This article was downloaded by: [UQ Library]On: 23 June 2015, At: 04:08Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Click for updates

Inverse Problems in Science andEngineeringPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gipe20

Computational algorithms for solvingthe coefficient inverse problem forparabolic equationsP.N. Vabishchevicha & V.I. Vasil’evb

a Nuclear Safety Institute, Russian Academy of Sciences, Moscow,Russia.b Institute of Mathematics and Information Science, North-EasternFederal University, Yakutsk, Russia.Published online: 22 Dec 2014.

To cite this article: P.N. Vabishchevich & V.I. Vasil’ev (2014): Computational algorithms forsolving the coefficient inverse problem for parabolic equations, Inverse Problems in Science andEngineering, DOI: 10.1080/17415977.2014.993984

To link to this article: http://dx.doi.org/10.1080/17415977.2014.993984

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering, 2014http://dx.doi.org/10.1080/17415977.2014.993984

Computational algorithms for solving the coefficient inverse problemfor parabolic equations

P.N. Vabishchevicha∗ and V.I. Vasil’evb

aNuclear Safety Institute, Russian Academy of Sciences, Moscow, Russia; bInstitute of Mathematicsand Information Science, North-Eastern Federal University, Yakutsk, Russia

(Received 10 July 2014; final version received 26 November 2014)

Among inverse problems for partial differential equations, we distinguish coeffi-cient inverse problems, which are associated with the identification of coefficientsand/or the right-hand side of an equation using some additional information.When considering time-dependent problems, the identification of the coefficientdependences on space and on time is usually separated into individual problems.In some cases, we have linear inverse problems (e.g. identification problemsfor the right-hand side of an equation); this situation essentially simplify theirstudy. This work deals with the problem of determining in a multidimensionalparabolic equation the lower coefficient that depends on time only. To solvenumerically a non-linear inverse problem, linearized approximations in timeare constructed using standard finite difference approximations in space. Thecomputational algorithm is based on a special decomposition, where the transitionto a new time level is implemented via solving two standard elliptic problems.

Keywords: inverse problem; identification of the coefficient; parabolic partialdifferential equation; finite difference scheme

AMS Subject Classifications: 65M06; 65M32; 80A23

1. Introduction

In the theory and practice of inverse problems for partial differential equations (PDEs), muchattention is paid to the problem of the identification of coefficients from some additionalinformation.[1,2] Particular attention should be given to inverse problems for PDEs.[3,4] Inthis case, a theoretical study includes the fundamental questions of uniqueness of the solutionand its stability both from the viewpoint of the theory of differential equations [4,5] andfrom the viewpoint of the theory of optimal control for distributed systems.[6] Many inverseproblems are formulated as non-classical problems for PDEs. To solve approximately theseproblems, emphasis is on the development of stable computational algorithms that take intoaccount peculiarities of inverse problems.[7,8]

Much attention is paid to the problem of determining the right-hand side, lower andleading coefficients of a parabolic equation of second order, where, in particular, the right-hand side and the coefficients depend on time only. An additional condition is most oftenformulated as a specification of the solution at an interior point or as the average value

∗Corresponding author. Emails: vab@ibrae.ac.ru, vabishchevich@gmail.com

© 2014 Taylor & Francis

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

2 P.N. Vabishchevich and V.I. Vasil’ev

that results from integration over the whole domain. The existence and uniqueness of thesolution to such an inverse problem and well posedness of this problem in various functionalclasses are examined, for example, in the works [9–12].

Numerical methods for solving problems of the identification of the right-hand side,lower and leading coefficients for parabolic equations are considered in many works.[13–17] In view of the practical use, we highlight separately the studies dealing with numericalsolving inverse problems for multidimensional parabolic equations.[8,18,19] To constructcomputational algorithms for the identification of the lower coefficient of a parabolicequation, there is widely used idea of transformation of the equation by introducing newunknowns that result in a linear inverse problem. General transformations of some classesof inverse problems for parabolic equations are discussed in the paper [20]. In the works[21,22], on the basis of such transitions to non-classical problems for unsteady PDEs,computational algorithms are constructed for the identification of the leading coefficient ofthe second-order parabolic equation, which depends on time only.

Approximation in space is performed using the standard finite differences [23,24] orfinite elements.[25,26] The main features of the non-linear inverse problem are taken intoaccount via a proper choice of the linearized approximation in time. In this paper, for amultidimensional parabolic equation, we consider the problem of determining the lowercoefficient that depends on time only. Linear problems at every time level are solved on thebasis of a special decomposition into two standard elliptic problems.

The paper is organized as follows. Statements of direct and inverse problems for second-order parabolic equation are given in Section 2. We consider identification problem ofthe lower coefficient of equation in rectangular domain with additional information aboutsolution of boundary value problem for inner point of computing domain. Approximationin space is considered in Section 3. Difference approximations of self-adjoint ellipticoperator with homogeneous Neumann boundary condition on uniform rectangular gridare used. Issues of approximation in time are discussed in Section 4. Focus is made onlinearized schemes, when explicit–implicit approximations are used with assigning thelower coefficient on new time layer and taking solution from the lower time level. Section5 is devoted to description of non-iterative computational algorithm of solution of inverseproblem. We use special decomposition of solution of the equation, in which the transition toa new time layer is based on solution of two standard grid elliptic problems. We introduce thealgorithm applicability conditions, which can be provided in the planning of experimentaland computational works. Results of computational experiment for model boundary valueproblem are represented in Section 6. Estimation of accuracy of identification problemsolution is evaluated by benchmark solution, as which we use numerical solution of prob-lem on sufficiently fine grid. Taking into account the specifics of the problem, we mostextensively investigate the influence of time discretization step. Calculation results showthe convergence of approximated solution with first order by time. In Section 7, we buildlinearized schemes of second degree approximation in time for estimated problem solution,which computational implementation is carried out similarly to first-order schemes. Weintroduce computational results, which demonstrate convergence of approximated problemwith second order by time. Finally, in Section 8, we demonstrate the efficiency of developedcomputational algorithm in general terms, when conditions of its applicability are violated.In this case, we recover desired coefficient of equation for almost the entire time interval,expect for a minor surrounding of violation of computational algorithm applicability.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering 3

Note that inverse problems are commonly belong to class of ill-posed ones. Therefore,when developing the computational algorithms of their approximated solutions, we focuson problems with inaccurate input data. The considered problem of identification of thelower coefficient of parabolic equation is referred to well-posed non-classical problems forpartial differential equations. So we do not represent the results of solution of our inverseproblem with noisy input data in this work.

2. Problem formulation

For simplicity, we restrict ourselves to a two-dimensional problem in a rectangle. Letx = (x1, x2) and

� = {x | x = (x1, x2) , 0 < xα < lα, α = 1, 2}.The direct problem is formulated as follows. We search u(x, t), 0 ≤ t ≤ T, T > 0 suchthat it is the solution of the parabolic equation of second order:

∂u

∂t− div(k(x)gradu) + p(t)u = f (x, t), x ∈ �, 0 < t ≤ T . (1)

The boundary and initial conditions are also specified:

k(x)∂u

∂n= 0, x ∈ ∂�, 0 < t ≤ T, (2)

u(x, 0) = u0(x), x ∈ �, (3)

where n is the normal to �. The formulation (1)–(3) presents the direct problem, where theright-hand side, coefficients of the equation as well as the boundary and initial conditionsare given.

Let us consider the inverse problem, where in Equation (1), the coefficient p(t) isunknown. An additional condition is often formulated as:∫

�

u(x, t)ω(x)dx = ϕ(t), 0 < t ≤ T, (4)

where ω(x) is a weight function. In particular, choosing ω(x) = δ(x − x∗) (x∗ ∈ �),where δ(x) is the Dirac δ-function, from (4), we get

u(x∗, t) = ϕ(t), 0 < t ≤ T . (5)

The model equation of unsteady diffusion-reaction (1) is the base for the consideration ofmany physical processes. In particular, in the modeling of diffusion processes with chemicalreactions, we search the reaction coefficient p(t), which depends only on the time.

We assume that the above inverse problem of finding a pair of u(x, t), p(t) fromEquations (1)–(3) and additional conditions (4) or (5) is well posed. The correspondingconditions for existence and uniqueness of the solution are available in the above-mentionedworks. In the present work, we consider only numerical techniques for solving these inverseproblems omitting theoretical issues of the convergence of an approximate solution to theexact one.

From the non-linear inverse problem, we can proceed to the linear one. Suppose

v(x, t) = χ(t)u(x, t), χ(t) = exp

(∫ t

0p(θ)dθ

).

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

4 P.N. Vabishchevich and V.I. Vasil’ev

Then from (1)–(3), we get

∂v∂t − div(k(x)gradv) = χ(t) f (x, t), x ∈ �, 0 < t ≤ T,

k(x) ∂v∂n + g(x)v = 0, x ∈ ∂�, 0 < t ≤ T,

v(x, 0) = u0(x), x ∈ �.

The additional conditions (4) and (5) to identify uniquely v(x, t), χ(t) take the form∫�

v(x, t)ω(x)dx = χ(t)ϕ(t), 0 < t ≤ T,

u(x∗, t) = χ(t)ϕ(t), 0 < t ≤ T .

The above transition from the non-linear inverse problem to the linear one is in common usefor numerically solving problems of identification. In our work, we focus on the originalformulation of the inverse problem (1)–(4) (or (1)–(3), (5)) without going to the linearproblem.

3. Semi-discrete problem

To solve numerically the time-dependent convection–diffusion problem, we introduce theuniform grid in the domain �:

ω = {x | x = (x1, x2) , xα =(

iα + 1

2

)hα,

iα = 0, 1, . . . , Nα, (Nα + 1)hα = lα, α = 1, 2}.For grid functions, we define the Hilbert space H = L2 (ω), where the scalar product andnorm are given as follows:

(y, w) ≡∑x∈ω

y (x)w (x) h1h2, ‖y‖ ≡ (y, y)1/2 .

The difference operator for the diffusion transport D has the following additive repre-sentation:

D =2∑

α=1

Dα, α = 1, 2, x ∈ ω, (6)

where Dα, α = 1, 2 are associated with the corresponding differential operator in onespatial direction.

For all nodes except adjoining the boundary, and for sufficiently smooth diffusioncoefficients k(x), the grid operator D1 can be written as:

D1 y = − 1

h21

k(x1 + 0.5h1, h2)(y(x1 + h1, h2) − y(x))

+ 1

h21

k(x1 − 0.5h1, h2)(y(x) − y(x1 − h1, h2)),

x ∈ ω, x1 �= 0.5h1, x1 �= l1 − 0.5h1.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering 5

At the nodes adjoining the boundary, approximation should take into account the boundarycondition (2):

D1 y = − 1

h21

k(x1 + 0.5h1, h2)(y(x1 + h1, h2) − y(x)),

x ∈ ω, x1 = 0.5h1,

D1 y = 1

h21

k(x1 − 0.5h1, h2)(y(x) − y(x1 − h1, h2)),

x ∈ ω, x1 = l1 − 0.5h1.

The grid operator D2 is constructed in a similarly way. Direct calculations yield (see, e.g.[24,27]):

Dα = D∗α ≥ 0, α = 1, 2.

This grid operator of diffusion approximates the corresponding differential operatorwith an accuracy of O

(|h|2). As in the differential case, the difference operator of diffusivetransport (6) is self-adjoint and positive definite in H :

D = D∗ ≥ 0. (7)

In view of (7), we can obtain the corresponding a priori estimates for the solution of theboundary value problem (1)–(3) in H that ensure the stability of the solution with respectto the initial data and the right-hand side.

After discretization in space, from the problem (1)–(3), we arrive at the Cauchy problemfor the semi-discrete equation:

dy

dt+ Dy + p(t)y = f (t), 0 < t ≤ T, (8)

y(0) = u0. (9)

We consider the case, where the lower coefficient in the parabolic Equation (1) is negative:

p(t) < 0, pm = max0≤t≤T

|p(t)|. (10)

Then, we have

‖y(t)‖ ≤ exp(pmt)‖u0‖ +∫ t

0exp(pm(t − θ))‖ f (θ)‖dθ, (11)

i.e. the norm of the solution of the homogeneous equation ( f (x, t) = 0 in (1)) maygrow exponentially with time. The a priori estimate (11) holds in the Banach space ofgrid functions L∞(ω), where

‖ · ‖ = ‖ · ‖∞, ‖y‖∞ ≡ maxx∈ω

|y|.

This fact can be established on the basis of the maximum principle for grid functions andthe relevant comparison theorems [24] taking into account the diagonal dominance of thematrix (operator) D.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

6 P.N. Vabishchevich and V.I. Vasil’ev

4. Time-stepping techniques

Let us define a uniform grid in time tn = nτ, n = 0, 1, . . . , N , τ M = T and denoteyn = y(tn), tn = nτ . We start with discretization in time for the numerically solving directproblem (8), (9). To solve numerically boundary value problems for transient diffusion–reaction Equation (1), approximation in time is carried out depending on the sign of thecoefficient p(t).[28,29] In particular, for p(t) ≥ 0, unconditionally stable schemes areconstructed using the implicit approximation for the lower coefficient, where

yn+1 − yn

τ+ Dyn+1 + pn+1 yn+1 = f n+1.

If p(t) ≤ 0, then we need to focus on the explicit approximation:

yn+1 − yn

τ+ Dyn+1 + pn+1 yn = f n+1, n = 0, 1, . . . , M − 1. (12)

The initial condition (9) yields

y0 = u0. (13)

For the coefficient p(t) with alternating signs, unconditionally stable schemes are based onthe explicit–implicit approximation for the lower term. If we apply the explicit approxima-tion (12) for the problem (8), (9), then, for p(t) ≥ 0, the stability of the scheme (12), (13)holds with rather weak restrictions on the time step.[29]

Under the assumptions (10), the difference solution of the problem (12), (13) satisfiesthe following level-wise estimate in L∞(ω):

‖yn+1‖ ≤ (1 + τpm)‖yn‖ + τ‖ f n+1‖, n = 0, 1, . . . , M − 1. (14)

The estimate (14) is a discrete analogue of the estimate (11) for the solution of the problem(8)–(10). Here, we prove the stability estimate (14).

From Equation (12), we have

(E + τ D)yn+1 = (1 + τpn+1)yn + τ f n+1,

where E is the identity operator. For the norm of the left-hand side of the equality, in thestandard way, we obtain

‖yn+1‖ ≤ ‖(E + τ D)yn+1‖.To prove this inequality, we can apply the maximum principle for grid functions.[24,30]The second possibility to check the a priori estimate (14) is associated with the use of theconcept of the logarithmic norm.[31,32] For the right-hand side, we have

‖(1 + τpn+1)yn + τ f n+1‖ ≤ (1 + τpm)‖yn‖ + τ‖ f n+1‖.Thus, we arrive at the estimate (14).

5. Algorithm for solving the inverse problem

For the fully discretized (both in space and in time) direct problem (1)–(3), we can solvethe inverse problem of the identification of the lower coefficient p(t). We restrict ourselves

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering 7

to the case, where an additional information on the solution is defined (see (5)) at someinterior node x∗ ∈ ω of the grid:

yn+1(x∗) = ϕn+1, n = 0, 1, . . . , M − 1. (15)

This inverse problem is non-linear, but for discretization in time, we can use the explicit–implicit scheme (12). This approximation in time leads to linear problems at each timelevel.

For the approximate solution of the problem (12), (13), (15) at the new time level yn+1,we introduce the following decomposition (see, e.g. [8,33]):

yn+1(x) = vn+1(x) + pn+1wn+1(x), x ∈ ω. (16)

To find vn+1(x), we employ the equation

vn+1 − yn

τ+ Dvn+1 = f n+1, n = 0, 1, . . . , M − 1. (17)

The function wn+1(x) is determined from

1

τwn+1 + Dwn+1 = −yn, n = 0, 1, . . . , M − 1. (18)

Using the decomposition (16)–(18), Equation (12) holds automatically for any pn+1.Recall that the grid operator D is constructed taking into account the boundary conditions

(2). This corresponds to the fact that the same homogeneous boundary conditions of secondkind are also formulated for auxiliary grid functions vn+1(x) and wn+1(x).

To evaluate pn+1, we apply the condition (15). The substitution of (16) into (15) yields

pn+1 = 1

wn+1(x∗)(ϕn+1 − vn+1(x∗)). (19)

The fundamental point of applicability of this algorithm is associated with the conditionwn+1(x∗) �= 0. The auxiliary function wn+1(x) is determined from the grid elliptic Equation(18). The property of having fixed sign for wn+1(x) is followed, in particular, from the sameproperty of the solution at the previous time level un(x). Such constraints on the solutioncan be provided by the corresponding restrictions on the input data of the inverse problem.

It is relatively easy to establish that the solution of the inverse problem for u0 > 0,f n+1 > 0, n = 0, 1, . . . , M − 1 in the class pn+1 < 0, n = 0, 1, . . . , N − 1 may berepresented in the form (16)–(19). Correctness of the computational algorithm follows fromfulfilling the condition wn+1(x∗) �= 0.

On the basis of the maximum principle, for the solution of the grid problem (18) withyn > 0, we have wn+1 < 0. Thus, it is sufficient to show that yn+1 < 0. The proof isperformed by induction. We have y0 > 0 and let yn > 0. From (18), we obtain wn+1 < 0.Similarly, from (17), we have vn+1 > 0.

For yn+1 − vn+1, from (12) and (17), we get

yn+1 − vn+1

τ+ D(yn+1 − vn+1) = −pn+1 yn .

Because of this, we have yn+1 − vn+1 > 0 and therefore yn+1 > 0. It should be noted thatϕn+1 − vn+1(x∗) > 0 and for the definition of the coefficient according to (18), we havepn+1 < 0. Thus, we remain in the class of negative coefficients.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

8 P.N. Vabishchevich and V.I. Vasil’ev

If the point of observation x∗ is not a grid point, then it is necessary to use someinterpolation. It is convenient to consider the simplest piecewise linear approximations,which ensure the maximum principle for the interpolated quantities. In our case, for arectangular computational grid, the observation point x∗ belongs to the right triangle whosevertices are the nodes of the mesh x∗

k , k = 1, 2, 3. Then, instead of (15), we use thecondition

3∑k=1

μk yn+1(x∗) = ϕn+1, n = 0, 1, . . . , M − 1.

Here, μk, k = 1, 2, 3 ≥ 0 are the weights of the piecewise linear approximation. Theseeking lowest coefficient of the parabolic equation is determined using the formula

pn+1 =(

3∑k=1

μkwn+1(x∗

k)

)−1 (ϕn+1 −

3∑k=1

μkvn+1(x∗

k)

).

The correctness of the computational algorithm follows from the mentionednon-negativeness of the interpolation coefficients μk, k = 1, 2, 3.

6. Numerical examples

To demonstrate possibilities of the above linearized schemes for solving the coefficientidentification problem for the parabolic equation, we consider a model problem. In theexamples below, we consider the problem in the unit square (l1 = l2 = 1). Suppose

k(x) = 1, f (x, t) = βx1x2, u0(x) = 1, x ∈ �.

For the considered initial state for a positive coefficient β, the solution of problem willbe remain positive. Therefore, performance of the following algorithm is guaranteed:denominator in right-hand side (19) does not vanish. For negative β, there can be someproblems.

The problem is considered on the grid N = N1 = N2 = 51. Note that for consideredinverse problems the influence of computational grid by space practically is not observedwhen we consider problems on sequence of grids from N = 11, 21, 41, 81 and more. Theobservation point is located at the square center (x∗ = (0.5, 0.5)). The coefficient p(t) istaken in the form

p(t) = − 1000t

1 + exp(γ (t − 0.5T )). (20)

For large values of the parameter γ (see Figure 1), the function p(t) is approached todiscontinuous at point of function t = 0.5T .

The solution of the direct problem (1)–(3) at the observation point is depicted inFigure 2 for the base case: β = 1, γ = 100. The solution at the final time moment ispresented in Figure 3.

To study the influence of parameters of the computational algorithm, we need to use thesame input data. In our case, as the input data we use the numerical solution of the directproblem obtained using a very fine grid in time. The solution of the direct problem obtainedwith M = 1000 is employed as the input data (the function ϕ(t) in the condition (5)). It iseasy to see that the approximate solution of the inverse problem converges with decreasingthe time step. The results of solving the inverse problem with various grids in time are shown

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering 9

Figure 1. Coefficient p(t) for various values of parameter γ .

Figure 2. The solution of the direct problem at the point of observation.

in Figure 4. Error in determination of p(t) on separate time moments is evaluated by valueε(t) = p̃(t) − p̄(t), where p̃(t) – approximated solution, and p̄(t) – benchmark solution(solution of problem for M = 1000). The error of approximate solution is consistent witherror of approximation in time for used scheme – first order of approximation in τ .

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

10 P.N. Vabishchevich and V.I. Vasil’ev

Figure 3. The solution of the direct problem for t = T .

Figure 4. Solution error of the inverse problem.

It make sense to observe the accuracy of identification of less smooth coefficient. Withinthe framework of quasi-real experiment, we consider the inverse problem in terms when γ =1000 in (20). Thus, situation of approximate solution of inverse problem with discontinuous

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering 11

Figure 5. Solution error for exact solution with γ = 1000.

coefficient is modelled. Accuracy of identification of coefficient p(t) is shown in Figure 5.Obviously, the accuracy is dropped in the area of strong variation of coefficient. But the factof convergence of approximate solution when τ → 0 is observed with about first order.

The computational algorithm of solution of inverse problem that we use is based onexplicit–implicit schemes of lower members of equation (see (12)): coefficient itself istaken from upper time layer, and solution – from lower one. Such approximation leadsto absolutely stable scheme of direct problem for non-positive coefficient p(t). For non-negative p(t), such approximations in time lead us, generally, to conditionally convergentschemes: convergence is guaranteed for sufficiently weak restrictions for time step. Theefficiency of the proposed computational algorithm for p(t) ≥ 0 is demonstrated bycomputational data, which is given in Figure 6. On this example

p(t) = 1000t

1 + exp(γ (t − 0.5T )),

when γ = 100.

7. Scheme of the second-order accuracy

The non-linear inverse problem (1)–(4) is characterized by a quadratic non-linearity. Whenusing the scheme with linearization (12), (13), the non-linear term is approximated with thefirst order with respect to τ . It is possible to apply the linearized scheme of second order.Let us consider the approximation

a(tn+1/2)b(tn+1/2) = 1

2a(tn+1)b(tn) + 1

2a(tn)b(tn+1) + O(τ 2).

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

12 P.N. Vabishchevich and V.I. Vasil’ev

Figure 6. Solution error for problem of identification of non-negative coefficient.

Approximation of Equation (8) with the boundary conditions (9) using the Crank–Nicolsonscheme yields the linearized scheme

yn+1 − yn

τ+ D

yn+1 + yn

2+ 1

2pn+1 yn + 1

2pn yn+1 = f n+1/2, n = 0, 1, . . . , M − 1,

(21)where for example, f n+1/2 = f (tn + 0.5τ). The scheme (13), (21) belongs to the classof linearized schemes. In comparison with the scheme (12), (13), it has a higher order ofaccuracy in time.

To implement (13), (21), we again use the decomposition (16). In this case, for vn+1(x),we have

vn+1 − yn

τ+ 1

2Dvn+1 + 1

2pnvn+1 = f n+1/2 − 1

2Dyn, n = 0, 1, . . . , M − 1. (22)

The auxiliary function wn+1(x) is defined as the solution of the equation

wn+1

τ+ 1

2Dwn+1 + 1

2pnwn+1 = −1

2yn, n = 0, 1, . . . , M − 1. (23)

Further, as in the case of the first-order scheme, we employ (15).The Crank–Nicholson scheme for numerical solving the direct problems for parabolic

equations is not very often used in computational practice. It is inferior to the fully implicitscheme in sense of conservation of monotonicity (fulfilment of the maximum principle forthe grid problem). We note in this regard, that robustness justification of our computationalalgorithm is based on the application of maximum principle for solution of appropriate gridproblems on new time layer. Results of solution of inverse problem for base case (γ = 100in (20)) are shown in Figure 7. Accuracy of the solution of problem is significantly high,

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering 13

Figure 7. Solution error for second-order schemes.

Figure 8. Solution error for second-order schemes when γ = 1000.

in comparison with usage of first-order scheme (12), (13) (see Figure 4). However, it isnecessary to note the non-monotonic behaviour of error, which is more strongly pronouncedfor problems with more high variation of coefficient p(t) (see Figure 8).

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

14 P.N. Vabishchevich and V.I. Vasil’ev

Figure 9. Auxiliary function w(x, t) at the point of observation.

Figure 10. Direct problem solution at the point of observation.

8. Solution with alternating signs

The computational algorithm is based on computation of coefficient in new time layeraccording to (19). This algorithm applicability is ensured under conditions, that denominatordoes not vanish. As we mentioned above, this can be achieved with constant signs solution

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering 15

Figure 11. The solution of the direct problem at t = T , β = −100.

Figure 12. Solution error of identification problem for β = −100.

– when right-hand side of equation got constant sign (18). Such behaviour of solution ofproblem can be reached by designing of experiment. Here, we show, how the algorithmworks under conditions, when sign constancy of solution is not guaranteed.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

16 P.N. Vabishchevich and V.I. Vasil’ev

Figure 13. Auxiliary function w(x, t) at the point of observation for β = −100.

Figure 9 shows the calculated value of auxiliary function w(x∗, t) at the point ofobservation x = x∗ at different moments of time t = tn for base case, when β = 1and γ = 100. Auxiliary function w(x, t) found as solution of grid problem (18). Thereis pronounced approximately linear dependence on time step. It is important that the signof function w(x∗, t) does not change, i.e. denominator in the right-hand side (19) remainnegative.

We model the situation with problem sign variation by defining the negative coefficientβ in right-hand side of equation. The solution of direct problem at the point of observationfor various β is given in Figure 10. For example, initial positive solution for β < 100changes its sign in the part of computational domain. Solution on the final time moment isshown in Figure 11.

For β = −100, the accuracy of inverse problem solution is presented in Figure 12.Comparison with Figure 13 shows that error of approximated solution sharply grows invicinity of sign change of auxiliary function w(x∗, t). Beyond this vicinity, the identificationaccuracy of coefficient p(t) is high.

9. Conclusion

(1) To find the approximate solution to the problem of the lower coefficient identifi-cation for the multidimensional parabolic equation that depends on time only, weemploy the explicit–implicit approximation in time for the reaction term. For thispart of the equation, the searched coefficient is taken at the upper time level, whereasthe solution is referred to the lower time level. A similar approximation is used toconstruct schemes with the second-order accuracy in time.

(2) A decomposition of the approximate solution at the new time level is proposed. It isbased on solving two standard elliptic problems. The conditions for the correctness

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

Inverse Problems in Science and Engineering 17

of the algorithm for determining the lower coefficient are formulated. They employthe maximum principle for the grid functions.

(3) The results of numerical experiments demonstrate the stability of the computationalalgorithm and convergence of the approximate solution of the inverse problemduring the grid refinement both in space and time.

(4) The proposed algorithm can be applied for solving more general problems. Inparticular, using this algorithm, we can solve problems in irregular computationaldomains using finite element or finite volume approximations in space. Next, it ispossible to employ it for solving more general parabolic equations, e.g. the equationsof convection–diffusion reaction, or for identification of the varying in time right-hand side and/or coefficients of the parabolic equation.

AcknowledgementsThis research was supported by the Russian Foundation for Basic Research (project 14-01-00785).We express our deep gratitude to Prof. George Dulikravich for constructive suggestion about worktext.

References

[1] Alifanov OM. Inverse heat transfer problems. Berlin: Springer; 2011.[2] Aster RC, Borchers B, Thurber CH. Parameter estimation and inverse problems. Amsterdam:

Elsevier Science; 2011.[3] Lavrent’ev MM, Romanov VG, Shishatskii SP. Ill-posed problems of mathematical physics and

analysis. Providence (RI): American Mathematical Society; 1986.[4] Isakov V. Inverse problems for partial differential equations. New York (NY): Springer; 1998.[5] Prilepko AI, Orlovsky DG, Vasin IA. Methods for solving inverse problems in mathematical

physics. New York (NY): Marcel Dekker; 2000.[6] Maksimov VI. Dynamical inverse problems of distributed systems. Boston (MA): VSP; 2002.[7] Vogel CR. Computational methods for inverse problems. Philadelphia (PA): Society for Industrial

and Applied Mathematics; 2002.[8] Samarskii AA, Vabishchevich PN. Numerical methods for solving inverse problems of

mathematical physics. Berlin: De Gruyter; 2007.[9] Prilepko AI, Orlovskii DG. Determination of evolution parameter of an equation, and inverse

problems in mathematical physics, part I and II. Differ. Equat. 1985;21:119–129, 694–701.Russian.

[10] Prilepko AI, Solo’ev VV. Solvability of the inverse boundary value problem of finding acoefficient of a lower order term in a parabolic equation. Differ. Equat. 1987;23:136–143.Russian.

[11] MacBain JA, Bednar JB. Existence and uniqueness properties for the one-dimensionalmagnetotellurics inversion problem. J. Math. Phys. 1986;27:645–649.

[12] Cannon JR, Lin Y. An inverse problem of finding a parameter in a semi-linear heat equation. J.Math. Anal. Appl. 1990;145:470–484.

[13] Wang S, Lin Y. A finite-difference solution to an inverse problem for determining a controlfunction in a parabolic partial differential equation. Inverse Probl. 1989;5:631–640.

[14] Cannon JR, Lin Y, Xu S. Numerical procedures for the determination of an unknown coefficientin semi-linear parabolic differential equations. Inverse Probl. 1994;10:227–243.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015

18 P.N. Vabishchevich and V.I. Vasil’ev

[15] Dehghan M. Identification of a time-dependent coefficient in a partial differential equationsubject to an extra measurement. Numer. Methods Partial Differ. Equat. 2005;21:611–622.

[16] Ye C, Sun Z. On the stability and convergence of a difference scheme for an one-dimensionalparabolic inverse problem. Appl. Math. Comput. 2007;188:214–225.

[17] Wang W, Han B, Yamamoto M. Inverse heat problem of determining time-dependent sourceparameter in reproducing kernel space. Nonlinear Anal. Real World Appl. 2013;14:875–887.

[18] Dehghan M. Numerical methods for two-dimensional parabolic inverse problem with energyoverspecification. Int. J. Comput. Math. 2001;77:441–455.

[19] Daoud DS, Subasi D. A splitting up algorithm for the determination of the control parameter inmulti dimensional parabolic problem. Appl. Math. Comput. 2005;166:584–595.

[20] Cannon JR, Yin HM. Numerical solutions of some parabolic inverse problems. Numer. MethodsPartial Differ. Equat. 1990;6:177–191.

[21] Azari H, Allegretto W, Lin Y, Zhang S. Numerical procedures for recovering a time dependentcoefficient in a parabolic differential equation. Dyn. Continuous, Discrete Impulsive Syst. Ser.B: Appl. Algorithms. 2004;11:181–199.

[22] Liao W, Dehghan M, Mohebbi A. Direct numerical method for an inverse problem of a parabolicpartial differential equation. J. Comput. Appl. Math. 2009;232:351–360.

[23] LeVeque RJ. Finite difference methods for ordinary and partial differential equations. Steady-state and time-dependent problems. Philadelphia (PA): SIAM; 2007.

[24] Samarskii AA. The theory of difference schemes. New York (NY): Marcel Dekker; 2001.[25] Quarteroni A, Valli A. Numerical approximation of partial differential equations. Berlin:

Springer; 2008.[26] Brenner SC, Scott LR. The mathematical theory of finite element methods. New York (NY):

Springer; 2008.[27] Samarskii AA, Nikolaev ES. Numerical methods for grid equations. Vols. i, ii. Basel: Birkhauser

Verlag; 1989.[28] Ascher UM, Ruuth SJ, Wetton BTR. Implicit–explicit methods for time-dependent partial

differential equations. SIAM J. Numer. Anal. 1995;32:797–823.[29] Vabishchevich PN, Vasil’eva MV. Explicit–implicit schemes for convection–diffusion–reaction

problems. Siberian J. Num. Math. 2012;15:359–369.[30] Samarskii AA, Gulin AV. Stability of difference schemes. Moscow: Nauka; 1973. Russian.[31] Hundsdorfer WH, Verwer JG. Numerical solution of time-dependent advection–diffusion–

reaction equations. Berlin: Springer Verlag; 2003.[32] Afanas’eva NM, Churbanov AG, Vabishchevich PN. Unconditionally monotone schemes for

unsteady convection–diffusion problems. Comput. Methods Appl. Math. 2013;13:185–205.[33] Borukhov VT, Vabishchevich PN. Numerical solution of the inverse problem of reconstructing

a distributed right-hand side of a parabolic equation. Comput. Phys. Commun. 2000;126:32–36.

Dow

nloa

ded

by [

UQ

Lib

rary

] at

04:

08 2

3 Ju

ne 2

015