Bui Viet Huong Tap chi KHOA HOC & CONG NGHE 135(05): 139- 143 DETERMINATION OF A TIME-DEPENDENT TERM IN THE RIGHT HAND
SIDE OF LINEAR PARABOLIC EQUATIONS
Bill Viet HfrOng*
College of Science, Thainguyen University, Vietnam ABSTRACT
We propose a variational mel;hod for determining a time-dependent term in the right hand side of parabolic equations from integral observations. It is proved that the functional to be minimized is Fr^chet differentiable and a formula for its gradient is derived via an adjoint problem. The problem is discretized by the finite difference methods and then he conjugate gradient method in coupling with Tikhonov regular- ization is applied for numerically solving it. Numerical results are presented showing that our technique is efficient.
Keywords: Inverse problems, ill-posed problems, integral observations, finite differ- ence method, conjugate gradient method.
1 Introduction
Let Si be an open bounded domain in R". De- note by 9fi the boundaiy of fi. Q .= O x (0, T], and S •= dUx {0,T]. Consider the following problem
(
Ut-Au ^ f{t)<f{x,t)+ip{x,t),{x,t) £Q, u{x,Q) — uo(x). xen,du
J - = 9 ( i , t ) . ( i , ( ) e S . ou
(11) Here, u is the outer normal to dfl. And ip € i°°(g), ip(x,t) G L^iQ), uo(x) e L^{a), g{x,t) e L^(S). Furthermore, it is assumed that V > r > 0, with if being a given con- stant.
To introduce the concept of weak solution, we use the standard Sobolev spaces i/'(il), H(5(rJ), H^-'^iQ) and H^'HQ) [3, 5, 6). Fur- ther, for a Banach space B, we define
£2(0, T;S) = {it u{t) e B a.e ? e (0,T) and ||u||i,2[o,r;B) < oo}, with the norm
ll"lll^(o,r;S) = / ll"Wlll'^^' In the sequel, we shall use the space
VV{O.T) defmed as W{0,T) - {v : u € L^{0,T-H^(il)),ut <= L'^(0,T-,{H\n)y)}, equipped with the norm ||iiIlu/|o 7-) = Defmition 1. The function u e W(0,T) is said to be a weak solution of (1.1) if for all 7/ € L'^{Q,T;H^(n)) satisfying Ji{-T) - 0, the following identity holds
Jo
j I SJuS? 'ndxdt+ I ag(x. t)ri{x. t)dsdt
-Ii
(/(i)v(i. () + i/'(i, i)) >7(i, t)dxdt, (1.2) and u\t^o = UQ. It is proved in [5| that there exists a unique solution in W(0, T) of the prob- lem (1.1). Furthermore, there is a positive con- stant c^ > 0 independent of f,ip,ili.g and UQ such thatll"llw(0,r) < ^diW f^\\L2(Q) + \\^'\\L^(Q) + II31IL2{S) + II"OIIL2(SO)' In this paper, we will consider the inverse prob- lem of determining the time-dependent term
Biii Viet Huong Tgp chi KHOA HQC & C 6 N G N G H $ 135(05): 139-143 /(() from an integral observation of the solu-
tion u. Namely, we try to reconstruct f(t) from the observation
lu{x,t):= I (j(x)u{x.t)dx = h(t), t€{0,T),
JU
(1.3) where u}{x) is a weight function. We suppose that w e U"{il), nonnegative almost every- where in SI and j^u{x)dx > 0. The obser- vation data h is supposed to be in i^(0, T).
We will reformulate the inverse problem in our setting to a variational problem and prove that the functional to be minimized is Frechet differentiable and derive a formula for it. To solve the variational problem numerically we discretize it by the Fmite difference method, and then use the conjugate gradient method (CG) in coupling with Tikhonov regularization to stabilize the solution and then test the al- gorithm on computer.
2 M a i n r e s u l t
2.1 Variational formulation From now on, to emphasize the dependence of the solution u on the unknown function / , we write u(/) or u{x,t-,f) instead of u. Fol- lowing the least-squares approach [1, 2, 4), we estimate the unknown function /(() by mini- mizing the objective functional
•/o(/) = ^ ! I M / ) - / ' | [ i , ( o , r ) (2-1) over L2{0,r).
To stabilize this variational problem, we min- imize the Tikhonov functional
MJ) = 5ii'«(/)-''iii.(o,T)+|il/-riii=(o,,., (2.2) with 7 being a regularization parameter which has to be properly chosen and /* an estimation of / which is supposed in L^(0, T). We have a result:
Kdu
Theorem 1 The functional J-, is Pr€chet dif- ferentiable and its gradient VJ'^(/) at f has the form
V A ( / ) - / p{x, t)-p{x, t)dx + 7{/(0 - /*(()), (2.3) where p{x. t) satisfies the adjoint problem
f
-pt - Ap = w(x) {lu[x, t\ f) - h{t)) in Q, p{x,T) - 0 , x e S i ,0,ix,t)es.
(2.4) Proof. For an infinitesimally small variatioa 6f of / , we have
Mf + Sf) - Mf) = l\Mf + if) - /.]|i^(,j.|
-l\Mf)'lt\^l^^„_r)
= {Uu{f)Mf) -h) + il|H«C/)lli,(o,T).
where 6u{f) is the solution to the problem
f
J t i , - A f u =5/(t)v5(i,i),(i,f) E g , i5u(3:,0) = 0 , i e ! i , asu= 0, (i, t) s s.
. du
(2.5) We have the estimate, ||''^u(/)||L/or) o{\\Sf\\LHQ.T)) as ¥f\\mQ.T) -> 0.
We have
Mf + Sf)-Hf)
= {ifa,iu-h>-fo(||«/||i.,„,3.))
= I { ! i^Sudx^ {tu -h)dt + o(||i/||i,!(o.T))
= j [j "<S"('" - h)dx)dt + o(||d7||L.(o,T))
= j j "<!"('« - t^didt + o(||«/|li2,„,„).
Using Green's formula (see, i.e. [5, Theorem 3.18]) for (2.5) and (2.4), we have
/ / u&u{lu - h)dj:dt = 1 1 Sfifipdxdt.
Jo Jii Jo Jn
Biii Viet Huong Tgp chi KHOA HOC & CONG NGHE 135(05): 139-143 Hence,
Joif + Sf)-Mf)
= {j^f{^;t.)p{^,t)dx,6f) + oi\\Sf\\L^^o_T^).
Consequently, Jo is FVechet differentiable and its gradient has the form
^Mf)= I <p{x,t)p{x,t)dx.
Jn
From this equality, we immediately arrive at (2.3). The proof is complete.
The direct and inverse problem are solved us- ing the fmite difference method. Now we intro- duce the conjugate gradient method for solving the variational problem.
5 i e p S ; F o r n ^ 0,1,2....
2.1. Calculate Ari„ = ^(7"(dn) Ji^ujix)U"{x,t:d„)dx, here U''{x,t;dn) is the solution of the direct problem.
2.2. Calculate o W'^kWL^jO.T) ll^*^llL2Co,r) + -^ll'^lli2(o,r) 2.3. Update /„+i = fn + Onrf-i-
2.4. Calculate the residual r„+i = fn + "'n/^dn.
2.5. Calculate the gradient r„+i given in (2.3) by solving the adjoint
f - ? ? + ' - Ap"+" = u{x) (iu(x, i; /") - /.{()) J p " + i { i , r ) = o , i e ! i .
S I / • = 0, ( l , ( ) E S .
2.6. Calculate ff„ = 2.2 C o n j u g a t e g r a d i e n t m e t h o d
Step 1: 1.1. Given an initial approximation /» e K».
1.2. Calculate C/°(/°) is the solution ofthe di- rect problem
(
t/f - At/» = f'itMx, t) + it(x. ()in Q, t/»(x,0) = uoW,.r £!!,""- ^ = s{x.t).{x,t}es.
1.3 Calculate the residual ro = lU°{f] - h = J„u{x)U°{x,t; f'')dx - h with t.i{x) is the ^gj j - , weight and /i = Jjj i.j{x)uex{x, t\ f)dx, for u^^
is the exact solution of (1.1).
1.4. Calculate rp = - V J , ( / ° ) given in (2.3) by solving the adjoint
[ -p? - ApO = ^{x) {lu{x, t; /°) - h(t)) in Q,
||'"n-|-l|li2(o,T)
\K\\l-2(^0,T) 2.7. Update d„+i = r„ -|- /?„d„.
2.3 N u m e r i c a l simulation In this we present some numerical examples showing that our algorithm is efficient. Let fl — (0,1). We reconstruct the function / from the system for x€ (0,1), i e (0,1)
ut - u^^ ^ f(t)<p(x,t) -^ g{x,t),
u{x,0) =uo{x), (2,6) -u^iO,t) = gi{t),u^{l,t) = g2{t).
1, we test our algorithm for recon- structing the functions in three cases: the first f(t) is smooth, the second f{t] not differen- tiable at ^ — 0.5 and the last one is discontin- uous.
Example 1: /(() = sin(7r().
(2i if 0 < £<0.5 1 2 ( 1 - 0 i f 0 5 < ( < l .
1.5. Set do = ro.
Example 2: fit) =
Example 3: f{t) = if0.25<£<0.75 otherwise .
BCii Viet Huong Tap chi KHOA HOC & CONG NGHE 135(05): 139-143 We take U{T. f) = siii(Tr/) cos(3- - 1 ) , uo(x) = 0,
gi(t) = sin(7ri)sin(-£), 92(i) = sm{iTt)sm{l - t), ^{x.t) - (x^-\-l)it'^ + 1) a n d then p u t one of the above functions / into t h e system to get ij{.r. f). In the observation lu (2.6) we take t h e following weight functions
w(:r) - .1-2 + 1 (2.7)
T h e numerical results for these tests are pre- sented in Figures 1-3. F i o m these results we see t h a t the numerical results in the one di- mensional cases are verj' good, although the noise level is 10%.
Figure 1. Example 1: T h e exact solution in comparison with t h e numerical solution with noise level ^ 0.1 (left) and noise level ^ 0.01 (right). T h e weight function w is given by (2.7).
Figure 2 Example 2: T h e exact solution in comparison with t h e numerical solution with noise level = 0.1 (left) and noise level — 0 01 (right). T h e weight function w is given by (2.7).
Figure 3. Example 3: T h e exact solution in comparison with t h e numerical solution with noise level — 0.1 (left) and noise level — 0.01 (right). T h e weight function uj is given by (2.7).
References
[1] Dinh Nho Hao, Bui Viet Huong, P h a n Xuan T h a n h . D. Lesnic: Identification
of nonlinear heat transfer laws from b o u n d a r y observations, Applicable Anal- ysis, DOI 10.1080/00036811.2014.948425. _
B£ii Vi^t Huong Tgp chf KHOA HQC & CONG NGHE 135(05): 139-143 [2] Dinh Nho Hao, Bui Viet Huong, Nguyen in the right hand side of hnear parabolic
Thi Ngoc Oanh, and Phan Xuan Thanh, equations, Acta Mathematica Vietnamica Determination of a term in the right-hand (to appeal).
side of parabolic equations. Preprint 2015.
[5] TVbltzsh F.,Optimal Control of Partial [3) Ladyzhenskaya O.A., The Boundary Differential Equations: Theory, Methods Value Problems of Mathematical Physics. and Applications, Amer. Math. Soc, Springer-Verlag, New York, 1985. Providence, Rhode Islajid, 2010.
[4] Nguyen Thi Ngoc Oanh, Bui Viet Huong, [6] Wloka J., Partial Differential Equations.
Determination of a time-dependent term Cambridge Univ. Press, Cambridge, 1987.
TOM TAT
XAC DINH M O T T H A N H P H A N P H U T H U O C VAO THCJl GIAN TRONG vh P H A I CUA P H U d N G T R I N H PARABOLIC
Biii Viet HUdng*
Trudng Dai lioc Kiwa hge, Dgi hoc Tiidi Nguyen - TNU
Chiing toi de xuat phudng phap bien phan cho bai toan xac dinh mot thanh phan phu thupc thai gian trong phUdng trinh parabolic tii cac quan sat tich phan. Chung toi chiing minh phiem ham cin toi thieu hoa kha vi Frecliet va dUa ra cSng thiic xac dinh no qua bai toan lien hdp. Bai toan dudc rdi rac bing phUOiig phap sai phan va dUdc giai so bang phUdng phap gradient lien hdp kit hdp vdi phudng phap chinh Tikhonov. Mot so vi du bing so thUc hien tren may tinh da th& hien tmh higu qua ciia phUdng phap.
Tit kh6a : Bai todn ngUOc, hai toan dgt khdng chinh, quan sdt tich phan, phuang phdp sai phan, phiidng phap gradient lien hop.
°*Tc\. 0913162829, E-mail: [email protected]
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