DSpace at VNU: An improved regularization method for initial inverse problem in 2-D heat equation

DSpace at VNU: An improved regularization method for initial inverse problem in 2-D heat equation tài liệu, giáo án, bài...

To appear in: Appl Math Modelling

Received Date: 1 October 2012

Revised Date: 18 November 2013

Accepted Date: 16 May 2014

Please cite this article as: N.H Tuan, T.T Binh, N.D Minh, T.T Nghia, An improved regularization method for

initial inverse problem in 2-D heat equation, Appl Math Modelling (2014), doi: http://dx.doi.org/10.1016/j.apm.2014.05.014

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An improved regularization method for initial inverse

problem in 2-D heat equation

Nguyen Huy Tuan 1,2∗, Tran Thanh Binh 3 Nguyen Dang Minh1, Truong Trong Nghia 4,

School of Computing, University of Utah, USA

Abstract The main purpose of this article is to present a new method to regularize the initialinverse heat problem with inhomogeneous source This problem is well known to be severely ill-posed.There are many regularization methods with error estimator of logarithmic order An improvedregularization method is proposed The error estimates of H¨older type are obtained Some numericaltests illustrate that the proposed method is feasible and effective

Keywords and phrases: Backward heat problem, Ill-posed problem, Quasi-boundary valuemethods, Quasi-Reversibility methods

Mathematics subject Classification 2000: 35K05, 35K99, 47J06, 47H10

(1)

where g(x, y) and f (x, y, z) are given This problem is well known to be severely ill-posed and ization methods for it are required (See [10]) As is known, the above problem is severely ill-posed, i.e.its solutions do not always exist and in the case of existence they do not depend continuously on thegiven data In fact, from small noise contaminated from physical measurements, the correspondingsolutions have large errors That makes difficult to numerical calculations Hence, a regularization

regular-is a need Authors such as K A Ames [1], R Lattes and J L Lions [13], R E Showalter [21], K.Miller [17], L.E Payne [19] have approximated the Problem (1) by quasi-reversibility method andquasi-boundary value method In [32], T Schr¨oter and U Tautenhahn established an optimal errorestimate for the homogeneous case of (1) A mollification method has been studied by D N Hao

in [11] S M Kirkup and M Wadsworth used an operator-splitting method in [15] A method ofhyperbolic equation for backward heat has been considered by K Masood et al [16] Very recently,the homogeneous problem of (1) was also investigated by Y.C Hon et al [12], T.Wei et al in [20], J.Rashidinia et al in [23], P Daripa et al in [24], Jin-Ru Wang in [35]

Although there are many papers on the homogeneous case of the initial inverse heat problem, weonly find a few results on the non-homogeneous case, especially the two-dimensional non-homogeneouscase is very rare For the case, we refer the reader to some recent works of X.L Feng et al [7], M Li

et al[14], P.T.Nam et al [18], Trong et al [25, 27, 30] Physically, g is measured data with an error ofparameter  Let u and v be the exact solution and the approximated solution of the backward heatproblem respectively In [27, 29] , the errors are of order 1

 )

1− t T

For the literature on non-homogeneous backward heat,

we refer the reader to the results in Fu et al [7, 8, 9] However, the error estimates in the mentionedpapers are still of logarithmic order

Very recently, P.T Nam et al [18] regularized the Problem (1) by truncation method and obtainedthe error estimate which is of order q, 0 < q < 1 Using this method, N.H Tuan et al [31] considered

a general version of the Problem (1) with similar results The truncation method introduced in [18, 31]

is simple and effective to solve the backward heat problem with good estimates However, in practice,the computation of the approximation solution (by the truncation method) is impossible and difficultwhen we consider the problem in a general two-dimensional domain (See [4]) If the spectral problem

of operator −∆ in this domain is unknown then the truncation method is seems to be useless This

is may be a disadvantage point of papers [18, 31] Motivated by this reason, in the present paper,

we provide another regularization method to established the H¨older estimates Our method is similar

to quasi-boundary value method (or non-local boundary value problems method, see [3, 5, 30]) but

it seems to be in a new direction In few words, we explain why this method is new By a naturalway, to approximate the solution of the Problem (1), in many previous methods, we usually propose

a regularized solution u for t ∈ [0, T ], then estimate the error ku(., t) − u(., t)k (norm in L2) for all

t ∈ [0, T ] The method in this paper is first to compute the regularized solution for t ∈ [0, T +(h−1)T ]where h ≥ 1, then use the resulting solution at t + (h − 1)T to approximate the exact solution at t.Under some suitable conditions on the exact solution, we will introduce the error which is of order

qln(1)−p for p > 0, 0 < q < 1 This type of error is not introduced in many related results.The paper is structured as follows In Section 2, we present the solution of the 2-D initial inverseheat problem In Section 3, motivated by the idea coming from [30], we establish stability results forour problem and propose a new strategy with H¨older estimates The proofs of the main theoreticalresults will be given in Section 4 In Section 5, the numerical results of our regularized method arepresented, which proved the effectiveness of our method

Throughout this paper, we denote < , >, k.k by the inner product and the norm in L2 spectively Let us first make clear what a weak solution of the Problem (1) is We call a function

re-u ∈ C [0, T ]; L2(I)
∩ C1((0, T ); L2(I)) to be a weak solution for the Problem (1) if

d

dthu(., , t), W i − h∆u(., , t), W i = hf(., , t), W i , (2)

for all functions W (x, y) ∈ H2(I) ∩ H1

0(I) In fact, it is enough to choose W in the orthogonal basis{π2 sin(px) sin(qy)}p,q≥1 and the formula (2) is equivalent to

Else if (5) holds then the Problem (1) has a unique solution

Proof Suppose the Problem (1) has a solution u ∈ C([0, T ]; H01(I)) ∩ C1((0, T ); L2(I)), then u isdefined by

Since (8), we see that v ∈ L2(I).

Consider the problem







ut− uxx− uyy = f (x, y, t),u(0, y, t) = u(π, y, t) = u(x, 0, t) = u(x, π, t) = 0, t ∈ (0, T )u(x, y, 0) = v(x, y), (x, y) ∈ (0, π) × (0, π)

Let us recall (g, f ) ∈ L2(I) × L2(0, T : L2(I)) be the exact data Assume that the noisy data(g, f) ∈ L2(I) × L2(0, T : L2(I)) satisfies

≤ 

In this paper, we establish a method to regularize the Problem (1) Infact, letting h ≥ 1 be a fixednumber, we denote Th = hT, Th−1 = (h − 1)T Let β be a positive constant (is called parameterregularization) which depends on  such that lim→0β = 0 We consider the following well-posedproblem

Let g ∈ L2(I) be the function such that kgxx+ gyyk < ∞.

(i)If the regularized solution u(x, y, Th−1) converges in L2(I), then the Problem (1) has a uniquesolution u Furthermore, u(x, y, t + Th−1) converges to u(x, y, t) uniformly in t as  tends to zero (ii)If there is a positive constant A1 such that

T

2 As we know, the convergence rate of T +γγ

ln



T

 hT



T

 hT

T +γ

−1h

is faster thanthe H¨older type estimate k, for any 0 < k < Tγ+γ when  → 0 This implies that the error (15) iseffective and useful

3 The method in this paper is inherently restricted to the square domain, and does not apply to moregeneral domain due to its reliance on the Fourier method We hope that in the future, we can derivesimilar estimates without resorting to the Fourier method The method can be applied to fairly generaldomains is very difficult and will be presented in next reports

where the function H(m, n) is defined by H(m, n) = n1−mn

Proof The proof of part (a) of this lemma can be found in [27]

The Problem (12) has a unique solution v ∈ C [0, T ]; H1

0(I)
∩ C1((0, T ); L2(I)) satisfying

, 0 ≤ t ≤ T (20)

Proof

The proof of this lemma is divided into two steps

Step 1.The existence, the uniqueness and the stability of a solution of (12) can be found in thepaper [30](Theorem 2.1, p 875)

Step 2 We shall prove (20)

In fact, from u, v are solutions of Problem (1) corresponding to the exact data g and noisy data

By replace t with t = t + Th−1, we obtain

2kg− gk2+ T kf(., t) − f(, t)k2L 2 (0,T :L 2 (I))

,

≤ (2 + T )2β2t−2ThT hT

1 + ln(hTβ )

!2T −2t hT

.The lemma is proved

Now we are in a position to prove the main result

4.1 Proof of Theorem 3.1

4.1.1 Proof of Part (i) Assume that lim

→0u(x, y, Th−1) = w(x, y) ∈ L2(I) exists We putu(x, y, t) =

We shall prove that u(x, y, t) is the unique solution of the Problem (1) First, it is clear to see that

u satisfies the system

(

ut− uxx− uyy = f (x, y, t), (x, y, t) ∈ I × (0, T )u(0, y, t) = u(π, y, t) = u(x, 0, t) = u(x, π, t) = 0, (x, y, t) ∈ I × (0, T ) (24)Next, we show that u(x, y, T ) = g(x, y) In fact, we have the formula of u(x, y, t + Th−1)

(25)

Combining (23) and (25), we obtain

< u(x, y, t + Th−1) − u(x, y, t), sin(px) sin(qy) >

< u(x, y, t + Th−1) − u(x, y, t), sin(px) sin(qy) > = e−t(p2+q2)upq(Th−1) − wpq(t)

−

Z t

0

βe(s−t)(p2+q2)(p2+ q2)β(p2+ q2) + e−T h (p 2 +q 2 )fpq(s)ds

By using the inequality (17), we get

|< u(x, y, t + Th−1) − u(x, y, t), sin(px) sin(qy) >| ≤ 

u



pq(Th−1) − wpq(t)+

Z t

0

Thln(Th

It gives lim

→0u(x, y, t + Th−1) = u(x, y, t) Thus lim

→0u(x, y, Th) = u(x, y, T ) On the other hand,

Using the triangle inequality, we get

ku(., , 0) − v(., , Th−1)k ≤ ku(., 0) − u(., , Th−1)k + ku(., , Th−1) − v(., , Th−1)k (28)First, from (20) in Theorem 2, we estimate

e(s−T )(p2+q2)fpq(s)ds = e−T(p2+q2)< u(x, y, 0), sin(px) sin(qy) > (32)

Combining (30) and (32), we obtain

< u(x, y, Th−1), sin(px) sin(qy) >

−Th−1(p 2

+q 2 )

It follows from (17) that

4.1.3 Proof of Part (iii)

It follows from (21) that

β(p2+ q2) + e−Th(p 2 +q 2 )fpq(s)ds

!sin(px) sin(qy)

!sin(px) sin(qy)

≤ H(γ, Th)β

γ Th

ln(Th

ln(Th

ln(Th

ln(Th

β )

2γ

Th −2

A22.Thus

ku(., , t + Th−1) − u(., , t)k ≤ H(γ, Th)β

γ Th

ln(Th

β )

 γ

Th −1

≤ H(γ, Th)T +γγ

ln

T

≤ H(γ, Th)T +γγ

ln

T

T +γhT

)

t−T hT

.(36)

If 0 < γ ≤ (h − 1)T , then

ln



T

 hT



T

 hT

T

#

If (h − 1)T < γ ≤ hT then (36) becomes

ku(., , t) − v(., , t + Th−1)k

≤ T +γγ

ln

T

#

In reality, we do not know exact solution of the problem, hence, the priori assumptions (ii) and(iii) become implicit and may be skipped in practice Thus, the regularized solution is expected that it

is a reasonable solution of the problem At the time, its certificate of use depends on its convergence

We consider two examples of Problem (1) in the region I = (0, π) × (0, π):







ut− uxx− uyy= f (x, y, t), (x, y, t) ∈ I × (0, 1)u(0, y, t) = u(π, y, t) = u(x, 0, t) = u(x, π, t) = 0, (x, y, t) ∈ I × (0, 1)u(x, y, 1) = g(x, y), (x, y) ∈ I

(37)

Let gρ(x, y) be the disturbed measure data such that ||g(x, y) − gρ(x, y)|| ≤ ρ

a) In the first example, we take:

g(x, y) = e−xysin(2x) sin(y),

gρ(x, y) = g(x, y) + ρ

π · rand()

f (x, y, t) = −e−txy xy + t2y2+ t2x2− 5
sin y sin 2x −

− 2tx cos y sin 2x − 4ty cos 2x sin y

fρ(x, y, t) = −e−txy xy + t2y2+ t2x2− 5
sin y sin 2x −

− 2tx cos y sin 2x − 4ty cos 2x sin y+ ρ

π · rand()

The problem has exact solution u(x, y, t) = e−txysin(2x) sin(y) Select h = 1.0 + 10−5, γ = h/100.For each regularization parameter β1 =  (from (ii)) and β2 = 1+γh (from (iii)) we have the relativeerror for regularized solution is

δk,(ρ) = ||uk,− u||

||u||

where uk, is the regularized solution by parameter βk Let’s observe the convergent behavior of theregularized solution regarding the changes of parameter  and magnitude ρ of data error Using ourproposed method with two types of regularization parameter βk, we set up the computation to findregularized solution at time t = 0, ρ = 10−r, r = 0, 1, 3, 5 and parameter  = 10−k, k = 1 6 Figure

1 visualizes the exact solution at t = 0 The computational result is showed in type of relative error

in Table 1 and in 3D graphs in Figure 2 We see that the second method yields more accurate resultthough not very significant Regularized solution converges as  decreases but diverges when  becomessmaller than ρ Therefore, we make an assumption that the best regularized solution for problem withperturbed data is obtained when  ≈ ρ

 δ 1, (ρ) δ 2, (ρ) δ 1, (ρ) δ 2, (ρ) δ 1, (ρ) δ 2, (ρ) δ 1, (ρ) δ 2, (ρ)

10 −1 9.867E-01 9.870E-01 9.807E-01 9.831E-01 9.866E-01 9.869E-01 9.867E-01 9.870E-01

10 −2 8.812E-01 8.859E-01 8.306E-01 8.540E-01 8.807E-01 8.852E-01 8.812E-01 8.859E-01

10 −3 4.258E-01 4.426E-01 7.145E-01 6.652E-01 4.233E-01 4.395E-01 4.258E-01 4.426E-01

10 −4 6.884E-02 7.493E-02 6.735E+00 5.675E+00 8.963E-02 8.511E-02 6.879E-02 7.491E-02

10 −5 1.061E-02 1.069E-02 6.020E+01 4.699E+01 4.754E-01 4.371E-01 1.041E-02 1.297E-02

10 −6 6.526E-02 5.770E-02 diverged diverged 3.947E+00 3.863E+00 6.572E-02 8.443E-02

0 0.5 1 1.5 2 2.5 3 3.5

X

0 0.5 1 1.5 2 2.5 3 3.5

Y

-1 -0.5 0 0.5 1

U

-1 -0.5 0 0.5

b) Consider the second example with given functions:



f (x, y, t) = x(π − x) 12y(π − y) − 2t

+ y(π − y) 12x(π − x) − 2t

+ ρ

π · rand()

It is easily to check that the exact solution u(x, y, t) = tx(π − x)y(π − y) Hence, we haveu(x, y, 0) = 0 We will find the regularized solution at t = 0 as the approximation to plane z = 0.Using presumption in the first example, we will set  = ρ and choose β = β1 =  With noise amplitude

ρ = 10−r, r = 1, 3, 5 the calculated relative errors for regularized solution are 0.349, 0.2172 and 0.1154respectively We also visualize 3D graphs of these regularized solutions in Figure 3 As expected, theyconverge smoothly to the exact solution z = 0

0 0.5 1 1.5 2 2.5 3 3.5

X

0 0.5 1 1.5 2 2.5 3 3.5

(a) u 1, ,  = 10 −3

0 0.5 1 1.5 2 2.5 3 3.5

X

0 0.5 1 1.5 2 2.5 3 3.5

Y

-1 -0.5 0 0.5 1

U

-1 -0.5 0 0.5

(b) u 2, ,  = 10 −3

0 0.5 1 1.5 2 2.5 3 3.5

X

0 0.5 1 1.5 2 2.5 3 3.5

(c) u 1, ,  = 10 −4

0 0.5 1 1.5 2 2.5 3 3.5

X

0 0.5 1 1.5 2 2.5 3 3.5

Y

-1 -0.5 0 0.5 1

U

-1 -0.5 0 0.5 1

(d) u 2, ,  = 10 −4

0 0.5 1 1.5 2 2.5 3 3.5

X

0 0.5 1 1.5 2 2.5 3 3.5

(e) u 1, ,  = 10 −5

0 0.5 1 1.5 2 2.5 3 3.5

X

0 0.5 1 1.5 2 2.5 3 3.5

Y

-1 -0.5 0 0.5 1 1.5

U

-1 -0.5 0 0.5 1 1.5

(f) u 2, ,  = 10 −5

In this paper, we considered an improved regularization method for initial inverse heat problem

in 2D case In the theoretical results, we obtained the error estimation of H¨older type mlnn(1k)for m, n, k > 0 based on the smooth assumptions of the exact solution These estimates improvethe results in many earlier works Finally, in the numerical experiments, two numerical examplesare presented to show that our proposed method is effective In future work, we will consider theregularized problem for the following problem



x+c(x, y, t)uy



y+ f (x, y, t), (x, y, t) ∈ I × (0, T )u|∂Ω = 0, t ∈ (0, T )

[3] G W Clark, S F Oppenheimer; Quasireversibility methods for non-well posed problems, Elect

J Diff Eqns., 1994 (1994) no 8, 1-9

[4] J Cheng, J.J Liu, A quasi Tikhonov regularization for a two-dimensional backward heat problem

by a fundamental solution, Inverse Problems 24 (2008), no 6, 065-012

[5] M Denche, K Bessila, A modified quasi-boundary value method for ill-posed problems,J.Math.Anal.Appl, Vol 301, 2005, pp.419-426

[6] L.C Evans, Partial Differential Equation, American Mathematical Society,Providence, RhodeIsland Volume 19 (1997)

[7] X.L Feng, Z Qian, C.-L Fu, Numerical approximation of solution of non-homogeneous backwardheat conduction problem in bounded region Math Comput Simulation 79 (2008), no 2, 177–188[8] C.-L Fu , Z Qian , R Shi (2007), A modified method for a backward heat conduction problem,Applied Mathematics and Computation, 185 564-573

In this paper, we considered an improved regularization method for initial inverse. .. backwardheat conduction problem in bounded region Math Comput Simulation 79 (2008), no 2, 177–188[8] C.-L Fu , Z Qian , R Shi (2007), A modified method for a backward heat conduction problem, Applied... regarding the changes of parameter  and magnitude ρ of data error Using ourproposed method with two types of regularization parameter βk, we set up the computation to findregularized