Solving forward and inverse problems in a non-linear 3D PDE via an asymptotic expansion based approach

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Solving forward and inverse problems in a non-linear 3D PDE via an asymptotic

expansion based approach ∗

Dmitrii Chaikovskii†and Ye Zhang‡

Abstract. This paper concerns the use of asymptotic expansions for the eﬃcient solving of forward and inverse

problems involving a nonlinear singularly perturbed time-dependent reaction–diﬀusion–advection

equation. By using an asymptotic expansion with the local coordinates in the transition-layer region,

we prove the existence and uniqueness of a smooth solution with a sharp transition layer for a

three-dimensional partial diﬀerential equation. Moreover, with the help of asymptotic expansion,

a simpliﬁed model is derived for the corresponding inverse source problem, which is close to the

original inverse problem over the entire region except for a narrow transition layer. We show that

such simpliﬁcation does not reduce the accuracy of the inversion results when the measurement

data contain noise. Based on this simpler inversion model, an asymptotic-expansion regularization

algorithm is proposed for eﬃciently solving the inverse source problem in the three-dimensional case.

A model problem shows the feasibility of the proposed numerical approach.

Key words. Singular perturbed partial diﬀerential equation; Reaction–diﬀusion–advection equation; Asymp-

totic expansion; Inverse source problem; Regularization; Convergence.

MSC codes. 65M32, 35C20, 35G31

1. Introduction. The main goal of this paper is to develop a new framework for solving

the following three-dimensional inverse source problem eﬃciently.

(IP): Given full noisy data {uδ, uδ

x, uδ

y, uδ

z}of {u, ux, uy, uz}or partial noisy data {uδ}of

{u}at the p·q·vlocation points {xi, yj, zk}p,q,v

i,j,k=0 and at the time point t0, ﬁnd the source

function f(x, y, z) such that (u, f) satisﬁes the following dimensionless nonlinear problem:











µ∆u−∂u

∂t =−u∂u

∂x +∂u

∂y +∂u

∂z +f, (x, y)∈R2, z ∈(−a, a)≡Ω, t ∈(0, T ]≡ T ,

u(x, y, z, t, µ) = u(x+L, y, z, t, µ) = u(x, y +M, z, t, µ),(x, y, z, t)∈R2×¯

Ω×¯

u(x, y, −a, t, µ) = u−a(x, y), u(x, y, a, t, µ) = ua(x, y),(x, y, t)∈R2×¯

u(x, y, z, 0, µ) = uinit(x, y, z, µ),(x, y, z)∈R2×¯

Ω,

(1.1)

where u(x, y, z, t) represents the dimensionless value (e.g., of the temperature), µ1 is a small

parameter (also called the diﬀusion coeﬃcient), and f(x, y, z) is the source function. Suppose

that (i) the function f(x, y, z) is L-periodic in the variable x,M-periodic in the variable y, and

suﬃciently smooth in the region (x, y, z) : R2×¯

Ω (Ω ≡(−a, a)), (ii) the functions u−a(x, y)

∗Corresponding Author: Ye Zhang.

Funding: The work was funded by the National Natural Science Foundation of China (No. 12171036) and

Beijing Natural Science Foundation (Key project No. Z210001).

†Shenzhen MSU-BIT University, 518172 Shenzhen, China (dmitriich@smbu.edu.cn).

‡School of Mathematics and Statistics, Beijing Institute of Technology, 100081 Beijing, China (

ye.zhang@smbu.edu.cn).

arXiv:2210.05220v2 [math.NA] 14 Feb 2023

2 DMITRII CHAIKOVSKII AND YE ZHANG

and ua(x, y) are L-periodic in x,M-periodic in y, and suﬃciently smooth in (x, y)∈R2, and

(iii) uinit(x, y, z, µ) is a suﬃciently smooth function in (x, y, z) : R2×Ω, is L-periodic in xand

M-periodic in y, and satisﬁes uinit(x, y, −a, µ) = u−a(x, y), uinit(x, y, a, µ) = ua(x, y).

The reaction–diﬀusion–advection (RDA) system with small parameters plays an impor-

tant role in the quantitative study of many scientiﬁc problems, including chromatography

[1,2], biology [3], and environmental problems [4]. This paper is focused on the solutions of

RDA system (1.1) with a large gradient in a certain region. This region is called the inner

transition layer, whose width is usually small compared to that of the entire region. The pres-

ence of a small parameter accompanying the highest derivative makes the equation singularly

perturbed, i.e., one cannot simply ignore the term with a small parameter in the diﬀerential

equation because doing so would lead to a totally diﬀerent solution that would not reﬂect the

physical phenomenon being studied. RDA problems with inner transition layers are often used

for the mathematical modeling of the density distributions of liquids or gases or temperature

in the presence of spatial inhomogeneities [5,6] or in nonlinear acoustics [7,8,9], and such

problems include the temperature behavior in the near-surface layer of the ocean [10], the

carrier wave functions in Si/Ge heterostructures [11], and propagation of an autowave front

in a medium with barriers [12,13]. Determining existence conditions for and the stability

of nonstationary solutions with large gradients is important for creating adequate models of

processes with nonstationary distributions of ﬁelds of physical quantities [14,15]. Analytical

studies also make it possible to create eﬃcient numerical methods for solving equations with

inner transition layers [16,17,18,19,20]. The technique of asymptotic analysis is remark-

ably good for overcoming the aforementioned diﬃculties, i.e., the issues of (i) existence and

uniqueness of smoothing solutions and (ii) high qualiﬁed resolution of approximate analytical

solutions.

On the other hand, because the inverse source problem (IP) is ill-posed (i.e., small noise

in the measurement data may lead to arbitrarily large changes in the classical approximate

solution, which is far from the ground truth; see [21,22] for details), a regularization tech-

nique should be developed for a stable solution of (IP). Following the framework of Tikhonov

regularization, (IP) can be converted to the following minimization problem with partial

diﬀerential equation (PDE) constraints:

(1.2) min

i=0

j=0 (hu(xi, yj, zk, t0)−uδ(xi, yj, zk, t0)i2

+∂u

∂x(xi, yj, zk, t0)−∂uδ

∂x (xi, yj, zk, t0)2

+∂u

∂y (xi, yj, zk, t0)−∂uδ

∂y (xi, yj, zk, t0)2

+∂u

∂z (xi, yj, zk, t0)−∂uδ

∂z (xi, yj, zk, t0)2)+εR(f),

where usolves the nonlinear PDE (1.1) with a given f,R(f) is the regularization term, and

ε > 0 is the regularization parameter.

There are two essential diﬃculties when employing the solution model (1.2). First, it

is diﬃcult to select an appropriate regularization term Rand regularization parameter εin

practice. By the standard argument of the regularization theory of inverse problems, the

ASYMPTOTIC EXPANSION REGULARIZATION FOR 3D INVERSE SOURCE PROBLEMS 3

optimal choices of these two quantities depend on the ground truth f, which is unknown in

real-world problems. Second, even if both Rand εare given, the numerical realization of the

PDE-constrained optimization problem (1.2) is a very hard task because it is a nonconvex

optimization with a nonlinear PDE constraint. Therefore, the main purpose of this work is

to ﬁnd a new model to replace (1.2), one that is much simpler but still suﬃciently accurate.

Fortunately, this can be achieved by using the theory of asymptotic analysis, which is also a

main motivation of this work.

Indeed, several previous studies have used asymptotic analysis to solve inverse problems

involving singularly perturbed PDEs; for instance, in [23], a coeﬃcient inverse problem for the

one-dimensional RDA equation was solved using the position of the moving front, which was

obtained approximately using asymptotic analysis, and in [20], an inverse problem of deter-

mining the position of a moving front for a two-dimensional reaction–diﬀusion equation was

studied via asympototic analysis. Actually, the present paper can be viewed as an extension

of our previous work [24], which dealt with one-dimensional inverse source problems under

some a priori structure assumptions about the source function. The novelty of the present

work is combining asymptotic expansions and local coordinates to solve the forward problem,

which in turn allows us to reduce the inverse problem of ﬁnding the source function in the

original PDE with a large derivative to a simpler equation and obtain suﬃciently accurate

results. Although local coordinates have been used previously to study some two-dimensional

forward [25] and inverse [20] problems, to the best of our knowledge this is the ﬁrst time

that the combination of local coordinates and Vasil’eva’s algorithm for asymptotic expansion

in a small parameter [26] has been used. Moreover, the mathematical analysis in a three-

dimensional setting—including the existence and uniqueness of the forward problem and the

inversion method for (IP)—has never been investigated before.

This paper is organized as follows. Section 2presents the asymptotic analysis for the

forward problem. Section 3presents the main theoretical results, while Section 4presents their

technical proofs. Section 5presents some experiments for the forward and inverse problems.

Finally, Section 6presents concluding remarks.

2. Asymptotic analysis. We investigate the solution of problem (1.1), which has the

form of a moving front that at each moment of time is close to ϕ(−)(x, y, z) for −a≤z≤

h(x, y, t), close to ϕ(+)(x, y, z) for h(x, y, t)≤z≤a, and changes sharply from ϕ(−)(x, y, z)

to ϕ(+)(x, y, z) in a neighbourhood of the surface z=h(x, y, t). In this case, the solution to

problem (1.1) has an inner transition layer in the vicinity of this surface.

First, we list all the main assumptions used throughout the paper.

Assumption 2.1. u−a(x, y)<0,ua(x, y)>0, and ua(x, y)−u−a(x, y)>2µ2for all (x, y)∈

R2.

Assumption 2.2. For any (x, y, z)∈R2×¯

Ω,

u−a(x−a−z,y−a−z)2>−2Za+z

f(x−a−z+s,y−a−z+s,−a+s)ds,

(ua(x+a−z,y+a−z))2>2Z0

z−a

f(x+a−z+s,y+a−z+s,a+s)ds.

4 DMITRII CHAIKOVSKII AND YE ZHANG

Assumption 2.3. −a<h0(x, y, t)< a for any (x, y, t)∈R2× T , where h0is the zero

approximation of h[see (2.12)] and max

(x,y)∈R2, t∈¯

T∂h0(x, y, t)

∂x +∂h0(x, y, t)

∂y <1.

Assumption 2.4. uinit(x, y, z) = Un−1(x, y, z, 0) + O(µn), where the asymptotic solution

Un−1is constructed later; see Theorem 3.1, for example.

Under Assumptions 2.1 and 2.2,ϕ(−)(x, y, z) and ϕ(+)(x, y, z)—which are used to con-

struct the zeroth-order asymptotic solution [cf. (2.23)]—can be calculated explicitly from the

reduced stationary equations (2.21) and (2.22) by using the method of characteristics:











ϕ(−)=−s(u−a(x−a−z,y−a−z))2+ 2 Za+z

f(x−a−z+s,y−a−z+s,−a+s)ds,

ϕ(+) =s(ua(x+a−z,y+a−z))2−2Z0

z−a

f(x+a−z+s,y+a−z+s,a+s)ds.

(2.1)

The surface z=h(x, y, t) at each moment of time divides the region ¯

Ω into two parts:

Ω(−)={z:z∈[−a;h(x, y, t)]}and ¯

Ω(+) ={z:z∈[h(x, y, t); a]}. Let us introduce the local

coordinates r, l, m and write the equation of the normal to the surface h(l, m, t):

x−l

hl(l, m, t)=y−m

hm(l, m, t)=z−h(l, m, t)

−1.

The equation for the distance from the point with coordinates (r, l, m) to the surface h(l, m, t)

along the normal to it has the following form:

r2= (x−l)2+ (y−m)2+ (z−h)2= (z−h)2(1 + h2

l+h2

m).

For a detailed description of the solution in the inner transition layer, in the vicinity of

surface h(l, m, t) we proceed to the extended variable

(2.2) ξ=r

and to the local coordinates (r, l, m) using the relations

(2.3) x=l−rhl

q1 + h2

l+h2

, y =m−rhm

q1 + h2

l+h2

, z =h(l, m, t) + r

q1 + h2

l+h2

We assume r > 0 in the domain R2×¯

Ω(+) and r < 0 in the domain R2×¯

Ω(−), and we note

that if z=h(l, m, t), then we have r= 0, l=x, and m=y; the derivatives of the function

h(l, m, t) in (2.3) are also taken for l=xand m=y.

The asymptotic approximation U(x, y, z, t, µ) of the solution of problem (1.1) is construc-

ted separately in regions ¯

Ω(−)and ¯

Ω(+), i.e.,

(2.4) U(x, y, z, t, µ) = (U(−)(x, y, z, t, µ),(x, y, z, t)∈R2×¯

Ω(−)× T ,

U(+)(x, y, z, t, µ),(x, y, z, t)∈R2×¯

Ω(+) × T ,

ASYMPTOTIC EXPANSION REGULARIZATION FOR 3D INVERSE SOURCE PROBLEMS 5

as the sums of two terms

(2.5) U(∓)= ¯u(∓)(x, y, z, µ) + Q(∓)(ξ, l, m, h(l, m, t), t, µ),

where ¯u(∓)(x, y, z, t, µ) are the functions describing the outer region and Q(∓)(ξ, l, m, h(l, m, t),

t, µ) are the functions describing the inner transition layer.

Each term in (2.5) is represented as an expansion in powers of the small parameter µ:

¯u(∓)(x, y, z, µ) = ¯u(∓)

0(x, y, z) + µ¯u(∓)

1(x, y, z) + . . . ,(2.6)

Q(∓)(ξ, l, m, h, t, µ) = Q(∓)

0(ξ, l, m, h, t) + µQ(∓)

1(ξ, l, m, h, t) + . . . .(2.7)

We assume that z=h(x, y, t) is the surface on which the solution u(x, y, z, t, µ) to problem

(1.1) at each time ttakes on a value equal to the half-sum of the functions ¯u(−)(x, y, z) and

¯u(+)(x, y, z):

(2.8) φ(x, y, h(x, y, t), µ) := 1

2¯u(−)(x, y, h(x, y, t), µ) + ¯u(+)(x, y, h(x, y, t), µ).

For the zeroth-order approximation, it takes the form

(2.9) φ0(x, y, h(x, y, t)) := 1

2ϕ(−)(x, y, h(x, y, t)) + ϕ(+)(x, y, h(x, y, t)).

The functions U(−)(x, y, z, t, µ) and U(+)(x, y, z, t, µ) and their derivatives along the nor-

mal to the surface z=h(x, y, t) are matched continuously on the surface h(x, y, t) at each

moment of time t, i.e., the following two conditions hold:

U(−)(x, y, h(x, y, t), t, µ) = U(+)(x, y, h(x, y, t), t, µ) = φ(x, y, h(x, y, t), µ),(2.10)

∂U(−)

∂n (x, y, h(x, y, t), t, µ) = ∂U(+)

∂n (x, y, h(x, y, t), t, µ),(2.11)

where the function φ(x, y, h(x, y, t), µ) is deﬁned in (2.8).

The surface z=h(x, y, t) is also sought in the form of an expansion in powers of a small

parameter, i.e.,

(2.12) h(x, y, t) = h0(x, y, t) + µh1(x, y, t) + µ2h2(x, y, t) + . . . ,

and we introduce a notation for the approximation of h(x, y, t) with expansion terms up to

order n, i.e.,

(2.13) ˆ

hn(x, y, t) =

i=0

µihi(x, y, t),(x, y)∈R2, t ∈¯

We introduce the vectors x= (x, y, z, t)Tand r= (r, l, m, t)T, where r=r(x, y, z, t), l =

l(x, y, z, t), m =m(x, y, z, t), and x, y, z are deﬁned in (2.3).

The total diﬀerential of the vector ris dr=∇xrdx. On the other hand, if we look at

the inverse problem of determining the diﬀerential changes in our original coordinate system

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Solvingforwardandinverseproblemsinanon-linear3DPDEviaanasymptoticexpansionbasedapproachDmitriiChaikovskiiyandYeZhangzAbstract.Thispaperconcernstheuseofasymptoticexpansionsfortheecientsolvingofforwardandinverseproblemsinvolvinganonlinearsingularlyperturbedtime-dependentreaction{diusion{advectioneq...

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Solving forward and inverse problems in a non-linear 3D PDE via an asymptotic expansion based approach

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