from structured data to evolution linear partial di ...elorin/paperbvp.pdf · from structured data...
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From structured data to evolution linear partial differential
equations
E. Lorina,b
aSchool of Mathematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6bCentre de Recherches Mathematiques, Universite de Montreal, Montreal, Canada, H3T 1J4
Abstract
This paper is devoted to the derivation of computational methods for constructing partialdifferential equations from data. Following some recent works [6, 12, 13, 17], we proposea methodology based on symbolic calculus [7, 8, 11], pseudospectral methods [2, 3] andstochastic processes [5], in order to determine non-constant coefficients of linear evolutionPartial Differential Equations (PDEs), from a set of structured data constituted by solutions,at given times and positions of the unknown PDE. Based on machine learning paradigm,the refinement of the reconstructed coefficients requires additional data.
Keywords: Machine learning, partial differential equations, numerical approximation,operator symbols, pseudospectral methods.
1. Introduction
Machine learning is one of the most active research areas in applied mathematics, statis-tics and computer science. Among fundamental problems in this field is the reconstructionof differential equations (DE) from experimental data [6, 12, 13, 17]. In this work, wepropose simple and direct methods for this problem. More specifically, we assume that aset of “data” corresponding to the solution of an unknown equation, are given, along withthe corresponding times and positions. This is typically what “in-lab” experiments wouldprovide, making the proposed approach relevant from a practical point of view. A similarparadigm was used in [6] with techniques distinct, from the ones develiped in this paper.This problem is very complex, as the differential equation could be in principle, of any kind,high order, fractional, nonlinear, non-homogeneous, etc. In this work, we will assume thatthe searched equation is a linear evolution equation, with non-constant coefficients. To acertain extent, quasilinear equations will also be considered. The latter assumption is themain drawback of the presented approach, as the “general form” of the searched model isimposed, which may lead to an inaccurate equation. However, in modeling in classical orquantum physics/chemistry/engineering [4], the general form of the searched equation is of-ten preliminarily known, but not necessarily their coefficients.
Email address: [email protected] (E. Lorin)
Preprint submitted to Elsevier September 18, 2018
From a set of given experimental data: i) vn := {vnj }j∈Zd,n∈N, ii) at locations D = {xj}j∈Zd ,iii) and times {tn}n∈N, we plan to construct a partial differential equation, assuming thatthe data are approximate solution to some PDE, at {tn}n∈N and at {xj}j∈Zd . We denoteunj = u(tn,xj) (≈ vnj ), xj =
(xj1 , · · · , xjd
)and j = (j1, · · · , jd), where u is the solution to
the searched PDE. We will denote along the paper, Kd (resp. N ) a finite subset of Zd (resp.N), for d ∈ N∗.
1.1. Random perturbation
In practice, we assume that the given data are perturbed by measurement. In otherwords, we assume that the given data are perturbations of the exact solution values to thePDE. We will then introduce a stochastic process {ε(t,x, ζ) : (t, x) ∈ R+ × Rd}, withuniform distribution U(0, ε∞), for ε∞ > 0 small enough. We define
ε(·, ·, ζ) : R+ × Rd 7→ S
where S is the state space. We will then assume that, for any n and j ∈ Zd
vn;`j := unj
(1 + εn;`
j
), (1)
where εn;`j = ε(tn,xj , ζ`), where ` is the realization index. This random perturbation models
the uncertainty in the structure and estimate of the given data. As a consequence for eachset of data, the predicted coefficients/equations are dependent on the random process. Theproposed algorithm then needs to be repeated and by an average process. Notice that the
limK→+∞1
K
∑K`=1 v
n;`j = unj limK→+∞
1
K
∑K`=1
(1 + εn;`
j
)= unj (1 + ε)
where ε is the mean of the discrete random variable, which is assumed constant (mean(n, j)-independent). This is a crucial property which allows to justify convergence of theproposed reconstruction algorithm. In order to accelerate the convergence, we could alsouse MultiResolution Analysis (MRI) algorithms [10]. The motivation is that the randomnoise on the data, are a priori spatial high frequency and without any particular structure.In this hand, we expect that the frequency of the operator symbol or of the coefficients,do not possess high frequencies as the symbols of the searched (relatively low order) PDEare polynomial functions (of relatively low degree). Notice, direct MRI on the data may bemisleading as the solution itself can possess high frequencies coming from the initial data.
1.2. Filtering
In order to refine the procedure described in this paper, it may be useful to filter theapproximate operator symbols. The idea is based on the fact that i) relevant/practical math-ematical models in physics, chemistry, etc, involve relatively low order operators (otherwise,the model may not be relevant for practical purposes, that is numerical or analytical predic-tions), and as a consequence ii) the associated symbols have a relatively low homogeneitydegree [11]. A filtering could then be applied to remove from the approximate symbol the
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very large spatial frequencies, which do not exist in the exact one. However, by construction,the approximate symbol is itself constructed as the ratio of the Fourier transform of the givendata. Although, the data themselves could contain high frequencies (depending for instanceon the initial data), the ratio of the Fourier transform of consecutive data should mainly con-tain low frequencies. The random perturbation will only pollute the high frequency part ofthe reconstructed symbol. For small enough wavenumbers, the approximate symbol shouldthen provide a good approximation of the exact symbol. As an example, we report in Fig. 1a comparison to the exact, approximate and filtered approximate symbols, using the methoddeveloped below (see details in Section 2 and Subsection 5.1) for the searched equation
∂tu− 5× 10−2ux − 5∂xxu− 10−1∂xxxu+ 4∂xxxxu− 10−3∂xxxxxu = 1 . (2)
We remove from the approximate symbol the high frequencies which come from the randomnoise.
−5 −4 −3 −2 −1 0 1 2 3 4 5
100
101
102
Spatial frequency
Modulus of filtered reconstructed symbol
Modulus of reconstructed symbol
Modulus of exact symbol
Figure 1: Spatial spectrum: exact, approximate and filtered approximate symbols, for Equation 2.
1.3. Organization of the paper
This paper is organized as follows. Section 2 is devoted to a symbol-based methodfor reconstructing time-dependent coefficients of homogeneous and non-homogeneous par-tial differential equations, Subsections 2.1, 2.2. Fractional equations are also considered inSubsection 2.3. In Section 3, we present the methodology for constructing space-dependentcoefficients to unknown PDEs from randomly perturbed data. The case of first order in-timeequations is first addressed in Subsection 3.1. The general case (Nth order equation) is pre-sented in Subsection 3.3. The corresponding methodology is mainly based on pseudospectral
3
methods, but finite differences can also be used, see Subsection 3.2. Extension to quasilinearequations is also discussed in Subsection 3.4. One of the most important questions, whichis, how to determine the order of the searched equation, is addressed in Section 4. One- andtwo-dimensional numerical experiments are presented in Section 5, illustrating the derivedmethodologies. We finally conclude in Section 6.
2. Symbolic method: first order in-time linear equation
In this section, we consider the following order N(∈ N) equation, which is assumed in ddimension, of the form
∂tu(t,x) +∑|α|6N
a(α)(t)Dαxu(t,x) = f(t,x), (t,x) ∈ (0, T )× Rn (3)
with N 6 1 and α = (α1, · · · , αn) ∈ Nn is a multi-index with n ∈ N∗. Fractional equationscould as well be considered, see Subsection 2.3. We intend to determine the operator P =∑|α|6k a
(α)(t)Dαx and the function f from the data. We denote by u the Fourier transform
/ x, of u
u(t, ξ) =1
(2π)d
∫Rd
u(t,x)e−ix·ξdx, (t, ξ) ∈ [0, T ]× Rd
We then denote by p the partial /x (polynomial) symbol of P [7, 8], p(t, ξ) = σx(P (t,Dx)
),
such that
p(t, ξ)u(t, ξ) =1
(2π)d
∫Rd
P (t,Dx)u(t,x)e−ix·ξdx .
That is, p(t, ξ) =∑|α|6N a
(α)(t)(iξ)α
. The equation is Fourier space reads
∂tu(t, ξ) + p(t, ξ)u(t, ξ) = f(t, ξ), (t,x) ∈ (0, T )× Rn . (4)
The objective is i) to construct an approximation p of p, from which ii) we will deduce an
approximate operator P . We construct the discrete Fourier transform in space for consecutivetimes, of {vn+k
j }j∈Zd , with k ∈ {1, · · · , }. The corresponding DFT are denoted {vn+kk }k∈Zd ,
and are defined as follows.
vn+kk =
∑j∈Zd
e−2iπk·xjvn+kj
2.1. Homogeneous equation
We assume here that f = 0. That is, we search for a homogeneous equation. Taking theFourier transform in space of (3), we get
u(t, ξ) = exp(−∫ t
0
p(s, ξ)ds)u0(ξ)
4
where u0 ∈ L2(Rd). Thus, from two consecutive “times” tn < tn+1
u(tn+1, ξ) = exp(−∫ tn+1
tn
p(s, ξ)ds)u(tn, ξ) .
Denoting ∆tn := tn+1 − tn and u(tn, ξ) 6= 0, this is formally equivalent to
1
∆tn
∫ tn+1
tn
p(s, ξ)ds = −1
∆tnlog( u(tn+1, ξ)
u(tn, ξ)
).
At the discrete level and for ` fixed, we denote by {pn;`k }k the sequence computed from given
data {vn;`j }j , which are assumed of the form vn;`
j = u(tn,xj)(1 + εn;`j ) with k ∈ Zd, and
vn;`k 6= 0, where
pn;`k := −
1
∆tnlog( vn+1;`
k
vn;`k
).
This sequence is dependent on the random process (`-dependence). In order to refine thedata, we then need to repeat the process, and we define
pnk := limK→+∞
1
K
K∑`=1
pn;`k
where, the index ` refers to the actual realization. The more data, the better the accuracyof the developed algorithm. The latter result is justified by the linearity of the equation.Assume for the sake of simplicity that εn;` is only time-dependent. We denote {unk}k theFourier transform of {un;`}j . We formally have
1
K
∑K`=1 p
n;`k =
1
K
∑K`=1 log
( vn+1;`k
vn;`k
)=
1
K
∑K`=1 log
( un+1;`k + un+1;`
k εn+1;`
un;`k + un;`
k εn;`
)= log
( un+1k
unk
)+
1
K
∑K`=1 log
(1 + εn+1;`
1 + εn;`
)As |εn;`| 6 ε∞ is small enough, we get
1
K
K∑`=1
pn;`k − log
( un+1k
unk
)=
1
K
K∑`=1
εn+1;` −1
K
K∑`=1
εn+1;` +O(ε2∞)
As
limK→+∞
{ 1
K
K∑`=1
εn+1;` −1
K
K∑`=1
εn;`}
= 0,
5
we expect that for K large
∣∣∣ 1
K
K∑`=1
pn;`k − σx
((P (t,Dx)
)tn,k)
∣∣∣to be small.
Remark 2.1. An alternative approach, consists in taking the average of the realization be-fore implementing the method presented above. In this case, we benefit from the fact that themean is space-time independent (1). Thus, vnk = (1 + ε)unk, and we can define
pnk :=vn+1k
vnk=un+1k
unk.
For K large enough, this approach is likely more accurate than the one developed above.
From the knowledge of pnk for each n in a subset N ⊂ N, and k in a subset Kd ⊂ Zd, wecan reconstruct an approximation of Operator P . More specifically, we define by p : (t, ξ) ∈[0, T ]×K 7→ p(t, ξ) a polynomial function, interpolating the sequence
{pnk}k,n
such that, for
any k ∈ Kd and any n ∈ N
p(tn, ξk) = pnk .
That is
p(t, ξ) = Π[{pnk}(k,n)∈Kd×N
](t, ξ)
where Π is a polynomial interpolation operator. From this function, we define the operatorP (t,Dx) from its symbol σ−1
ξ
(p(t, ·)
). Thus, P is defined by
P (t,Dx)u(t,x) :=
∫Rd
p(t, ξ)u(t, ξ)eix·ξdξ
and the reconstructed equation reads
∂tu(t,x) + P u(t,x) = 0, (t,x) ∈ (0, T )× Rn , (5)
which is such that for k ∈ Kd and n ∈ N
σx(P (t,Dx)
)(tn, ξk) = σx
((P (t,Dx)
)tn, ξk) .
By construction, ‖p − p‖∞ is bounded by a constant depending on CardKd, CardN , theregularity of p and the size of the convex hull of Kd×N . B-splines can as well be employedto further reduce ‖p − p‖∞, see [9]. Notice that the methodology does not require anyassumption of the spatial order of the searched equation.
6
Remark 2.2. If we know the general form of the search equation and its order, the strategycan be refined. Say we search for b(α) such that
∂tu(t,x) +∑|α|6N
b(α)(t)Dαxu(t,x) = 0, (t,x) ∈ (0, T )× Rn . (6)
We denote by t 7→ A(α)(t) the antiderivative of t 7→ a(α)(t)
u(tn+1, ξ) = exp(−∑|α|6N
(iξ)α(A(α)(tn+1)− A(α)(tn)
))u(tn, ξ) .
We then expect that {pnk}k will provide an approximation to {p(tn, ξk)}k. In order to deter-mine t 7→ a(α)(t), we search for t 7→ B(α)(t) approximating t 7→ A(α)(t)∑
|α|6N
(iξk)α(B(α)(tn+1)−B(α)(tn)
)= log
( un+1k
unk
), (7)
where we have denoted un+1k = u(tn+1,xk), and for unk 6= 0. For each n, we then determine
B(α)(tn), by considering a sufficiently large number of equations (7), for different values ofk.
Remark 2.3. In the case of equations with constant coefficients a(α), (7) is solved, by con-sidering again the equation ∑
|α|6N
α(α)(iξk)α =1
∆tnlog( un+1
k
unk
)for a sufficiently large number values of k (depending on N).
2.2. Non-homogeneous equation
The same approach can still be applied to non-homogeneous equation, as an explicitsolution can be explicitly constructed in Fourier space, say for u0 and f in L2(Rd)
u(t, ξ) = exp(−∫ t
0
p(s, ξ)ds)( ∫ t
0
f(s, ξ) exp( ∫ s
0
p(τ, ξ)dτ)ds)u0(ξ)
Thus, from two consecutive “times” tn < tn+1
u(tn+1, ξ) = exp(−∫ tn+1
tn
p(s, ξ)ds)( ∫ tn+1
tn
f(s, ξ) exp( ∫ s
tn
p(τ, ξ)dτ)ds)u(tn, ξ) .
Unlike the homogeneous case, a second unknown must be determined. The case f(t, ξ) will
be treated in a more general framework. Assuming instead that f is t-independent, f(ξ), weformally get
− log( u(tn+1, ξ)
u(tn, ξ)
)= log
(f(ξ)
)+
∫ tn+1
tn
p(s, ξ)ds+ log(∫ tn+1
tn
exp( ∫ s
tn
p(τ, ξ)dτ)ds)
7
for u(tn, ξ) 6= 0. A possible algorithm can then be established from
pn;`k −
1
∆tnlog f `k = −
1
∆tn
(log( vn+1;`
k
vn;`k
)+ log(∆tn)
).
A second equation is needed, as there are now 2 unknowns pn;`k and f `k. In practice, we
can instead solve a 3-equation system. Assuming p(·, ξ) (that is a(·, ξ)) smooth enough:pn±1;`k = pn;`
k ±∆tnpn;`t,k +O((∆tn)2), where
pn;`t,k =
∑j∈Zd
e−2iπk·xj (∂tp)n;`j .
The unknowns are now: pn;`k , pn;`
t,k and f `k. The second and third equations which are proposed
are then, for vn±1;`k 6= 0,
pn;`k ±∆tnpt,k −
1
∆tnlog f `k = −
1
∆tn±1
(log( vn+1±1;`
k
vn±1;`k
)+ log(∆tn±1)
).
As proposed above, an average process allows for the construction of {pnk}k,n, {pnt,k}k,n and
{fk}k
pnk := limK→+∞
1
K
K∑`=1
pn;`k , pnt,k := lim
K→+∞
1
K
K∑`=1
pn;`t,k, fk := lim
K→+∞
1
K
K∑`=1
f `k .
We then construct approximate f , and P using the exact same interpolation strategy as thehomogeneous case, to get an approximate equation:
∂tu(t,x) + P u(t,x) = f(x), (t,x) ∈ (0, T )× Rn, (8)
such that, for k ∈ Kd ⊂ Zd and n ∈ N ⊂ N: p(tn, ξk) = pnk, f(ξk) = fk. Notice thatthe procedure described above, still does not require the a priori knowledge of the order thesearched equation. An expression of p can also be obtained, using sparse regression from afeature library, see [6].
2.3. Fractional equations
The methodology which was presented above remains essentially valid for fractional equa-tions, α = (α1, · · · , αN) ∈ Qd and order Q ∈ Q [16]
∂tu(t,x) +∑
αi∈Q, |α|6Q
a(α)(t)Dαxu(t,x) = f(t,x), (t,x) ∈ (0, T )× Rn . (9)
The corresponding symbol is a fractional function in ξ,
p(t, ξ) =∑
αi∈Q, |α|6Q
a(α)(t)(iξ)α.
8
As in the partial differential case, we take the Fourier transform in space, of the (3) we getfor two consecutive “times” tn < tn+1 and denoting ∆tn := tn+1 − tn
1
∆tn
∫ tn+1
tn
p(s, ξ)ds = −1
∆tnlog( u(tn+1, ξ)
u(tn, ξ)
).
We denote by t 7→ A(α)(t) the antiderivative of t 7→ a(α)(t)
u(tn+1, ξ) = exp(−
∑αi∈Q, |α|6Q
(iξ)α(A(α)(tn+1)− A(α)(tn)
))u(tn, ξ) .
At the discrete level, we proceed as for the partial differential case. We then expect that{pnk}k∈Kd
will provide an approximation of the function {p(tn, ξk)}k∈Kd. From this sequence,
fractional interpolation allows for the construction of an approximate symbol (t, ξ) 7→ p(t, ξ).
Finally, we define P (t,Dx) = σ−1ξ
(p(t, ξ)
)such that p(tn, ξk) = p(tn, ξk) for all n ∈ N ⊂ N
and k ∈ Kd ⊂ Zd. From the approximate symbol, we can again reconstruct a differentialoperator. However, unlike the PDE case, if we do not know the exact structure of thesearched fractional equation, we can hardly reconstruct an approximate symbol, as a verylarge number of configurations is possible (depending on α).
3. Linear systems with non-constant coefficients
We now consider more general equations. We assume that the searched equation in d > 1dimension reads:
o∑i=1
α(i)(x)∂itu(t,x) +∑|α|6N
β(α)(x)Dαxu(t,x) = 0, (t,x) ∈ (0, T )× Rn .
The coefficients {α(`)}
16`6o, and {β(α)
}16|α|6N are assumed smooth. This time, it is no more
possible to directly apply the Fourier transform, as the coefficients are space dependent, andthe symbol σx(P ) = p(t, ξ) reads
p(x, ξ) =∑|α|6N
β(α)(x)(iξ)α .
3.1. First order equations
Data {vnj }j are collected at different times steps tn−1 < tn < tn+1. We first assume thatthe searched system is a first order system of the form:
∂tu(t,x) +d∑i=1
β(i)(x)∂xiu(t,x) = 0, (t,x) ∈ (0, T )× Rn (10)
9
where x = (x1, · · · , xd). The unknown coefficients, we intend to determine are the functions{β(i)}16i6d. In this goal, we propose the following formal discretization to (10)
un+1j − unj
∆tn+
∫ tn+1
tn
d∑i=1
β(i)j [[∂xiu(t)]]jdt = 0, (11)
where [[∂xiun]] =
{[[∂xiu
n]]j}j
is an approximation of
[[∂xiun]] ≈ F−1
x
(Fx(∂xiu)
)= F−1
x
(iξFx(u)
)which is defined in Appendix. We intend to determine the sequences β
(i)j , for all 1 6 i 6 d,
from (11)
−∫ tn+1
tn
d∑i=1
β(i)j [[∂xiu(t)]]jdt = un+1
j − unj . (12)
The idea consists in mimicking (12) using the data {vn;`j }j approximating {unj}j , and assumed
of the form vn;`j := unj
(1 + εn;`
j
), where εn;`
j is a random perturbation, see Subsection 1.1. Forany j:
1. We numerically compute [[∂xivn]]j .
2. We assume that {vn;`j }j satisfies
∆tn
d∑i=1
β(i);`j [[∂xiv
n;`]]j = vn+1;`j − vn;`
j . (13)
For each j ∈ Zd, we have d unknowns β(1);`j , · · · β(d);`
j , we then need d equations. As thecoefficients are only x−dependent, we then consider d linear equations for 1 6 k 6 d.
∆tn+k−1
d∑i=1
β(i);`j [[∂xiv
n+k−1;`]]j = vn+k;`j − vn+k−1;`
j (14)
which is equivalent to An;`j X
`j = bn;`
j where
An;`j =
(∆tn+k−1[[∂xiv
n+k−1;`]]j)
16k,i6d∈ Rd×d, bn;`
j =(vn+k;`j − vn+k−1;`
j
)16k6d
.
The unknown vector is given by
X`j =
(β
(i);`j
)16i6d
. (15)
The computational complexity for solving this system, depends on the dimension d. Al-though in principle this can be computed once for each j ∈ Zd, for a given n ∈ N, the
10
accuracy can be increased by solving the system for several realizations, and to statisticallyget a better estimation of β
(i)j . Finally, we propose the following average process:
Xj = limK→+∞
1
KX`j .
This is justified by the fact that, by linearity of the equation, we get
limK→+∞
1
K
K∑`=1
bn;`j = (1 + ε)bnj
where bn;`j =
(un+kj − un+k−1
j
)16k6d
. Similarly
limK→+∞
1
K
K∑`=1
An;`j = (1 + ε)An
j ,
where
Anj =
(∆tn+k−1[[∂xiu
n+k−1]]j)
16k,i6d∈ Cd×d .
This comes from the fact that
limK→+∞1
K
∑K`=1[[∂xiv
n;`]]j = [[limK→+∞1
K
∑K`=1 ∂xiv
n;`]]j
= (1 + ε)[[∂xiun]]j .
This justifies that, the more realizations, the better the accuracy in the reconstruction ofthe equation coefficients.
3.2. Finite difference approach
The same exact idea can be implemented using finite difference approximation. That israther than using the DFT, we can approximate ∂xiu at xj using the following approximation,at (tn,xj) and ζ`. For ` ∈ N, we denote
τ±`i j := (j1, · · · , ji−1, ji ± `, ji+1, · · · , jd) .
vn;`
τ1i j− vn;`
τ−1i j
∆xji+1 + ∆xji.
Then, we consider
vn+1;`j − vn;`
j
∆tn+
d∑i=1
β(i)
τ−1i j
vn;`
τ1i j− vn;`
τ−1i j
∆xji+1 + ∆xji= 0 . (16)
11
As above, in order to compute X`j (15) knowing bn;`
j for given j, we have to solve a linear
system of d equations An;`j X
`j = bn;`
j , where this time
An;`j =
(∆tn+k−1
vn;`
τ1i j− vn;`
τ−1i j
∆xji+1 + ∆xji
)16k,i6d
∈ Cd×d .
Example 3.1. For instance for first order equation is 3d
ut +3∑i=1
β(i)(x)∂xiu = 0 .
We use the following equations for k = 0, 1, 2
2(vn+k+1;`j − vn+k;`
j
)∆x1∆x2∆x3 = ∆tn+k∆x2∆x3
(vn+k;`
j+1
− vn+k;`
j−1
)β
(1);`j
+∆tn+k∆x1∆x3
(vn+k
j+2
− vn+k;`
j−2
)β
(2);`j
+∆tn+k∆x1∆x2
(vn+k;`
j+3
− vn+k;`
j−3
)β
(3);`j .
For each j ∈ Z3, this 3-equation system is to be solved, to estimate β(1);`j , β
(2);`j and β
(3);`j .
3.3. N th order linear equations
The generation of Nth order linear equations with non-constant coefficients is straight-forward, but naturally requires in principle the solution to larger linear systems. In d > 1dimensions, we assume that the searched system reads
o∑i=1
α(i)(x)∂itu(t,x) +∑|α|6N
β(α)(x)Dαxu(t,x) = 0, (t,x) ∈ (0, T )× Rn .
We also assume that α(o) = 1. In the case of homogeneous equation, the total number ofunknown functions is (up to)
∑Nk=1 d
k + o− 1 = (dN − 2d+ 1)/(d− 1) + o. More specifically
for each |α| = k 6 N , dk equations are required to determine β(α)j and an additional o − 1
equations are necessary for determining α(i)j , with i ∈ {1, · · · , o− 1}. This is done thanks to
approximations. Formally, for vn;` given, we consider the following approximation, for eachj ∈ Zd
o∑i=1
α(i);`j Π
[vn;`j , · · · , vn+i;`
j ; ∆tn, · · · ,∆tn+i−1
]+∑|α|6N
β(α);`j [[∂(α)vn;`]]j = 0
where Π[vn;`j , · · · , vn+i;`
j
]is a finite difference approximation to ∂itu(tn+i,xj). As (dN − 2d+
1)/(d− 1) + o equations are needed, we consider, for k ∈{
1, · · · , (dN − 2d+ 1)/(d− 1) + o}
o∑i=1
α(i);`j Π
[vn+k;`j , · · · , vn+i+k;`
j ; ∆tn, · · · ,∆tn+i−1
]j
+∑|α|6N
β(α;`)j [[∂(α)vn+k]]j = 0 .
12
This can again be rewritten as a linear system of the form An;`j X
`j = bn;`
j , where An;`j ∈
R(dN−2d+1)/(d−1)+o×(dN−2d+1)/(d−1)+o, and bn;`j ∈ R(dN−2d+1)/(d−1)+o read
X`j =
(α
(i);`j , β
(α);`j
)16i6o−1,16k6(dN−d)/(d−1)
, bn;`j = −
(Π[vn;`j , · · · , vn+o;`
j
])16i6o−1,16k6(dN−d)/(d−1)
As proposed above, we will eventually define Xj = limK→+∞∑K
`=1X`j/K. In practice the
larger K, the more accurate the reconstruction.
Example 3.2. Assume that we search for a 2nd order system of the form (t,x) ∈ (0, T )×Rn
∂2t u(t,x) + α(x)∂tu(t,x) +
d∑i=1
β(i)(x)∂xi(t,x) +d∑i=1
d∑j=1
γ(i,j)(x)∂xixju(t,x) = 0 . (17)
The proposed formal discretization is
∑di=1 β
(i);`j [[∂xiv
n;`]]j +∑d
i=1
∑dl=1 γ
(i,l);`j1,··· ,jd [[∂xixjv
n;`]]j = −vn+1;`j − 2vn;`
j + vn−1;`j
∆tn∆tn−1
−αjvn+1;`j − vn;`
j
∆tn.
(18)
For each j ∈ Zd, we have 1 + d+ d2 unknowns β(1);`j , · · · β(d);`
j , we then need 1 + d+ d2 equa-tions. As the coefficients are only x−dependent, we proceed as follows, from the knowledgevn−1;`,vn;`,vn+1;`
1. We numerically compute [[∂xivn;`]]j, [[∂xixlv
n;`]]j.
2. We assume that vn;` := {vn;`j }j satisfies
vn+k;`j − 2vn+k−1;`
j + vn+k−2;`j = ∆tn+k−2α
`j(v
n+k`j − vn+k−1;`
j )
+∆tn+k−2∆tn+k−1
∑di=1 β
(i);`j [[∂xiv
n+k−1;`]]j
+∆tn+k−2∆tnk−1
∑di=1
∑dl=1 γ
(i,l);`j [[∂xixjv
n+k−1;`]]j
(19)
We consider 1 6 k 6 1 + d + d2 independent linear equations (19) . This system can berewritten in the form An;`
j X`j = bn;`
j , where
An;`j =
(∆tn+k−2v
n+k;`j − vn+k−1;`
j ,∆tn+k−2∆tn+k−1[[∂xivn+k−1;`]]j ,
∆tn+k−2∆tnk−1[[∂xixjvn+k−1;`]]j
)16i,k61+d+d2
and
bn;`j =
(vn+k;`j − 2vn+k−1;`
j + vn+k−2;`j
)16k61+d+d2 (20)
and the unknown vector is given by
X`j =
(α`j , β
(1);`j , · · · , β(d);`
j , γ(1,1);`j , · · · , γ(d,d);`
j
)∈ R1+d+d2
. (21)
Remark 3.1. In the case of non-homogeneous systems with source term of the form x 7→f(x), an additional equation is necessary in order to determine f `j for each j ∈ Zd.
13
3.4. Quasilinear equationsA forthcoming paper will be dedicated to the treatment of time-dependent coefficients
and nonlinear equations. However, some techniques developed here can be extended to aclass of standard quasilinear PDEs. Assume that the searched equation is of the form (firstorder in time):
ut(t,x) +d∑i=1
β(i)(u)∂xiu(t,x) = 0 (22)
where β(i) are smooth functions. We are given the data vn;` := {vn;`j }j , locations {xj}j , and
times {tn}n∈N, and propose the following formal discretization of (22):
vn+1j − vn;`
j
∆tn+
d∑i=1
β(i)(vnj )[[∂xivn;`]]j = 0 .
In order to reconstruct u 7→ β(i)(u) from(vn;`j , [[∂xiv
n;`)]]j), we can not use the exact same
procedure to determine, as above, as the unknown
Xn;`j =
(β(1)(vn;`
j ), · · · , β(d)(vn;`j ))
is t-dependent. As we have d unknown functions, we need for each at least d equations, fork = 1, · · · , d
vn+k;`j − vn+k−1;`
j
∆tn+k−1
+d∑i=1
β(i);`(vn+k−1;`j )[[∂xiv
n+k−1;`]]j = 0 .
It is well-known that nonlinear hyperbolic equations, such as the one which is searchedusually have non-differentiable weak solutions, called shock waves [15, 14, 1]. The procedurewhich is proposed can then be applied for any j ∈ Zd, if β(i) is smooth enough and in regimesavoiding shock-waves. The latter can easily be identified as they correspond to data withspatial discontinuity. In this case:
β(i);`(vn+1;`j ) = β(i)(vn;`
j ) + β(i);`(vn;`j )(vn+1;`
j − vn;`j ) +O
((vn;`j − v
n+1;`j )2
).
As a consequence, for |vn+1;`j − vn;`
j | small enough, we need to determine β(i)(vn;`j ) and
β(i)(vn;`j ). This can be done by doubling the size of the linear system, compared to the
linear case. That is for k, k′ = 1, · · · , d, we solve the 2d-equation linear system:
vn+k;`j − vn+k−1;`
j
∆tn+k−1
=∑d
i=1 β(i);`(vn+k−1;`
j )[[∂xivn+k−1;`]]j
vn+k′+1;`j − vn+k′;`
j
∆tn+k′=
∑di=1 β
(i);`(vn+k′−1;`j )[[∂xiv
n+k′;`]]j
+β(i);`(vn+k′−1;`j )[[∂xiv
n+k′;`]]j(vn+k′;`j − vn+k′−1;`
j ) .
This system can be equivalently rewritten An;`j X
n;`j = bn;`
j , where
bn;`j =
(vn+1;`j − vn;`
j , · · · , vn+d;`j − vn+d−1;`
j ,(vn+2;`j − vn+1;`
j , · · · , vn+d+1;`j − vn+d;`
j
),
X`j =
(β(1);`(vn;`
j ), · · · , β(d);`(vn;`j ), β(1);`(vn;`
j ), · · · , β(d);`(vn;`j )).
(23)
14
4. How to determine the order of the linear equation ?
Beyond the construction of the β(α) and α(i) functions, one of the main problems is theevaluation of the overall order of the searched PDE. This is a crucial question, as a wrongorder will lead to an inaccurate equation. Assume that the searched equation Eo;N , reads
o∑i=1
α(i)(x)∂itu(t,x) +∑|α|6N
β(α)(x)Dαxu(t,x) = 0, (t,x) ∈ (0, T )× Rn .
In this goal, we consider an equation Eo;N , of the form
o∑i=1
α(i)(x)∂itu(t,x) +∑|α|6N
β(α)(x)Dαxu(t,x) = 0, (t,x) ∈ (0, T )× Rn
where in principle N 6= N , o 6= o, and as a consequence ‖β(α) − β(α)‖∞ and ‖α(i) − α(i)‖∞are expected to be large. Nine situations can occur N = N , N < N , N > N , with o = o,o < o, o > o.
4.1. Iterative approach
In order to reach convergence of the overall algorithm, a natural approach is to proceedby induction in o and N . The algorithm looks as follows for given data vn;`, · · · ,vn+k;` alongwith {xj}j and {tn+k}k>0, assuming that the maximal order in t (resp. x) is omax (resp.
Nmax). Start with o = 0.
Step 0. (Re)set N = 1 and o← o+ 1.Step 1. Solve linear system on β(α), and α(i): An;`
j X`j = bn;`
j .Step 2. Analysis of the results.
• If no solution to linear system: N ← N + 1.
– If N < N∞ go back to Step 1.
– If N = N∞ go back to Step 0.
• If solution exists: same test with vn;`′ to validate the model.
– If model Eo;N validated, stop the algorithm.
– If model not validated, go back to Step 0.
4.2. Direct approach
An alternative approach is to start by default, with an high order PDE, where a priori,omax > o and Nmax > N . That is the searched equation is of the form
o∑i=1
α(i)(x)∂itu+omax∑i=o+1
α(i)(x)∂itu+∑|α|6N
β(α)(x)Dαxu+
∑N<|α|6Nmax
β(α)(x)Dαxu = 0
15
where o and N are unknown. We expect that the approximate coefficients α(i), β(α) of orderstrictly greater than i > o and |α| > N would be found null or inconsistent, and otherwords the matrix An
j must have a null determinant. In this case, the order of the searchedequation must be reduced. This is a crucial criterion in order to determine the order ofthe searched equation. Alternatively the corresponding coefficients can be found very smallsuggesting that the contribution of the corresponding differential operators can be neglected.A recursive algorithm can again be established to iteratively, reducing the overall equationorder, discriminate the negligible operators.
5. Numerical experiments
We propose in this section, several numerical experiments to validate the methodologyderived above.
5.1. Symbolic method
In this first test, we plan to reconstruct the following equation, using a symbolic approach.
∂tu+ a(1)ux + a(2)∂xxu+ a(3)∂xxxu+ a(4)∂xxxxu+ a(5)∂xxxxxu = a0 (24)
where the coefficients are given by
a(0) = −10−1, a(1) = 1, a(2) = 10−1, a(3) = 10−2, a(4) = 5× 10−3, a(5) = −10−4 .
From a initial data, u0(x) = exp(−x2/2) cos(10x), we construct an approximate solution{unj }j computed in a domain [−5, 5], on a Nx = 8001-point grid with ∆x = 1/800 and ∆t =10−3. The final iteration is Nt = 400. We impose a stochastic process on the approximatesolution, with an uniform law U(0, ε∞), with ε∞ = 10−8, where vn;`
j = unj (1 + εn;`j ). The
non-perturbed data at final time is reported in Fig. 2.Case 1. Perturbation is null/cancelled. We first assume that the perturbation is null. This
case is also equivalent to perturbations which would be averaged, as discussed in Remark 2.1.In the latter situation, we indeed have vn+1
k /vnk = un+1k /unk . We construct the DFT {vNt−1
k }k,and {vNt
k }k from {vNt−1j }j and {vNt
j }j. We then construct the sequence, for vNt−1k 6= 0 ,
{pNtk }k
pNtk := −
1
∆tlog( vNt
k
vNt−1k
).
We compare in Fig. 13, the real and imaginary parts of the exact and reconstructed symbolsto (24). That is we ξk 7→ p(ξk) =
∑5`=0 a
(`)(iξk)` and ξk 7→ p(ξk). We get an excellent
agreement. However, we still need to reconstruct an approximate (continuous) symbol fromthese data. In this goal, we assume that the searched equation is ∂tu+
∑omax
l=0 α(l)∂lxu = 0, withomax unknown, and we implement a polynomial interpolation in order determine the unknowncoefficients of the PDE. That is, we search p(t, ξ) such that for all l ∈ {1, · · · omax + 1},
p(tn, ξk) = p(tn, ξk),
16
−5 −4 −3 −2 −1 0 1 2 3 4 5−1.5
−1
−0.5
0
0.5
1
1.5x 10
−4
Approximate solution
Initial data
Figure 2: Non-perturbed data at final time.
0 2 4 6 8 10 12 1410
−1
100
101
102
103
Real part of the reconstructed symbol
Real part of the exact symbol
Imaginary part of the reconstructed symbol
Imaginary part of the exact symbol
Figure 3: Real and imaginary parts of the exact and reconstructed symbols.
where ξk = k∆ξ, and ∆ξ = π/5, with k ∈ Z. We assume the maximal order is omax = 7.Searching for the coefficients is equivalent to solve a Vandermonde system. We picked 1different set of omax + 1 = 7 constraints. According to Table 1, we get estimates of the orderof the searched equation and of its coefficients. More specifically, the inconsistency of the
17
α(0) α(1) α(2)
−1.000081850888496× 10−1 9.999912470046054× 10−1 1.000042171075373× 10−2
α(3) α(4) α(5)
1.000097592645466× 10−2 5.000141820071519× 10−3 −1.000096443526068× 10−4
α(6)
4.257054797261438× 10−10
Table 1: Numerical values α(0), · · · , α(6), where ε is null.
computed α(6)’s suggest that o = 5. Next, the consistency of the other computed coefficientsallow for an accurate estimate of the PDE coefficients.Case 2. Space dependent random perturbation. In this case we assume that the randomprocess is t-independent, that is vn;`
j = unj (1 + ε`j), with ` denoting the realization index of
the stochastic process. We set pn;`k = −
1
∆tlog(vNt;`k /vNt−1;`
k
), where we have denoted v
N(`)t ;`
k
the DFT of the perturbed data at `th realization. The rest of the algorithm is as in the non-perturbed case. We assume that ξk → p`(ξk) is of the form p(`)(ξk) =
∑omax
l=1 α(l);`(iξk)l where
omax = 7, and solve a Vandermonde system to get approximates values of the coefficients.Next, it is necessary to implement an average process in order to get an estimate of the PDEcoefficients. We denote by {α(l);`}16l6omax the computed coefficients for each sample `. Wethen compute, (for K large enough, in practice).
α(l) = limK→+∞
1
K
K∑`=1
α(l);` .
We report in Fig. 4 (Left) the `2-norm of the error in logscale, between the exact andapproximate coefficients as a function of `, that is {`, log
(E(`)
)}
E(`) :=( omax∑
l=1
|α(l);` − a(l)|2)1/2
.
We also report in Tables 2, the found coefficients ` = 1 (first line), and the convergedcoefficients (second line).
Case 3. Space and time dependent random perturbation. In this case we assume that therandom process is such that vn;`
j = unj (1 + εn;`j ), with εmax = 10−9. We proceed as above,
performing the algorithms several times. We report in Fig. 4 (Right) the `2-norm of theerror in logscale, between the exact and approximate coefficients at stochastic realization `,that is {`, log
(E(`)
)}.
5.2. One-dimensional test. Linear equation.
In this subsection, the searched equation is
∂tu+ c(x)∂xu+ ν(x)∂xxu = 0, (t, x) ∈ [0, T ]× [−1, 1]
18
α(0) α(1) α(2)
−8.496905353740625× 10−2 9.914638147888091× 10−1 9.302849563998560× 10−2
−1.001897636979343× 10−1 9.997928029182725× 10−1 1.000833219501233× 10−1
α(3) α(4) α(5)
1.1290669346466807× 10−2 4.787951773996767× 10−3 −1.075221456936866× 10−4
1.001814167944528× 10−2 5.002455815685041× 10−3 −1.001455983589550× 10−4
α(6)
4.648585626177240× 10−7
6.266263919080856× 10−9
Table 2: Numerical values α(0);1, · · · , α(6);1 and α(0), · · · , α(6): ε(x, ζ) with εmax = 10−8.
α(0) α(1) α(2)
−3.887008083111291× 10−1 6.243784097545035× 10−1 1.952699801460228× 10−1
−9.031650266918659× 10−2 1.004654230531507 9.554460918355122× 10−2
α(3) α(4) α(5)
4.678156614586105× 10−2 7.108540944477762× 10−3 −4.248146685454826× 10−4
9.404316598788863× 10−3 4.863543299705904× 10−3 −9.337375897290761× 10−5
α(6)1
9.29794446838190× 10−6
3.547161290147765× 10−7
Table 3: Numerical values α(0);1, · · · , α(6);1 and α(0), · · · , α(6): ε(t, x, ζ) with εmax = 10−9.
19
100 200 300 400 500 600 700 800 900 1000
10−3
10−2
l2−norm error
Iteration
Err
or
in lo
gsca
le
20 40 60 80 100 120 14010
−2
10−1
l2−norm error
Iteration
Err
or
in lo
gsca
le
Figure 4: `2-norm error in logscale {`, E(`)} (Left) ε(x, ζ) with εmax = 10−8. (Right) ε(t, x, ζ) with εmax =10−9.
and {c(x) = 10 exp
(− 10x2 + 0.1
)×(1 + 0.2 sin(20x)
)ν(x) = 10−1 sin2(8πx)× exp(−10x2) .
The initial data is given by u0(x) = exp(−20x2). We accurately solve this equation, whichprovides the sequences {unj }|j|6J for several n 6 Nt andNx = 2001, ∆t = ∆x/ supx∈[−1,1] |c(x)|.We implement the method, assuming that we do not know the order in space of the equation,of the searched equation, that is
α(x)∂ttu+ ∂tu+ β(1)(x)∂xu+ β(2)(x)∂xxu+ β(3)(x)∂xxxu = 0 .
Case 1. We assume that for any j, εn;`j = 0, that is there is no perturbation on the data, in this
first test. For each |j| 6 J , we propose here a FDS approach, rather than a pseudospectral
one: we search for Xj = (αj, β(1)j , β
(2)j , β
(2)j ) such that ANt
j Xj = bNtj where
bNtj =
((vNtj − v
Nt−1j )/∆x, (vNt−1
j − vNt−2j )/∆x, (vNt−2
j − vNt−3j )/∆x, (vNt−3
j − vNt−4j )/∆x
)and ANt ∈ R4×4
ANtj;m =
(− (vNt+1−m
j − 2vNt−mj + vNt−m−1
j )/∆t2,−(vNt−mj+1 − vNt−m
j−1 )/2∆x,
(vNt−mj+1 − 2vNt−m
j + vNt−mj−1 )/∆x2 − (vNt−m
j+2 − 2vNt−mj+1 + 2vNt−m
j−1 − vNt−mj−2 )/2∆x3
)We respectively report the data {vNt
j }|j|6J , at final time T , {αj}|j|6J and {cj}|j|6J and in Fig.
5 and in Fig. 6 {β(2)}|j|6J , {νj}|j|6J , and {β(3)}|j|6J , {ηj}|j|6J . A very good agreement for thevelocity and diffusion coefficients is observed. We also obtain a dispersion coefficients closeto zero, as expected. More generally, this test suggests that it may be appropriate to directlysearch for an high order equation, and discriminate afterwards the negligible coefficients.
20
−0.6 −0.4 −0.2 0 0.2 0.4 0.60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data at final time T
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.01
−0.008
−0.006
−0.004
−0.002
0
0.002
0.004
0.006
0.008
0.01
Double derivative in t coefficient, α
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0
2
4
6
8
10
12
14
16
18
20
Velocity, β(1)
Reconstructed velocity
Exact velocity
Figure 5: (Left) Experimental data (Middle) Coefficients {αj}|j|6J (Right) Reconstructed and exact veloci-
ties ({β(1)j }|j|6J , {cj}|j|6J , ).
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.2
0
0.2
0.4
0.6
0.8
Diffusion coefficient, β(2)
Reconstructed diffusion
Exact diffusion
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−5
−4
−3
−2
−1
0
1
2
3
4
5x 10
−3 Dispersion coefficient, β(3)
Reconstructed dispersion
Exact dispersion
Figure 6: (Left) Reconstructed and exact diffusion ({β(2)j }|j|6J , {νj}|j|6J) (Right) Reconstructed dispersion
({β(3)j }|j|6J , {ηj}|j|6J).
The reconstruction was performed using the data at iterations Nt, · · ·Nt − 4, but couldhave been as well done using earlier iterations, in order to validate the results.Case 2. We assume here that the perturbation is of the form ε(x, ζ) (resp. ε(t, x, ζ)) whereζ is the random variable, where ‖ε‖∞ = ε∞ = 10−7 (resp. ‖ε‖∞ = ε∞ = 10−9). We reportthe reconstructed velocity and diffusion coefficients in Fig. 7. We report in Fig. 8, the `2-norm, as a function of `, for time independent (resp. dependent) perturbation (Left) (resp.(Right)).
E(`) =
√(∑j
|α`j|2 +∑j
(|β(1);` − cj|2 + |β(2);` − νj|2 + |β(3);`|2
)).
Again a very good agreement is obtained.Case 3. In this test, we assume that the searched equation is the following
∂tu+ c(x)∂x + ν(x)∂xxu+ η(x)∂xxxu = 0, (t, x) ∈ [0, T ]× [−1, 1]
21
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0
2
4
6
8
10
12
14
16
18
20
Velocity, β(1)
Reconstructed velocity
Exact velocity
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−5
−4
−3
−2
−1
0
1
2
3
4
5x 10
−3 Dispersion coefficient, β(3)
Reconstructed dispersion
Exact dispersion
Figure 7: (Left) Reconstructed and exact velocities ({β(1)j }|j|6J , {cj}|j|6J). (Middle) Reconstructed and
exact dispersion ({β(3)j }|j|6J , {ηj}|j|6J).
0 50 100 150 200 250
100
l2−norm error
Iteration
l2−
no
rm e
rro
r
10 20 30 40 50 60
101
l2−norm error
Iteration
l2−
no
rm e
rro
r
Figure 8: `2-norm error in logscale {`, E(`)}. (Left) ε(x, ζ) with εmax = 10−7. (Right) ε(t, x, ζ) withεmax = 10−9.
with c(x) = exp(− 10x2 + 10−1
)×(1 + 0.2 sin(20x)
),
ν(x) = 5× 10−1 sin2(2πx) exp(−10x2),η(x) = 10−3 exp(−50x2) sin2(8x) .
The initial data is given by u0(x) = exp(−20x2)| cos(x)|. Numerically the data are computedNx = 1001-point grid, ∆t = ∆x/ supx∈[−1,1] |c(x)|. We do not impose any perturbation inthis test. The convergence is then reached in 1 iteration. We respectively report Fig. 9{βj}(2)
|j|6J , {νj}|j|6J , and {βj}(3)|j|6J , {ηj}|j|6J . We notice again a very good agreement for the
velocity and diffusion coefficients. We also obtain a dispersion coefficients close to zero, asexpected. This suggests again that it may be appropriate to directly search for an high orderequation, and to discriminate afterwards the negligible operators.
22
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Diffusion coefficient, β(2)
Reconstructed diffusion
Exact diffusion
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−5
0
5
10
15x 10
−4 Dispersion coefficient, β(3)
Reconstructed dispersion
Exact dispersion
Figure 9: (Left) Reconstructed and exact diffusion ({β(2)j }|j|6J , {νj}|j|6J) (Right) Reconstructed dispersion
({β(3)j }|j|6J , {ηj}|j|6J).
5.3. First order quasilinear equationIn this test we assume that the searched equation is quasilinear
∂tu+ a(u)∂xu+ ν(u)∂xxu = 0, (t, x) ∈ [0, T ]× [−a, a]
where a(u) = 0.5u2 and ν(u) = u2| sin(u)|. The initial data, the data are constructed fromis given by u0(x) = exp(−5x2). Numerically these data are computed on Nx = 3001 pointgrid, ∆t = ∆x/ supx∈[−1,1] |a(u)|. We search the equation in the form
∂tu+ β(1)(u)∂xu+ β(2)(u)∂xxu = 0, (t, x) ∈ [0, T ]× [−a, a] .
In order to reconstruct the unknown function, we search for
XNtj = (β(1)(vNt
j ), β(2)(vNtj ), β(1)(vNt
j ), β(2)(vNtj )) ,
solving a 4 equations systems ANtj X
Ntj = bNt
j where
bNtj =
((vNtj − v
Nt−1j )/∆tNt , (v
Nt−1j − vNt−2
j )/∆tNt−1,
(vNt−2j − vNt−3
j )/∆tNt−2, (vNt−1j+1 − v
Nt−2j+1 )/∆tNt−1
)and ANt ∈ R4×4
ANtj;1 =
((vNt−1j+1 − v
Nt−1j−1 )/2∆x, (vNt−1
j+1 − 2vNt−1j + vNt−1
j−1 ))/∆x2
(vNt−1j − vNt−2
j )(vNt−1j − vNt−1
j−1 )/∆x, (vNt−1j − vNt−2
j )(vNt−1j+1 − 2vNt−1
j + vNt−1j−1 )/∆x2
)ANtj;2 =
((vNt−2j+1 − v
Nt−2j−1 )/2∆x, (vNt−2
j+1 − 2vNt−2j + vNt−2
j−1 ))/∆x,0, 0)
ANtj;3 =
((vNt−3j+1 − v
Nt−3j−1 )/2∆x, (vNt−3
j+1 − 2vNt−3j + vNt−3
j−1 ))/∆x2
−(vNt−3j − vNt−4
j )(vNt−3j+1 − v
Nt−1j−1 )/2∆x, (vNt−3
j − vNt−4j )(vNt−3
j+1 − 2vNt−3j + vNt−3
j−1 )/∆x2)
ANtj;4 =
((vNt−2j+2 − v
Nt−2j )/2∆x, (vNt−2
j+2 − 2vNt−2j+1 + vNt−2
j ))/∆x2
−(vNt−2j+1 − v
Nt−2j−1 )(vNt−2
j+2 − vNt−2j )/4∆x, (vNt−2
j+1 − vNt−2j−1 )(vNt−2
j+2 − 2vNt−2j+1 + vNt−2
j )/∆x2)
23
In Fig. 10, we respectively report the solution at final time T ,{β(1)(vNt
j )}|j|6J approxi-
mating {a(uNtj )}|j|6J and
{β(2)(vNt
j )}|j|6J approximating ν(uNt
j )|j|6J . We notice again a very
good agreement for the quasilinear velocity and diffusion. This example is however, from alimited interest as the exact order of the equation is known.
−0.6 −0.4 −0.2 0 0.2 0.4 0.60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data at final time T
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Velocity, β(1)
(u)
Reconstructed nonlinear velocity
Exact nonlinear velocity
−0.6 −0.4 −0.2 0 0.2 0.4 0.6−0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Nonlinear diffusion, β(2)
(u)
Reconstructed nonlinear diffusion
Exact nonlinear diffusion
Figure 10: (Left) Experimental data (Middle) Reconstructed and exact velocities ({β(1)(vNt
j )}|j|6J
,
{a(uNtj )}|j|6J) (Right) Reconstructed and exact diffusion (
{β(2)(vNt
j )}|j|6J
, ν(uNtj )|j|6J).
5.4. Fractional equation in 2-d
We assume that the searched equation is a first order fractional equation of the form
∂tu+4∑l=1
(a(l)∂l/4x u+ a(l+4)∂l/4y u
)+ a(9)∂1/2
x ∂1/2y u (25)
+a(10)∂3/4x ∂1/4
y u+ a(11)∂1/4x ∂3/4
y u+ a(12)∂1/2x ∂1/2
y u = 0 . (26)
The given data corresponds to the solution to (25), where
a(1) = 0.25, a(2) = −0.1, a(3) = 0.5, a(4) = 0.1, a(5) = −0.25, a(6) = 0.2,a(7) = 0.1, a(8) = 0.04, a(9) = 0.0, a(10) = 0.1, a11 = 0.05, a(12) = 0.075
(27)
and the initial data u0(x, y) = exp(− (|x|2 + |y|2)/20
)on [−20, 20]2. The given data are
approximations of the sequence {u(tn,xj)}j , with j = (k, l) ∈ Z2 and x = (x, y), whereu is solution to the unknown Equation (25). In this benchmark, there are preliminarilynumerically computed from (25) on aNx×Ny point grid, withNx = Ny = 401 and ∆t = 10−4.The data are assumed to be perturbed by a space dependent stochastic process
vn;`j = unj (1 + ε`j)
where ‖ε‖∞ 6 10−6. We construct the Fourier transform of {vNt;`j }j and {vNt−1;`
j }j , that is
{vNt;`k }k and {vNt−1
k }k.The objective is to estimate the symbol p of (25),
p(t, ξx, ξy) =∑4
l=1
(a(l)(iξx)
l/4 + a(l+4)(iξy)l/4)− a(9)i(ξxξy)
1/2
+a(10)(iξy)1/4(iξx)
3/4 + a(11)(iξx(i)1/4ξy)3/4 + a(12)i(ξxξy)
1/2 .
24
That is, we search for p(tNt , ·) such that for any k: p(tNt , ξk) = pNtk , with
pNt;`k = −
1
∆tlog( vNt;`
k
vNt−1;`k
).
where vNt−1;`k 6= 0. The approximate symbol then reads
p(t, ξx, ξy) =∑4
l=1
(α(l)(iξx)
l/4 + α(l+4)(iξy)l/4)− α(9)i(ξxξy)
1/2
+α(10)iξy(iξx)1/2 + α(11)iξx(iξy)
1/2 − α(12)ξxξy
such that p(tNt , ξk) = p(tNt , ξk). This leads to a 12-equation system, which can be solved for12 different values of ξk of `. We report in Tables 4, the computed values of the {α(l)}16l612,as well as the relative error (last line) compared to the exact coefficients in (25) and (27).We report in Fig. 11 the `2-norm of the error in logscale, between the exact and approximatecoefficients at stochastic iteration `, that is {`, log
(E(`)
)}, where
E(`) :=( 12∑l=1
|α(l);` − a(l)|2)1/2
.
A good agreement is again obtained for a small amplitude perturbation. In particular, the
10 20 30 40 50 60 70 80 90
10−1
l2−norm error
Iteration
Err
or
in lo
gsca
le
Figure 11: `2-norm error in logscale {`, E(`)} (Left) ε(x, ζ) with εmax = 10−8.
simulations suggest that α(0) should be taken null.
5.5. Linear equation with non-constant coefficients
In this second two-dimensional test, the searched equation is the following
∂ttu(t, x, y) + a(t)∂tu(t, x, y) = b(x)(x, y)∂xu(t, x, y) + b(xx)(x, y)∂xxu(t, x, y)+b(xy)(x, y)∂xyu(t, x, y) + a(0)(x, y)u(t, x, y) .
25
α(1) α(2) α(3)
2.545853246443962× 10−1 −1.026353837787546× 10−1 5.102543087095289× 10−1
α(4) α(5) α(6)
9.532077960495579× 10−2 −2.543654731841340× 10−1 2.064762283831498× 10−1
α(7) α(8) α(9)
1.001893005064210× 10−1 3.465727520278510× 10−2 1.432068478385523× 10−8
α(10) α(11) α(12)
9.559526009066791× 10−2 4.139113603023897× 10−2 7.337416044103000× 10−2
Table 4: Numerical values α(0);1, · · · , α(6);1 and α(0), · · · , α(6): ε(t, x, ζ) with εmax = 10−9.
where the coefficients are given by
a(0)(x, y) = exp(− |x|/2− 3|y|/4
), a(t)(x, y) = exp
(− |x|2/4− |y|2/2
)b(x)(x, y) =
1
2exp
(− |x|2/10− |y|2/10
)cos(2x) sin(2y), b(xx)(x, y) = exp
(− 2|x|2 − |y|2/2
)b(xy)(x, y) = exp
(− (|x|2 + |y|2)/10
).
(28)
We assume that the searched equation is a linear wave equation of the form
∂ttu(t, x, y) + α(t)∂tu(t, x, y) = β(x)(x, y)∂xu(t, x, y) + β(y)(x, y)∂yu(t, x, y)+β(xx)(x, y)∂xxu(t, x, y) + β(yy)(x, y)∂yyu(t, x, y)+β(xy)(x, y)∂xyu(t, x, y) + α(0)(x, y)u(t, x, y) .
We implement the method described in Section 3. The data {vnj }j are given with thecorresponding positions, for (at least) 7 consecutive times. In practice, these data wereobtained numerically on a finite domain [−5, 5]2, at final time T = 1.75, with a initialdata u0(x, y) exp(−|x|2 − |y|2). The time step is ∆t = 1.75 × 10−2, and the computationwas performed on a 101 × 101-point grid. From {vn;`}, with n ∈ {Nt − 11, · · · , Nt}, andNt = 100, we compute:
• [[∂xvn;`]], [[∂yv
n;`]], [[∂xxvn;`]], [[∂yyv
n;`]], [[∂xyvn;`]].
• For any j, we have to determine 7 coefficients α(0);`j , α
(t);`j , β
(x);`j , β
(y);`j , β
(xx);`j , β
(yy);`j , β
(xy);`j ,
from the equations (n-dependent)
0 = ∆tα(t);`j
(vn−1;`j + vn−2;`
j
)+ ∆t2
(β
(x);`j [[∂xv
n−1;`]]j + β(y);`j [[∂yv
n−1;`]]j
+β(xx);`j [[∂xxv
n−1;`]]j + β(yy);`j [[∂yyv
n−1;`]]j + β(xy);`j [[∂xyv
n−1;`]]j
)+α(0)vn−1;`
j − vn;`j − 2vn−1;`;`
j + vn−2;`j .
As for any j, we then need 7 equations, involving {vNt−11;`}, · · · , {vNt;`}. We then solve a7× 7 system, with An;`
j ∈ C7×7: An;`j X
`j = bn;`
j , where
X`j =
(α
(0);`j , α
(t);`j , β
(x);`j , β
(y);`j , β
(xx);`j , β
(yy);`j , β
(xy);`j
)bNt;`j =
(vNt−6;`j − 2vNt−7;`
j + vNt−8;`j , · · · , vNt;`
j − 2vNt−1;`j + vNt−2;`
j
).
26
Figure 12: Solution at final time.
Again, more data, would allow to refine the computed data. We represent in Fig. 13the reconstructed coefficients, when the perturbation ε is null, or cancelled following themethodology developed in Remark 2.1. The obtained coefficients are consistent with theexact coefficients (28).We also report the reconstructed coefficients, in the case of a perturbative spatial random
process of the data. That is: vn;`j = u(tn,xj)(1 + ε`j) with supj |εj| 6 10−4, see Fig. 14. In
particular, the numerical computations suggest, as expected, that β(y), and β(yy) should benull functions.
6. Conclusion
In this paper, we have presented a methodology for constructing linear evolution PDEfrom data: i) experimental values of the solution of the unknown equation, along withii) the corresponding positions and times. Using pseudospectral methods and symboliccomputation, we have solved this inverse problem by assuming that the data are randomlyperturbed. Although, some assumptions are still required to address this problem (linearityof the equation, order in time), some promising results are obtained, at least for sufficientlysmall amplitude perturbations. Extension to time-dependent and nonlinear equations iscurrently investigated.
27
Appendix A. Spectral approximation
In a bounded domain Πdi=1[−axi , axi ], we denote the grid-point set by
DNx1 ,··· ,Nxd={xj = (x1,j1 , · · · , xd,jd)
}j∈O
Πdi=1
Nxi
where j = (j1, · · · , jd) and with
OΠdi=1Nxi
={j ∈ Nd, : j1 = 0, · · · , Nx1 − 1; · · · ; jd = 0, · · · , Nxd − 1
}.
and
xki+1 − xki = ∆xi = 2axi/Nxi .
The corresponding discrete wavenumbers are defined by ξi := (ξi,1, · · · , ξi,d), where ξp =pπ/axi with p ∈ {−Nxi/2, · · · , Nxi/2− 1}. We can define the partial Fourier pseudospectral
approximations φ with, in the xi-direction
φk1
(t, x2, · · · , xd)eiξp(x1+ax1 ) =∑Nx1−1
k1=0 φk1(t, x2, · · · , xd)e−iξk1(xk1
+ax1)
· · · · · · · · ·· · · · · · · · ·
φkd(t, x1, · · · , xd−1)eiξp(x1+ax1 ) =∑Nxd
−1
kd=0 φkd(t, x1, · · · , xd−1)e−iξkd (xkd+axd)
where we have also denoted
φk1(t, x2, · · · , xd) = φ(t, x1,k1 , x2, · · · , xd)· · · · · · · · ·· · · · · · · · ·
φkd(t, x1, · · · , xd−1) = φ(t, x1, · · · , xd−1, xd,kd).
We finally define the approximate first-order partial derivatives, using the pseudospectralapproximation ∂xiφ(tn,x), by
[[∂xiφ]] :=1
Nxi
∑Nxi/2−1
ki=−Nxi/2iξpφk1
(t, x1, · · · , xi−1, xi+1, · · · , xd)eiξki (xki+axi )
Approximate higher (`th) order partial derivatives ∂(`)xi φ(tn,x), by
[[∂(`)xi φ]] :=
1
Nxi
∑Nxi/2−1
ki=−Nxi/2
(iξki
)`φk1
(t, x1, · · · , xi−1, xi+1, · · · , xd)eiξki (xki+axi )
We refer to [3] and [2] for details.
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Figure 13: Computed coefficients: (1st row) α(0), α(t). (2nd row) β(x), β(y) (3rd row) β(xx), β(yy). (4th row)β(xy).
31
Figure 14: Computed coefficients: (1st row) α(0), α(t). (2nd row) β(x), β(y) (3rd row) β(xx), β(yy). (4th row)β(xy).
32