text: segment

text: of

text: data

text: of

text: the

text: meridional

text: velocities

text: from

text: this

text: drifter

text: in

text: cm/s.

text: This

text: ﬁgure

text: is

text: produced

text: using

text: the

text: jLab

text: toolbox

text: (Lilly,

text: transform

text: S

text: also

text: known

text: as

text: the

text: spectral

text: density

text: of

text: X

text: X

text: t

text: S

text: (ω)

text: c

text: (τ)e

text: iωτ,

text: ω

text: π,

text: π].

text: X

text: 2π

text: X

text: τ=

text: The

text: spectral

text: density

text: S

text: (ω)

text: is

text: then

text: a

text: continuous

text: function

text: of

text: ω.

text: The

text: corresponding

text: inversion

text: formulae

text: is

text: given

text: for

text: X

text: all

text: integer

text: value

text: τ

text: by

text: π

text: c

text: (τ)

text: S

text: (ω)eiωτdω.

text: X

text: X

text: π

text: A

text: consequence

text: of

text: stationarity

text: is

text: that

text: the

text: quantities

text: in

text: question

text: can

text: be

text: stably

text: estimated

text: by

text: averaging

text: in

text: time

text: (Brockwell

text: and

text: Davis,

text: If

text: c

text: (t,

text: t

text: τ)

text: is

text: not

text: stationary,

text: but

text: is

text: slowly

text: varying

text: in

text: time,

text: then

text: it

text: can

text: be

text: X

text: estimated

text: by

text: dividing

text: the

text: observed

text: data

text: into

text: multiple

text: segments

text: and

text: performing

text: inference

text: on

text: each

text: segment

text: (Adak,

text: This

text: does

text: not

text: hold

text: in

text: settings

text: where

text: the

text: time

text: variation

text: is

text: too

text: rapid.

text: Our

text: goal

text: is

text: to

text: estimate

text: c

text: (t,

text: t

text: τ)

text: in

text: X

text: such

text: settings,

text: in

text: particular

text: when

text: a

text: parametric

text: speciﬁcation

text: is

text: made

text: for

text: the

text: function.

text: Classes

text: of

text: modulated

text: processes

text: Modulation

text: is

text: a

text: natural

text: and

text: simple

text: method

text: of

text: producing

text: a

text: nonstationary

text: process

text: (Parzen,

text: A

text: univariate

text: modulated

text: process

text: is

text: deﬁned

text: as

text: follows.

text: N}

text: Deﬁnition

text: (Modulated

text: process).

text: Let

text: X

text: t

text: be

text: a

text: Gaussian,

text: real-valued,

text: zero-mean

text: stationary

text: process.

text: N}

text: t

text: Let

text: g

text: t

text: be

text: a

text: given

text: bounded

text: real-valued

text: deterministic

text: sequence.

text: Then

text: a

text: modulated

text: process

text: is

text: deﬁned

text: as

text: t

text: one

text: taking

text: the

text: form

text: X

text: g

text: X

text: t

text: t

text: t

text: N

text: at

text: all

text: time

text: points

text: t

text: Herein

text: we

text: treat

text: g

text: as

text: a

text: known

text: deterministic

text: signal.

text: In

text: our

text: setting

text: the

text: process

text: X

text: which

text: is

text: referred

text: to

text: as

text: t

text: R

text: t

text: the

text: latent

text: process,

text: is

text: modelled

text: through

text: a

text: ﬁnite

text: set

text: of

text: parameters

text: θ

text: Θ

text: d,

text: where

text: d

text: is

text: a

text: positive

text: integer

text: and

text: Θ

text: is

text: the

text: parameter

text: space.

text: Usually

text: our

text: object

text: of

text: interest

text: is

text: θ,

text: the

text: particular

text: values

text: of

text: parameters

text: that

text: generated

text: the

text: observed

text: realization.

text: For

text: example,

text: if

text: the

text: latent

text: process

text: is

text: an

text: autoregressive

text: process

text: of

text: order

text: p,

text: we

text: then

text: have

text: d

text: p

text: if

text: the

text: mean

text: is

text: known

text: (p

text: regressive

text: parameters

text: and

text: the

text: variance

text: of

text: the

text: innovations).

text: We

text: denote

text: the

text: autocovariance

text: function

text: of

text: the

text: stationary

text: zero-mean

text: process

text: X

text: by

text: c

text: (τ),

text: or

text: c

text: (τ;

text: θ)

text: when

text: we

text: want

text: to

text: make

text: the

text: t

text: X

text: X

text: dependence

text: on

text: θ

text: explicit.

text: Its

text: Fourier

text: transform,

text: the

text: spectral

text: density,

text: is

text: denoted

text: S

text: (ω)

text: or

text: S

text: (ω;

text: θ),

text: respectively.

text: X

text: X

text: The

text: modulation

text: of

text: the

text: latent

text: process

text: X

text: is

text: a

text: convenient

text: mechanism

text: to

text: account

text: for

text: a

text: wide

text: range

text: of

text: nonstationary

text: t

text: processes.

text: In

text: particular

text: this

text: mechanism

text: has

text: been

text: widely

text: used

text: as

text: a

text: modelling

text: tool

text: for

text: missing

text: data

text: g

text: is

text: assigned

text: values

text: or

text: when

text: respectively

text: missing

text: or

text: a

text: data

text: point

text: in

text: time

text: (Jones,

text: t

text: To

text: understand

text: when

text: we

text: can

text: recover

text: the

text: parameters

text: controlling

text: the

text: latent

text: process

text: X

text: from

text: observing

text: X

text: t

text: t

text: need

text: to

text: put

text: further

text: conditions

text: in

text: place

text: on

text: g

text: The

text: series

text: X

text: /g

text: cannot

text: always

text: be

text: formed

text: as

text: g

text: may

text: be

text: zero

text: for

text: t

text: t

text: t

text: t

text: some

text: time

text: indices,

text: corresponding

text: to

text: missing

text: observations.

text: Another

text: reason

text: is

text: that

text: we

text: may

text: not

text: directly

text: observe

text: X

text: t

text: but

text: instead

text: we

text: may

text: observe

text: an

text: aggregated

text: process

text: X

text: Z

text: where

text: Z

text: is

text: a

text: stationary

text: process

text: (or

text: more

text: generally

text: t

text: t

text: t

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##