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Applied Mathematical Modelling xxx (2014) xxx–xxx Contents lists available at ScienceDirect Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation Quoc Viet Tran a, Huy Tuan Nguyen b, Van Thinh Nguyen a,⇑, Duc Trong Dang c a Department of Civil and Environmental Engineering, Seoul National University, Republic of Korea Institute for Computational Science and Technology at Ho Chi Minh City, Viet Nam c Department of Mathematics, University of Science, Vietnam National University at Ho Chi Minh City, Viet Nam b a r t i c l e i n f o a b s t r a c t Article history: Received 23 September 2012 Received in revised form 10 October 2013 Accepted March 2014 Available online xxxx Keywords: Cauchy problem Helmholtz Ill-posed Inhomogeneous source Regularization method In this paper, we solve the Cauchy problem for an inhomogeneous Helmholtz-type equation with homogeneous Dirichlet and Neumann boundary condition The proposed problem is ill-posed Up to now, most investigations on this topic focus on very specific cases, and with Dirichlet boundary condition Recently, we solve this problem in 2D for an inhomogeneous modified Helmholtz equation (2012) This work is a continuous expansion of our previous results Herein we introduce a general filter regularization (GFR) method, and then from the GFR we deduce two concrete filters, which are a foundation to implement a numerical procedure In addition, we develop a numerical model for solving this problem in three dimensional region The proposed filter method has been verified by numerical experiments Ó 2014 Elsevier Inc All rights reserved Introduction It is well known that the Helmholtz-type equations arise in many engineering applications related to propagating waves in different environments, such as acoustic, hydrodynamic and electromagnetic waves [1,2] Such applications are interdisciplinary regarding material science, aerospace, chemical, nuclear and environmental engineering, medical instrumentation, etc In most of real-world applications, the Helmholtz-type equations are usually solved in 3D domains with inhomogeneous sources, and particularly the boundary conditions are often incomplete A classical example of such problems is the Cauchy problem, in which boundary conditions for the solution and its normal derivative are given only on a part of the solution domain Therefore a need of investigation on the Cauchy problem for inhomogeneous Helmholtz-type equations with Dirichlet, Neumann or mixed boundary conditions in 3D regions has given a motivation for this study Let a; b; c > and X ẳ 0; aị 0; bị We consider the Cauchy problem for 3D Helmholtz-type equation to determine u satised following system Du ỵ k u ẳ f x; y; zị; uz x; y; 0ị ẳ gx; yị; x; yị X; z ẵ0; c; x; yị X; ð1Þ ð2Þ ⇑ Corresponding author Address: 599 Gwanak-ro, Gwanak-gu, Seoul 151-744, Republic of Korea Tel.: +82 880 7355; fax: +82 873 2684 E-mail address: vnguyen@snu.ac.kr (V.T Nguyen) http://dx.doi.org/10.1016/j.apm.2014.03.001 0307-904X/Ó 2014 Elsevier Inc All rights reserved Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx uðx; y; 0ị ẳ ux; yị; Bux; y; zị ẳ 0; x; yÞ X; ð3Þ ðx; yÞ @ X; z ẵ0; c; 4ị where the boundary condition B given by (4) is corresponding to the homogeneous Dirichlet boundary condition ux; y; zị ẳ 0; x; yị @ X; z ẵ0; c 5ị or the homogeneous Neumann boundary condition @u x; y; zị ẳ 0; @n x; yị @ X; z ẵ0; c; 6ị where n is the unit vector normal to @ X, and k C In the past, there were many studies on the Cauchy problem for specific forms of the Helmholtz-type Eq (1) In case of the pffiffiffiffiffiffiffi source function f ¼ and the wave number k ¼ k0 R or k ¼ k0 À1, Eq (1) is known as homogeneous Helmholtz or modified Helmholtz equation, which can be referred to [3–7] In case of the wave number k ¼ and the source function f ¼ 0, Eq (1) is known as Laplace equation, which can be referred to [8–14] Recently, Nguyen et al [15] considered Eq (1) in 2D for pffiffiffiffiffiffiffi modified Helmholtz equation with inhomogeneous source; i.e with k ¼ k0 À1 and f – The study on the Cauchy problem for inhomogeneous Helmholtz-type problem in 3D is still limited In this research, we continue to consider Eq (1) in 3D with both homogeneous Dirichlet and Neumann boundary conditions In addition, we also pffiffiffiffiffiffiffi solve the problem with k ¼ k0 R and k ¼ k0 À1 covering both Helmholtz and modified Helmholtz equations Before leading to the mathematical formulation of an ill-posed problem consisting of Eqs (1)–(4), we recall following fundamentals of mathematics First, the inner product and the norm of L2 ðXÞ are defined as follows: hu; v i ¼ Z a Z sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z aZ b kuk ¼ juj2 dx dy b uv dx dy; 0 ð7Þ and two complete orthonormal basis sets of L2 ðXÞ are given by /ss pq x; yị ẳ jpq sin /cc pq x; yị ẳ jpq cos ppx a ppx a sin qpy ; b p > 0; q > 0; ð8Þ qpy ; b p P 0; q P ð9Þ cos here jpq ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ÁÀ Á À À dp0 À d0q ; ab ð10Þ where dpq is Kronecker delta Second, the following eigenvalue problem ( ÀDxy w k w ẳ kw; x; yị X; B wx; yị ẳ 0; x; yị @ X has infinite countable solutions ðkpq ; wpq Þ according to the boundary condition B given by (5) as follows: ( kpq ẳ wpq ẳ pp2 a /ss pq ỵ qp2 b Àk ; for p > 0; q > ð11Þ ; for p P 0; q P 0: 12ị and given by (6) as follows: ( kpq ẳ wpq ẳ pp2 a /cc pq ỵ qp2 b k For simplification, we set È É A1 ¼ ðp; qị : kpq < ; ẩ ẫ A2 ẳ p; qị : kpq ẳ ; Henceforward, we denote Fourier coefficients of v pq ¼ hv ; wpq i ¼ Z a Z È É A3 ¼ ðp; qị : kpq > ; A ẳ A1 [ A2 [ A3 : v ẳ v x; yị by b v ðx; yÞwpq ðx; yÞ dx dy: ð13Þ Applying variable separable method, the solutions of problem (1)–(4) can be presented in the following form Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxxxxx X ux; y; zị ẳ upq ðu; g; zÞwpq ðx; yÞ; ð14Þ ðp;qÞ2A where pffiffiffiffiffiffiffi À pffiffiffiffiffiffiffiffiffiffiÁ sin ðz Àkpq Þ > > g pq > cos z kpq upq ỵ p > kpq > > > À p ffiffiffiffiffiffiffiffiffiffiÁ R z > > ỵ p fpq sị sin z sị kpq ds; if ðp; qÞ A1 ; > > Àkpq > > < Rz if ðp; qÞ A2 ; upq u; g; zị ẳ upq ỵ z g pq ỵ z sịfpq sị ds; > > pffiffiffiffiffi u p ffiffiffiffiffi > u g g > > ez kpq 2pq ỵ ppq ỵ ez kpq 2pq À ppqffiffiffiffiffi > > kpq kpq > > > pffiffiffiffiffi pffiffiffiffiffi > > > þ p1ffiffiffiffiffi R z eðzÀsÞ kpq À eðsÀzÞ kpq f ðsÞ ds; if ðp; qÞ A : : pq ð15Þ kpq pffiffiffiffiffi We can see that the instability of (15) is caused by the fast increasing of ez kpq as kpq tends to infinity Even though the exact Fourier coefficients ðupq ; g pq ; fpq ðsÞÞ may tend to zero rapidly, a calculation of (14) is impossible because a given data set is usually diffused by varied factors, such as round-off, measurement errors, etc A small perturbation in the data can arbitrarily deduce a large error in the solution Therefore, the problem is ill-posed and a regularization is in order The remainder of this paper is organized as follows In Section 2, the theorical foundations of the general filter regularization (GFR) method and two specific regularization filters deduced from the GFR are presented In Section 3, the numerical experiments including data input, numerical procedures, examples, results and discussions are described Finally, the conclusion is shown in Section A general filter regularization method In this section, we are constructing several regularization strategies for Eq (14) from measured data ðue ; g e Þ satisfy kue À uk e; kg e À g k e z for a positive number e The primary idea of this method is to replace e to approximate u by ua which is defined by ua ðx; y; zị ẳ X p kpq 16ị in Eq (15) by damping quantities Q ða; p; qÞe uapq ðue ; g e ; zÞwpq ðx; yÞ; pffiffiffiffiffi z kpq ð17Þ ðp;qÞ2A where uapq is given by pffiffiffiffiffiffiffi À pffiffiffiffiffiffiffiffiffiffiÁ sin ðz Àkpq Þ > > pffiffiffiffiffiffiffi v pq cos z þ Àk w > pq pq > Àkpq > > > p Rz > > p ỵ fpq ðsÞ sin ðz À sÞ Àkpq ds; if ðp; qÞ A1 ; > > Àkpq > > < Rz if ðp; qÞ A2 ; uapq ðw; v ; zị ẳ wpq ỵ z v pq ỵ ðz À sÞfpq ðsÞ ds; > > pffiffiffiffiffi pffiffiffiffiffi > v pqffiffiffiffiffi w v > Q a; p; qịez kpq wpq ỵ p > ỵ eÀz kpq 2pq À ppqffiffiffiffiffi > > kpq kpq > > > pffiffiffiffiffi pffiffiffiffiffi > R > > : ỵ p1 0z Q a; p; qÞeðzÀsÞ kpq À eðsÀzÞ kpq fpq ðsÞ ds; if ðp; qÞ A3 ð18Þ kpq for any z ð0; c and w; v L ðXÞ Herein the function Q is called a regularizing filter and a plays the role as the regularization parameter, which should be chosen with respect to e Under a priori assumption of u, the regularization strategy a ẳ aeị is admitted if aðeÞ ! and kua ðÁ; Á; zÞ À uðÁ; Á; zÞk ! as e ! for any z ð0; c Now we establish a general regularizing filter Q in the following theorems 2.1 Theorem (General regularization filter) Assuming that u is the exact solution of problem (1)–(4) and M is a real-valued function such that X ðp;qÞ2A3 2 2 E2 M2 kpq ị hu; ; zị; wpq i ỵ huz ðÁ; Á; zÞ; wpq i kpq ð19Þ for any z ð0; c, where E is a positive number Let Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx Q : ð0; 1Þ Â A3 ! R be a function in such a way that, for any a > there exist K ðaÞ and K ðaÞ satisfy jQ ða; p; qÞj K ðaÞeÀz pffiffiffiffiffi kpq ð20Þ ; jQ ða; p; qÞ À 1j K ðaÞMðkpq Þ for any ðp; qÞ A3 and z > Then ua ð21Þ defined by Eqs (17) and (18), fulfills the following estimate a ku ðÁ; Á; zÞ À uðÁ; Á; zÞk C ðK aị ỵ 1ịe ỵ EK aị; where C is a positive constant independent of e A choice a ¼ aðeÞ is admissable if the values of aðeÞ; ð22Þ eK ðaÞ and K ðaÞ tend to zero as e tends to zero Proof The proof is split into two steps: 2 1=2 P a e e a , now we estimate S1 Step 1: Assigning S1 ẳ p;qị2A upq u ; g ; zÞ À upq ðu; g; zÞ From Eq (18), by subtracting uapq ðu; g; zÞ from uapq ðue ; g e ; zÞ and using triangle inequality, we have 8 > ue u ỵ zg e À g ; if ðp; qÞ A1 [ A2 ; > < pq pq a ue Àu upq ðue ; g e ; zÞ À uapq ðu; g; zÞ pffiffiffiffiffi e Àg j j pq j jgp > z k > ỵ pq ; if p; qị A3 ; : jQ a; p; qịje pq ỵ 2 where we applied following elementary estimates eÀz pffiffiffiffiffiffiffiffiffiffi cos z Àkpq 1; pffiffiffiffiffi kpq kpq for ðp; qÞ A3 and Á sin Àzpffiffiffiffiffiffiffiffiffiffi Àkpq pffiffiffiffiffiffiffiffiffiffi z for ðp; qÞ A1 : Àkpq Set k ẳ minp;qị2A3 kpq > Combine the latter estimate with (20) in regard to K ðaÞ P 0, we get S21 ẳ X p;qị2A1 [A2 2 2 X a upq ðue ; g e ; zÞ À uapq ðu; g; zÞ þ uapq ðue ; g e ; zÞ À uapq ðu; g; zÞ ðp;qÞ2A3 12 0 2 X uepq À upq g epq À g pq e e @ A p ffiffiffiffiffiffi ffi u u ỵ z g g ỵ K a ị ỵ ị ỵ pq pq pq pq 2 kpq ðp;qÞ2A1 [A2 ðp;qÞ2A3 X 2 2 1 e 2 2 1ỵ ỵ c2 ỵ K aị ỵ 1ị2 upq upq ỵ g epq g pq C K aị ỵ 1ị kue uk ỵ kg e À g k k ðp;qÞ2A X C K aị ỵ 1ị2 e2 ; n p qo where C ẳ max ỵ c2 ; ỵ 1k Hence S1 C K aị ỵ 1ịe: Step 2: Assigning S2 ẳ P 2 1=2 a , we now estimate S2 ðp;qÞ2A upq ðu; g; zÞ À upq ðu; g; zÞ ð23Þ Combining Eqs (15) and (18), we have ! Z z pffiffiffiffiffi pffiffiffiffiffi upq g pq uapq u; g; zị upq u; g; zị ẳ ez kpq Qa; p; qị 1ị ỵ p ỵ pffiffiffiffiffiffiffi eÀs kpq fpq ðsÞ ds : 2 kpq kpq ð24Þ On other hand, for ðp; qÞ A3 , we have ! ! Z z pffiffiffiffiffi u pffiffiffiffiffi u pffiffiffiffiffi pffiffiffiffiffi g pq g pq pq pq kpq Àz kpq huðÁ; Á; zÞ; wpq i ẳ e ỵ p ỵ e p ỵ pffiffiffiffiffiffiffi eðzÀsÞ kpq À eðsÀzÞ kpq fpq ðsÞ ds: 2 kpq kpq kpq z ð25Þ Differentiating (25) with respect to z gives ! ! Z z pffiffiffiffiffi u pffiffiffiffiffi u pffiffiffiffiffi pffiffiffiffiffi huz ðÁ; Á; zÞ; wpq i g pq g pq pq pq p ẳ ez kpq ỵ p ez kpq p ỵ p ezsị kpq ỵ eszị kpq fpq ðsÞ ds: 2 kpq kpq kpq kpq ð26Þ Adding (25) and (26) yields Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx eÀz pffiffiffiffiffi kpq huðÁ; ; zị; wpq i ỵ ! Z z p upq huz ðÁ; Á; zÞ; wpq i g pq pffiffiffiffiffiffiffi ẳ ỵ p ỵ p es kpq fpq sị ds: kpq kpq kpq ð27Þ Combining (24) with (27), we have ! huz ðÁ; Á; zÞ; wpq i a pffiffiffiffiffiffiffi : upq ðu; g; zÞ À upq ðu; g; zÞ jQða; p; qÞ À 1j huðÁ; Á; zÞ; wpq i ỵ kpq 28ị Applying (19) and (21) to (28), we have S22 ¼ 2 2 X X uapq ðu; g; zÞ À upq ðu; g; zị ẳ uapq u; g; zị upq u; g; zÞ ðp;qÞ2A 62 X ðp;qÞ2A3 2 ! 2 huz ðÁ; Á; zÞ; wpq i huðÁ; Á; zÞ; wpq i ỵ E2 K 22 aị: kpq jQ ða; p; qÞ À 1j ðp;qÞ2A3 Hence S2 EK ðaÞ: ð29Þ From (14) and (17), applying (23) and (29) we deduce 2 X uapq ðue ; g e ; zÞ À upq ðu; g; zÞ a ku ; ; zị u; ; zịk ẳ !1=2 S1 ỵ S2 C K aị þ 1Þe þ EK ðaÞ: ðp;qÞ2A Theorem has been completely proved h Remark The priori assumption (19) is hold naturally in the case of Mkpq ị ẳ kpq Indeed, let u be exact solution of the problem (1)(4), and /sc pq x; yị ẳ jpq sin /cs pq x; yị ẳ jpq cos where ppx qpy cos ; a b ppx a sin qpy ; b p > 0; q P 0; ð30Þ p P 0; q > 0; ð31Þ jpq was given by (10) By applying the Green’s first identity, we have ( pp huz ; /ss pq i a ss qp huz ; /pq i b ¼ huzx ; /cs pq i ¼ huzy ; /sc pq i p > 0; q > ; with the boundary condition B given by (5), and ( pp huz ; /cc pq i a cc qp huz ; /pq i b ¼ Àhuzx ; /sc pq i ¼ Àhuzy ; /cs pq i ; p P 0; q P n o n o n o n o /cc /sc /cs with the boundary condition B given by (6) Since /ss pq ; pq ; pq ; pq are complete orthonormal basis sets of L ðXÞ, then applying the Parseval’s identity yields X 2 2 kpq huz ; wpq i kuzx k2 þ uzy þ jkj2 kuz k2 : ð32Þ ðp;qÞ2A3 Furthermore, applying Green’s second identity, we have 8À Á < pp hu; /ss i ¼ Àhuxx ; /ss i pq pq a ; : ÀqpÁ2 hu; /ss i ¼ Àhu ; /ss i yy pq pq b p > 0; q > 0; with the boundary condition B given by (5), and similarly we have 8À Á < pp hu; /cc i ¼ Àhuxx ; /cc i pq pq a ; : ÀqpÁ2 hu; /cc i ¼ Àhu ; /cc i yy pq pq b p P 0; q; P with the boundary condition B given by (6) Thus we get Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx 8ðp; qÞ A: kpq hu; wpq i ẳ huxx ỵ uyy ỵ k u; wpq i; Hence, using Parseval’s identity and Schwarz’s inequality yielded 2 2 k2pq hu; wpq i kuxx k2 ỵ uyy ỵ jkj4 kuk2 : X ð33Þ ðp;qÞ2A3 From (32) and (33), we deduce X 2 2 2 E2 kpq huðÁ; Á; zị; wpq i ỵ kpq huz ; ; zị; wpq i p;qị2A for any z ẵ0; c, where E ẳ Ezị ẳ r 2 2 kuxx k2 ỵ uyy ỵ kuzx k2 ỵ uzy ỵ jkj2 kuz k2 ỵ jkj4 kuk2 : ð34Þ In order to implement numerical procedure, we deduce from GFR two concrete filters Applying Theorem directly, we obtain Theorem in the following section 2.2 Theorem (Two regularization filters) Assuming the problem (1)–(4) has a unique solution u, there exist two filters and corresponding regularized solutions defined by Eqs (17) and (18), named u1;a and u2;a , as follows (1) Filter 1: (Quasi-boundary-type method [1619]) Let Q be Qa; p; qị ẳ 1 þ akpq ez pffiffiffiffiffi ; a > 0; kpq ðp; qÞ A3 : ð35Þ Then u1;a defined by (17) and (18) satisfies the following inequality 1;a u ðÁ; Á; zÞ À uðÁ; Á; zÞ C Ez ỵ 1ị2 ỵ e ỵ k ln 2ezm e1Àm ð36Þ for any z ð0; c, where E; C and k are independent of e The admissible regularization parameter is aeị ẳ em ; m ð0; 1Þ (2) Filter 2: (Truncation method) Let Q be ( Q a; p; qị ẳ 1; if kpq a12 0; if kpq > a12 ; a > 0; ðp; qÞ A3 : ð37Þ Then u2;a defined by (17) and (18) satisfies the following inequality 2;a À Á u ðÁ; Á; zÞ À uðÁ; Á; zÞ C e1nc ỵ e ỵ E 38ị n2 ln ð1= eÞ for any z ð0; c, where E and C are independent of e In this case, the admissible regularization parameter is aeị ẳ 1=n ln1=eịị; n 0; c1 ị Proof Let Mkpq ị ẳ kpq ; E is given by (34) Proof of Filter Set K aị ẳ 1=kaị and K aị ẳ z ỵ 1ị2 =ln follows Qa; p; qị ez pffiffiffiffiffi kpq ¼ eÀz 2À z Á , 2a where k ¼ minp2A3 kpq The constrain (20) is obtained as 1 p ẳ K aị: kpq þ akpq ak In order to verify constrain (21), we apply an elementary estimate z ỵ 1ị2 p ; ez kpq ỵ akpq aln 2za 8p; qÞ A3 : ð39Þ Indeed, let a > 0, set Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx hfị ẳ ezf ỵ af2 ; f > 0: Á 2 z À fezf : 2a We have h fị ẳ 2aezf ezf ỵ af It yields z 2a z 2a À fezf P ¼ 2za À ef1 ezf1 ; f f1 ; À fezf ¼ 2za À f2 fz2 ; f P f2 ; where f1 ¼ z ; ln zỵ1 2a f2 ẳ z zỵ1 : 2a We can see that hðfÞ is increased as f < f1 and decreased as f > f2 Thus, Eq (39) becomes & hfị max hfị ẳ max f2ẵf1 ;f2 f2ẵf1 ;f2 ' ezf ỵ af & max f2½f1 ;f2 ' af ẳ f21 a z ỵ 1ị2 Á: aln2 2za ¼ From (39), we yield jQða; p; qị 1j ẳ kpq a p Mkpq ị K aị: kpq e ỵ akpq z By substituting K and K into (22), we deduce (36) in the following manner 1;a u ðÁ; Á; zÞ À uðÁ; Á; zÞ C ðK ðaÞ þ 1Þe þ EK ðaÞ C þ1 ak eþ Eðz þ 1Þ2 6C 2À Á ln 2za Ez ỵ 1ị2 ỵ e ỵ 2À Á : k ln 2ezm e1Àm The regularization parameter is admissible as m ð0; 1Þ Proof of Filter Set K aị ẳ ec=a and K aị ẳ a2 We derive constrains (20) and (21) as follows Q a; p; qị ez p kpq ẳ ( pffiffiffiffiffi pffiffiffiffiffiffiffi ez kpq ; if kpq a1 pffiffiffiffiffiffiffi 0; if kpq > 1a ez=a ec=a ¼ K ðaÞ and ( jQ ða; p; qÞ À 1j ¼ pffiffiffiffiffiffiffi 0; if a kpq pffiffiffiffiffiffiffi 1; if a kpq > a2 kpq ¼ K ðaÞ Mðkpq Þ: Hence (38) is deduced by substituting K and K into (22) as follows 2;a À Á À Á u ðÁ; Á; zÞ À uðÁ; Á; zÞ C ðK aị ỵ 1ịe ỵ EK aị C ec=a þ e þ Ea2 C e1Ànc þ e þ The regularization parameter is admissible as n ð0; cÀ1 Þ Theorem has been proved E n2 ln ð1=eÞ : h Numerical experiments 3.1 Modelling data In this paper, we assume that the exact data ðu; gÞ were belong to H2 ðXÞ We apply an uniform rectangular grid with a resolution of m  n in the horizontal xy-plane, which includes nodal points ðxi ; yj Þ determined by xi ¼ ði À 1Þdx ; dx ¼ a ; mÀ1 i ¼ 1; m; ð40Þ yj ¼ ðj À 1Þdy ; dy ¼ b ; nÀ1 j ẳ 1; n: 41ị Denote ui;j ẳ uxi ; yj ị and g i;j ẳ gxi ; yj ị Let ðg ei;j ; uei;j Þ be measured data in the grid with some error levels of pointwise Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx Fig Quadrilateral 9-node element uei;j ẳ ui;j ỵ e1;i;j ; g ei;j ẳ g i;j ỵ e2;i;j ; 42ị where e1;i;j and e2;i;j play the role as random noise of measurement Now using ðg ei;j ; uei;j Þ, we approximate the exact data ðu; gÞ In order that we model analytical expressions of ðg e ; ue Þ in ðg ei;j ; uei;j Þ by quadrature interpolation scheme as shown in following steps: First, a merit partition of X contained 9-node rectangular elements is chosen as follows È É C ¼ Eij ẳ ẵx2i1 ; x2iỵ1 ẵy2j1 ; y2jỵ1 : i m0 ; j n0 ; where m0 ẳ m 1ị=2 and n0 ẳ n 1ị=2 (see Fig 1) ẩ É ~ be an interpolating function of a given data set v i;j in the manner that, for any ðx; yÞ X there exists an Second, let v element Eij and a local coordinate r; sị ẵ1; such that x; yị Eij ; xẳ X xk Nk r; sị; kẳ1 yẳ X yk Nk r; sị 43ị kẳ1 and v~ x; yị ẳ X v k Nk r; sị; 44ị kẳ1 where the shape functions N k were listed in Table 1, the group of fv k ; xk ; yk g is numbered in an element (see Fig 1) in a fashion that v ¼ v 2iÀ1;2jÀ1 ; v ẳ v 2iỵ1;2j1 ; v ẳ v 2iỵ1;2jỵ1 ; v ẳ v 2i1;2jỵ1 ; v ẳ v 2i;2j1 ; v ẳ v 2iỵ1;2j ; v ẳ v 2i;2jỵ1 ; v ẳ v 2i1;2j ; v ¼ v 2i;2j : To be more precise, we emphasize that Eq (43) can be rewritten as follows x ẳ x2i ỵ rdx ; y ẳ y2j ỵ sdy ð45Þ due to the 9-node rectangular element (see Fig 1) Table Nodes and shape functions on local rectangular ½À1; 1  ½À1; 1 k Nodal point Mk Shape function N k M1 1; 1ị N1 ẳ M2 1; 1ị N2 ẳ M3 1; 1ị N3 ¼ M4 ðÀ1; 1Þ N4 ¼ M5 ð0; 1ị N5 ẳ M6 1; 0ị N6 ẳ M7 0; 1ị N7 ẳ M8 1; 0ị N8 ¼ M9 ð0; 0Þ N ¼ ð1 À r Þð1 À s Þ ð1 À rÞð1 sịrs 14 ỵ rị1 sịrs ỵ rị1 ỵ sịrs 14 rị1 ỵ sịrs 12 r ị1 sịs 2 ỵ rị1 s ịr 2 r ị1 ỵ sịs À 12 ð1 À rÞð1 À s2 Þr 2 kN k kL2 ððÀ1;1Þ2 Þ 15 15 15 15 15 15 15 15 16 15 Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx ~ are interpolating functions of data sets ~ e Next, we define u ~ and g Now we assign g e ¼ g~e and ue ¼ u respectively From (42) and (45) and Table 1, we obtain ~k ¼ kue À u Z m À1 n À1 X X i¼1 ¼ where ~ j dx dy ¼ jue À u m À1 n À1 X X Eij j¼1 i¼1 Z X À1 j¼1 k¼1 Z n o È É ui;j and g i;j , m À1 n À1 X X e1;i;j 2 jNk ðr; sịj2 dx dy dr ds ẳ 64 e1;i;j 2 dx dy 25 i¼1 j¼1 À1 m0 À1 nX À1 64 X b 16 e1;i;j 2 a abe20 ; 25 i¼1 j¼1 m À n À 25 È É e0 ¼ maxi;j je1;i;j j; je2;i;j j Thus, applying the same estimate to g e , we get ~k kue À u pffiffiffiffiffiffi ab e0 ; kg e À g~k pffiffiffiffiffiffi ab e0 : On the other hand, the Bramble–Hilbert theorem (see Chapter of [20]) guarantees the following estimate qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi kv À v~ k Cðd2x þ d2y Þ kv xx k2 þ v yy ð46Þ v H ðXÞ, where the constant C ẳ CXị Hence, we deduce ~ k ỵ ku ~ À uk e; kg e À g k kg e g~k ỵ kg~ g k e; kue À uk kue À u for any where e¼ (rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi) pffiffiffiffiffiffi 2 2 ab e0 ỵ Cd2x ỵ d2y ị max kuxx k2 ỵ uyy ; kg xx k2 ỵ g yy : 3.2 Quadrature formula for Fourier coefficients In this section, we introduce numerical methods to calculate Fourier coefficients in Eq (18) Due to the fluctuation of wpq as p and q become large, the integrals require a special technique (see Chapter 13 of [21]) Herein we applied a key idea suggested by Filon [22], in which we first identify discrete data of the integrands by their own quadrature interpolating functions described in Section 3.1, and then obtain an exact formula of the integrals (Eq (13)) Let a; b > 0, according to wpq with regardless to jpq , the integrals (13) can be written as follows Z v sspq ¼ Z a Z v ccpq ¼ v~ ðx; yÞ sin ppx qpy sin dx dy; a b p; q N; ð47Þ v~ ðx; yÞ cos ppx qpy cos dx dy; a b p; q N [ f0g; ð48Þ b Z a b ~ is defined by (44) Then we have where v Z x2iỵ1 m n X X v sspq ẳ iẳ1 jẳ1 x2i1 Z x2iỵ1 m À1 n À1 X X v ccpq ẳ iẳ1 jẳ1 Z y2jỵ1 v~ x; yị sinpa xị sinðqb yÞ dx dy; ð49Þ v~ ðx; yÞ cosðpa xÞ cosqb yị dx dy; 50ị y2j1 Z x2i1 y2jỵ1 y2j1 where pa ¼ pp=a; qb ¼ qp=b and m0 ¼ m 1ị=2; n0 ẳ n 1ị=2 By substituting (44) and (45) into Eqs (49) and (50), and applying some trigonometric calculations, we obtain m À1 n À1 X X v sspq ¼ dx dy i¼1 i¼1 vk spi sqj J k1 ỵ spi cqj J k2 ỵ cpi sqj J k3 ỵ cpi cqj J k4 ; ð51Þ cpi cqj J k1 À cpi sqj J k2 À spi cqj J k3 ỵ spi sqj J k4 ; 52ị jẳ1 kẳ1 m À1 n À1 X X v ccpq ¼ dx dy X X vk j¼1 k¼1 where we denote J k1 ¼ J k3 ¼ Z À1 Z À1 Z N k ðr; sÞ cosðpd rÞ cosðqd sÞ dr ds; À1 Z À1 N k ðr; sÞ sinðpd rÞ cosðqd sÞ dr ds; J k2 ¼ J k4 ¼ Z Z Z À1 Z À1 N k ðr; sÞ cosðpd rÞ sinðqd sÞ dr ds; À1 Nk ðr; sÞ sinðpd rÞ sinðqd sÞ dr ds ð53Þ À1 Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 10 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx and pd ¼ pa dx ; spi ẳ sin pa x2i ị; cpi ¼ cos ðpa x2i Þ; qd ¼ qb dy ; À Á sqj ¼ sin qb y2j ; À Á cqj ¼ cos qb y2j : In order to accurately approximate RHS of (51) and (52), we need to calculate J kl exactly Referring to Table with respect to odd–even property of integrands, we can see following group relationships J 11 ¼ J 31 ; J 21 J 54 ¼ ¼ J 12 ¼ ÀJ 32 ; J 41 ; J 74 ; J 22 J 61 J 13 ¼ ÀJ 33 ; ÀJ 42 ; J 81 ; ¼ ¼ J 14 ¼ J 34 ; J 23 ¼ ÀJ 43 ; J 62 ¼ ÀJ 82 ; J 24 ¼ J 44 ; J 63 ¼ ÀJ 83 ; J 51 ¼ J 71 ; J 52 ¼ ÀJ 72 ; J 53 ¼ ÀJ 73 ; J 64 ẳ J 84 54ị are satised at any p; q N [ f0g Thus we need to estimate only five terms duced from the group (54) Let #sị ẳ bsị ẳ s2 2ị sin s ỵ 2s cos s s3 sin s À s cos s s2 J 1l ; J 2l ; J 5l ; J 6l ; J 9l l ẳ 1; 4ị and the others can be de- ð55Þ ; ð56Þ : Applying integration by part to the group (53), we have J 11 ẳ #pd ị#qd ị; J 12 ẳ #pd ịbqd Þ; J 13 ¼ Àbðpd Þ#ðqd Þ; J 14 ¼ bpd ịbqd ị; J 21 ẳ #pd ị#qd ị; J 22 ẳ #pd ịbqd ị; J 23 ẳ bpd ị#qd Þ; J 24 ¼ Àbðpd Þbðqd Þ; 4 J 51 ẳ bpd ị#qd ị; J 52 ẳ bpd Þbðqd Þ; J53 ¼ 0; J 54 ¼ 0; pd pd 4 J 61 ẳ #pd ịbqd ị; J 62 ẳ 0; J 63 ẳ bpd ịbqd ị; J 64 ¼ 0; qd qd 16 J 91 ¼ bðpd Þbðqd Þ; J 92 ¼ 0; J93 ¼ 0; J 94 ẳ 0: pd qd 57ị Some of J kl within the group (57) require to be defined at pd ¼ or qd ¼ 0, however the formulas (55) and (56) are not defined at s ¼ 0, fortunately the functions #sị; bsị and csị :ẳ bsị=s can be extended to be continuous functions on ẵ0; 1ị, particularly at s ẳ by following denitions: #0ị :ẳ lim #sị ¼ s!0 ; bð0Þ :¼ lim bðsÞ ¼ 0; s!0 c0ị :ẳ lim bsị s s!0 ẳ : cc Thus the group (57) is well-defined at any p; q N [ f0g Therefore, Fourier coefficients v ss pq and v pq from (51) and (52) can be determined Remark In order to calculate (18), we need not only to estimate integrations in form of (47) and (48), but also to calculate the following integrals fpq zị ẳ Z Z z a Z b f ðx; y; sÞwpq ðx; yÞ dx dy ds; 8z ½0; c: For a fixed value of z ½0; c, based on Fubini theorem and applying above Filon technique, the fpq ðzÞ can be integrated by ~ ðx; yÞ in Eqs (49) and (50) by replacing v Fx; y; zị ẳ Z z f x; y; sÞ ds: In order to approximate RHS of the latter equation, we use Gauss–Legendre quadrature method as follows Fðxi ; yj ; zị ẳ Z zX z wk f xi ; yj ; nk ỵ 1ị ; kẳ1 k z f xi ; yj ; sị ds where nk abscissas in ẵ1; 1 and wk are associated weights The computation of abscissas and weights is carried out by gauleg subroutine [21] Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx 11 3.3 Numerical test In this section, the proposed regularization methods are implemented numerically The efficiency of the methods is observed by comparing the result between numerical and exact solutions To that, we constructed Eqs (1)–(3), in such a way that they have a given exact solution Our numerical implementation is focused on calculation of Fourier coefficients in Eq (13) We consider Eqs (1)–(3), under three examples; the first and second examples are dealing with Dirichlet boundary condition (5) and Neumann boundary condition (6), respectively The third example is dealing with unbounded horizontal boundary and not-given boundary conditions Before implementing numerical procedure, we introduce a relative root mean square error between the regularized solutions uj;a ðÁ; Á; cÞ and exact solution uðÁ; Á; cÞ as follows qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi Pm Pn j;a ðxi ; yj ; cÞ À uðxi ; yj ; cị iẳ1 jẳ1 u q d qị ẳ 2 Pm Pn iẳ1 jẳ1 uxi ; yj ; cị j with 58ị j ẳ 1; In fact, dj is dependent on the regularization parameter a which were determined by Theorem ( aqị ẳ q0:9=c ; for u1;a of Filter 1; c ; 0:9 lnð1=qÞ for u2;a of Filter 2; ð59Þ where q is a positive number In order to illustrate the sensitivity of the computational accuracy to noise of the data, we repeated calculations with variety of perturbed data The perturbation was defined as e randðÁ; ÁÞ where each random term randðÁ; ÁÞ was uniformly determined on ½À1; 1, and e plays as an amplitude, i.e g ei;j ẳ gxi ; yj ị ỵ e randxi ; yj ị; 60ị uei;j ẳ uxi ; yj ị ỵ e randxi ; yj ị 61ị for any i ẳ 1; m; j ¼ 1; n, where xi ; yj and m; n were described by (40) and (41) The aim of the numerical experiments is to observe the relative errors dj ðqÞ as q tends to zero in two following cases of e: e ¼ The couple of ðuei;j ; g ei;j Þ play a role as exact data We expect that regularized solutions uj;a stably converge to the e ¼ 10Àr as r ¼ 2; 4; The couple of ðuei;j ; g ei;j Þ play as measured data with a random noise The regularized solutions uj;a were expected to be closed to the exact solution u as q tends to zero However, after a certain value of q ¼ qconv the exact solution u under a proper discretization numerical solution starts to diverge because the regularization was broken This tendency has been predicted as above theoretical results by the restriction (23) Consequently, the computational process will be stopped at qconv 3.3.1 Example Solving Eqs (1)–(3), in which f ; g and u have been chosen in order that ux; y; zị ẳ cosðxyzÞ sin px a sin py b is exact solution (see Fig 2(a)) In this case, the wave number k is a real number, therefore the problem (1)–(3) becomes Helmholtz problem with Dirichlet boundary condition, Fig The graph of exact solutions uðx; y; cÞ corresponding to each example Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 12 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx Table Example 1, a ¼ b ¼ 3; c ¼ 1; k ¼ 3, noise k¼3 q e ¼ m ¼ n ¼ 65 m ¼ n ¼ 129 m ẳ n ẳ 257 d qị d qị d ðqÞ d ðqÞ d1 ðqÞ d2 ðqÞ À1 10 3.519EÀ01 2.990EÀ01 3.519EÀ01 2.990EÀ01 3.519EÀ01 2.990EÀ01 10À2 1.671EÀ01 1.304EÀ01 1.671EÀ01 1.304EÀ01 1.671EÀ01 1.304EÀ01 10À3 6.547EÀ02 1.642EÀ02 6.547EÀ02 1.642EÀ02 6.547EÀ02 1.642EÀ02 10À4 1.734EÀ02 6.414EÀ03 1.734EÀ02 6.417EÀ03 1.734EÀ02 6.417EÀ03 10À5 5.839EÀ03 2.993EÀ03 5.839EÀ03 2.994EÀ03 5.839EÀ03 2.994EÀ03 10À6 2.857EÀ03 1.791EÀ03 2.855EÀ03 1.745EÀ03 2.855EÀ03 1.745EÀ03 10À7 1.739EÀ03 2.514EÀ03 1.685EÀ03 1.137EÀ03 1.684EÀ03 1.126EÀ03 Table Example 1, a ¼ b ¼ 3; c ¼ 1; k ¼ 3; m ¼ n ¼ 257, noise amplitude k¼3 e ¼ 10À2 q e ¼ 10Àr ; r ¼ 2; 4; e ¼ 10À4 e ¼ 10À6 d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ À1 10 3.518Ề01 2.990Ề01 3.519Ề01 2.990Ề01 3.519EÀ01 2.990EÀ01 10À2 1.668EÀ01 1.304EÀ01 1.671EÀ01 1.304EÀ01 1.671EÀ01 1.304EÀ01 10À3 6.809EÀ02 3.576EÀ02 6.546EÀ02 1.642EÀ02 6.547EÀ02 1.642EÀ02 10À4 1.633EÀ01 2.616EÀ01 1.735EÀ02 6.940EÀ03 1.734EÀ02 6.417EÀ03 10À5 Diverged Diverged 1.264EÀ02 3.588EÀ02 5.845EÀ03 3.012EÀ03 10À6 Diverged Diverged 7.341EÀ02 Diverged 3.020EÀ03 3.327EÀ03 10À7 Diverged Diverged Diverged Diverged 4.834Ề03 Diverged Table Example 1, a ¼ b ¼ 3; c ¼ 1; k ¼ 7, noise k¼7 e ¼ m ¼ n ¼ 65 m ¼ n ¼ 129 m ¼ n ¼ 257 q d ðqÞ d ðqÞ d ðqÞ d ðqÞ d1 ðqÞ d2 ðqÞ 2 10À1 1.828Ề02 1.971Ề02 1.828Ề02 1.971EÀ02 1.828EÀ02 1.971EÀ02 10À2 1.214EÀ02 9.908EÀ03 1.215EÀ02 9.910EÀ03 1.215EÀ02 9.910EÀ03 10À3 7.296EÀ03 5.075EÀ03 7.298EÀ03 5.078EÀ03 7.298EÀ03 5.078EÀ03 10À4 4.353EÀ03 3.123EÀ03 4.356EÀ03 3.127EÀ03 4.356EÀ03 3.127EÀ03 10À5 2.694EÀ03 1.897EÀ03 2.696EÀ03 1.899EÀ03 2.696EÀ03 1.899EÀ03 10À6 1.761EÀ03 1.224EÀ03 1.759EÀ03 1.192EÀ03 1.759EÀ03 1.192EÀ03 10À7 1.288EÀ03 2.027EÀ03 1.207EÀ03 8.285EÀ04 1.206EÀ03 8.177EÀ04 Table Example 1, a ¼ b ¼ 3; c ¼ 1; k ¼ 7; m ¼ n ¼ 257, noise amplitude e ¼ 10Àr ; r ¼ 2; 4; À2 e ¼ 10À4 e ẳ 106 kẳ7 e ẳ 10 q d1 qị d2 ðqÞ d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ 10À1 1.845EÀ02 1.972EÀ02 1.828EÀ02 1.971EÀ02 1.828EÀ02 1.971EÀ02 10À2 1.341EÀ02 1.082EÀ02 1.215EÀ02 9.911EÀ03 1.215EÀ02 9.910EÀ03 10À3 2.332EÀ02 1.890EÀ02 7.302EÀ03 5.083EÀ03 7.298EÀ03 5.078EÀ03 10À4 Diverged Diverged 4.514EÀ03 4.357EÀ03 4.355EÀ03 3.127EÀ03 10À5 Diverged Diverged 9.437EÀ03 2.085EÀ02 2.684EÀ03 1.922EÀ03 10À6 Diverged Diverged Diverged Diverged 1.852EÀ03 3.426EÀ03 10À7 Diverged Diverged Diverged Diverged 4.268EÀ03 Diverged uð0; y; zị ẳ ua; y; zị ẳ ux; 0; zị ¼ uðx; b; zÞ ¼ 0: Let a ¼ b ¼ and c ¼ The horizontal grid resolution of this calculation is 2s ỵ 1ị 2s ỵ 1ị 2s ị for s ẳ 6; and The numerical results with wave number k ¼ have been shown on Tables and 3, and with wave number k ¼ shown on Tables and Fig shows the graph of numerical solutions with the wave number k ¼ Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx Fig Example 1, k ¼ 7, noise amplitude 13 e ¼ 10À2 ; m ¼ n ¼ 257 Divergence phenomena occurs as q becomes less than e Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 14 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx 3.3.2 Example Similar to Example 1, the exact solution of (1)(3) was chosen as ux; y; zị ẳ cos ðzyðb À yÞÞ cos px a Table Example 2, a ¼ b ¼ 5; c ¼ 1; k ¼ 1, noise amplitude e ¼ k¼1 m ¼ n ẳ 65 q d1 qị d2 qị m ẳ n ẳ 129 d1 qị d2 qị m ẳ n ¼ 257 d1 ðqÞ d2 ðqÞ À1 10 4.961Ề01 4.842Ề01 4.986EÀ01 4.847EÀ01 4.999EÀ01 4.849EÀ01 10À2 3.321EÀ01 2.103EÀ01 3.300EÀ01 2.084EÀ01 3.290EÀ01 2.074EÀ01 10À3 1.595EÀ01 7.199EÀ02 1.568EÀ01 7.126EÀ02 1.554EÀ01 7.090EÀ02 10À4 5.062EÀ02 9.555EÀ03 4.902EÀ02 9.298EÀ03 4.833EÀ02 9.214EÀ03 10À5 1.416EÀ02 5.451EÀ03 1.254EÀ02 2.242EÀ03 1.221EÀ02 2.179EÀ03 10À6 1.017EÀ02 1.357EÀ02 3.778EÀ03 1.543EÀ03 3.376EÀ03 1.268EÀ03 10À7 3.909EÀ02 Diverged 3.134EÀ03 6.607EÀ03 1.335EÀ03 6.763EÀ04 Table Example 2, a ¼ b ¼ 5; c ¼ 1; k ¼ 1; m ¼ n ¼ 257, noise amplitude k¼1 e ¼ 10À2 q e ¼ 10Àr ; r ¼ 2; 4; e ẳ 104 e ẳ 106 d1 qị d2 ðqÞ d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ À1 10 4.999EÀ01 4.849EÀ01 4.999EÀ01 4.849EÀ01 4.999EÀ01 4.849EÀ01 10À2 3.286EÀ01 2.075EÀ01 3.290EÀ01 2.074EÀ01 3.290EÀ01 2.074EÀ01 10À3 1.548EÀ01 8.534EÀ02 1.555EÀ01 7.090EÀ02 1.554EÀ01 7.090EÀ02 10À4 1.140EÀ01 3.071EÀ01 4.835EÀ02 9.558EÀ03 4.833EÀ02 9.214EÀ03 10À5 4.995EÀ01 Diverged 1.350EÀ02 2.207EÀ02 1.221EÀ02 2.196EÀ03 10À6 Diverged Diverged 2.461EÀ02 Diverged 3.386EÀ03 1.806EÀ03 10À7 Diverged Diverged Diverged Diverged 1.543EÀ03 1.822EÀ03 Table Example 2, a ¼ b ¼ 5; c ¼ 1; k ¼ 7, noise amplitude k¼7 q e ¼ m ¼ n ¼ 65 m ¼ n ẳ 129 d qị d qị d1 qị d2 ðqÞ 3.296Ề02 3.745Ề02 3.258Ề02 3.710Ề02 3.239Ề02 3.691Ề02 1.950Ề02 1.116Ề02 1.908Ề02 1.097EÀ02 1.891EÀ02 1.088EÀ02 10À3 9.375EÀ03 4.769EÀ03 8.959EÀ03 4.605EÀ03 8.826EÀ03 4.552EÀ03 2.291Ề03 10 10À2 m ¼ n ¼ 257 d ðqÞ d ðqÞ À1 2 10À4 5.016Ề03 2.721Ề03 4.372Ề03 2.329EÀ03 4.265EÀ03 10À5 4.089EÀ03 3.562EÀ03 2.220EÀ03 1.353EÀ03 2.073EÀ03 1.308EÀ03 10À6 9.685EÀ03 Diverged 1.545EÀ03 1.905EÀ03 1.095EÀ03 5.562EÀ04 10À7 Diverged Diverged 2.847EÀ03 Diverged 7.258Ề04 5.218Ề04 Table Example 2, a ¼ b ¼ 5; c ¼ 1; k ¼ 7; m ¼ n ¼ 257, noise amplitude k¼7 e ¼ 10À2 q e ¼ 10Àr ; r ¼ 2; 4; e ¼ 10À4 e ¼ 10À6 d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ À1 10 3.240Ề02 3.692EÀ02 3.239EÀ02 3.691EÀ02 3.239EÀ02 3.691EÀ02 10À2 1.906EÀ02 1.166EÀ02 1.891EÀ02 1.088EÀ02 1.891EÀ02 1.088EÀ02 10À3 1.576EÀ02 3.189EÀ02 8.827EÀ03 4.560EÀ03 8.826EÀ03 4.552EÀ03 10À4 6.952EÀ02 Diverged 4.327EÀ03 3.310EÀ03 4.265EÀ03 2.292EÀ03 10À5 Diverged Diverged 3.895EÀ03 1.505EÀ02 2.073EÀ03 1.310EÀ03 10À6 Diverged Diverged 1.039EÀ02 Diverged 1.099EÀ03 6.377EÀ04 10À7 Diverged Diverged Diverged Diverged 7.623EÀ04 6.079EÀ04 Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx 15 (see Fig 2(b)) which satises Neumann boundary condition, ux 0; y; zị ẳ ux a; y; zị ẳ uy x; 0; zị ẳ uy x; b; zị ẳ 0: Fig Example 2, k ¼ 7, noise amplitude e ¼ 10À2 ; m ¼ n ¼ 257 Divergence phenomena occurs as q becomes less than e Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 16 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx Fig Example 3, Dirichlet condition, m ¼ m ¼ 257, noise amplitude e ¼ 10À4 Divergence phenomena occur as q becomes less than e Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx Fig Example 3, Neumann condition, m ¼ m ¼ 257, noise amplitude 17 e ¼ 10À4 Divergence phenomena occurs as q becomes less than e Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 18 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx Let a ¼ b ¼ and c ¼ The grid resolutions of this calculation are the same as in Example The numerical results are shown in Tables and with the wave number k ¼ 1, and Tables and with k ¼ Fig shows the graph of numerical solutions with the wave number k ¼ 3.3.3 Example Considering Eqs (1)–(3), in a specific case as the modified Helmholtz problem as follows Du À k u ¼ f ðx; y; zị; x; yị R2 ; z ẵ0; c; ux; y; 0ị ẳ ux; yị; x; yị R2 ; uz x; y; 0ị ẳ gx; yị; x; yị R2 ; where k ẳ 1; c > The same idea as in the Example 1, the functions f ; g and u were chosen in such a way that Pi0 cxxi ị2 ỵyyi ị2 ỵzzi ị2 Þ is an exact solution (see Fig 2(c)), where c ¼ 20; i0 ¼ 3; ðx1 ;y1 ;z1 Þ ¼ 1:125;1:125; 1:0ị; iẳ1 e ux; y;zị ẳ x2 ; y2 ; z2 ị ẳ 1:875;1:125;1:0ị and x3 ; y3 ;z3 Þ ¼ ð1:5;1:875; 1:0Þ In this test case, starting from an unbounded domain in R2 we try to reconstruct uðx; y;cị in the bounded region X ẳ 0;aị 0;aị (a ¼ b ¼ 3) in such a way that the values of g; u and f are infinitesimal outside of X and X 2c; 2cị with c ẳ Herein either homogeneous Dirichlet or Neumann boundary condition can be set up for this example Now we are seeking a solution of the following problem Du À k u ẳ f x; y; zị; x; yị X; z ẵ0; c; ux; y; 0ị ẳ ue x; yị; x; yị X; uz x; y; 0ị ẳ g e ðx; yÞ; ðx; yÞ X satisfied either Dirichlet (5) or Neumann (6) boundary conditions, where the disturbed data ðue ; g e Þ described in Section 3.1 satisfy Eqs (60) and (61) Each numerical result uj;a associated with a specific boundary condition is expected to be a reasonable regularized solution Figs and show the graph of numerical solutions with Dirichlet and Neumann boundary conditions, respectively The grid resolutions of this calculation are the same as in two above Examples Tables 2, 4, 6, 8, 10 and 12 show relative error estimate dj qị j ẳ 1; 2ị in case of e ẳ 0, i.e for exact data ðu; gÞ As q tends to zero shown in these tables, the numerical calculations of all grid resolutions are stable converged until q ¼ 10À6 , thereafter the approximated solution uj;a is still converged with finer grid resolutions (m ¼ n ¼ 129 and 257), however it almost started to diverge in most of cases with coarser grid resolutions (m ¼ n ¼ 65) (except in the Example 3, see Tables 10 and 12) Tables 3, 5, 7, 9, 11 and 13 show relative error estimate dj ðqÞ j ẳ 1; 2ị in case of e ẳ 10r r ẳ 2; 4; 6ị, i.e for disturbed data ue ; g e Þ In this case, the finest grid was used (m ¼ n ¼ 257) As shown in these tables, once q tends to zero and its value Table 10 Example 3, Dirichlet condition, a ¼ b ¼ 3; c ¼ 1; k ¼ 1, noise amplitude k ¼ 10 À1 q m ¼ n ¼ 65 e ¼ m ¼ n ¼ 129 m ¼ n ¼ 257 d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ À1 10 4.581Ề01 4.340Ề01 4.580Ề01 4.339Ề01 4.580EÀ01 4.339EÀ01 10À2 3.916EÀ01 3.804EÀ01 3.915EÀ01 3.802EÀ01 3.915EÀ01 3.802EÀ01 10À3 3.479EÀ01 3.416EÀ01 3.477EÀ01 3.415EÀ01 3.477EÀ01 3.414EÀ01 10À4 2.889EÀ01 2.424EÀ01 2.888EÀ01 2.422EÀ01 2.887EÀ01 2.422EÀ01 10À5 1.923EÀ01 1.479EÀ01 1.922EÀ01 1.478EÀ01 1.921EÀ01 1.478EÀ01 10À6 1.109EÀ01 1.011EÀ01 1.107EÀ01 1.012EÀ01 1.107EÀ01 1.012EÀ01 10À7 8.344EÀ02 9.158EÀ02 8.341EÀ02 9.178EÀ02 8.341EÀ02 9.179EÀ02 Table 11 Example 3, Dirichlet condition, a ¼ b ¼ 3; c ¼ 1; k ¼ 1; m ¼ n ¼ 257, noise amplitude k ¼ 10À1 e ¼ 10À2 q e ¼ 10Àr ; r ¼ 2; 4; e ¼ 10À4 e ¼ 10À6 d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ À1 10 4.578Ề01 4.340Ề01 4.580Ề01 4.339Ề01 4.580Ề01 4.339Ề01 10À2 3.916EÀ01 3.803EÀ01 3.915EÀ01 3.802EÀ01 3.915EÀ01 3.802EÀ01 10À3 3.512EÀ01 3.478EÀ01 3.477EÀ01 3.414EÀ01 3.477EÀ01 3.414EÀ01 10À4 4.551EÀ01 8.181EÀ01 2.889EÀ01 2.423EÀ01 2.887EÀ01 2.422EÀ01 10À5 Diverged Diverged 1.942EÀ01 1.566EÀ01 1.922EÀ01 1.478EÀ01 10À6 Diverged Diverged 1.886EÀ01 5.221EÀ01 1.109EÀ01 1.010EÀ01 10À7 Diverged Diverged 9.698EÀ01 Diverged 8.446EÀ02 9.754EÀ02 Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 19 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx Table 12 Example 3, Neumann condition, a ¼ b ¼ 3; c ¼ 1; k ¼ 1, noise amplitude k¼1 q m ¼ n ¼ 65 e ¼ m ¼ n ¼ 129 m ¼ n ẳ 257 d qị d qị d qị d ðqÞ d1 ðqÞ d2 ðqÞ À1 10 4.608Ề01 5.027Ề01 4.607Ề01 5.020EÀ01 4.607EÀ01 5.016EÀ01 10À2 3.895EÀ01 3.819EÀ01 3.894EÀ01 3.815EÀ01 3.894EÀ01 3.814EÀ01 10À3 3.490EÀ01 3.579EÀ01 3.489EÀ01 3.575EÀ01 3.489EÀ01 3.573EÀ01 10À4 2.876EÀ01 2.623EÀ01 2.875EÀ01 2.613EÀ01 2.875EÀ01 2.609EÀ01 10À5 1.933EÀ01 1.479EÀ01 1.931EÀ01 1.472EÀ01 1.931EÀ01 1.468EÀ01 10À6 1.114EÀ01 1.019EÀ01 1.112EÀ01 1.017EÀ01 1.112EÀ01 1.016EÀ01 10À7 8.312EÀ02 9.209EÀ02 8.306EÀ02 9.206EÀ02 8.305EÀ02 9.197EÀ02 Table 13 Example 3, Neumann condition, a ¼ b ¼ 3; c ¼ 1; k ¼ 1; m ¼ n ¼ 257, noise amplitude k¼1 e ¼ 10À2 q e ¼ 10Àr ; r ¼ 2; 4; e ¼ 10À4 e ¼ 10À6 d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ d1 ðqÞ d2 ðqÞ À1 10 4.607Ề01 5.016Ề01 4.607Ề01 5.016Ề01 4.607Ề01 5.016Ề01 10À2 3.896EÀ01 3.814EÀ01 3.894EÀ01 3.814EÀ01 3.894EÀ01 3.814EÀ01 10À3 3.519EÀ01 3.699EÀ01 3.489EÀ01 3.573EÀ01 3.489EÀ01 3.573EÀ01 10À4 4.493EÀ01 8.199EÀ01 2.877EÀ01 2.610EÀ01 2.875EÀ01 2.609EÀ01 10À5 Diverged Diverged 1.963EÀ01 1.558EÀ01 1.931EÀ01 1.467EÀ01 10À6 Diverged Diverged 1.916EÀ01 5.457EÀ01 1.112EÀ01 1.015EÀ01 10À7 Diverged Diverged 9.247EÀ01 Diverged 8.316EÀ02 1.011EÀ01 is still greater than or equal to e, the numerical solutions are still converged, however when q < e the numerical solutions are starting to diverge Tables 4, (Example 1) and 8, (Example 2) showed relative error estimate dj qị j ẳ 1; 2ị in case of wave number k ¼ The main difficulty in solving Eqs (1)–(3), is dealing with a high wave number k, consequently a large number of grid points must be used to resolve the short wave length Therefore, in order to test the sensibility of the numerical computation to a high wave number, in the exact data we increase the wave number k from to in Example and from to in Example The results are showed the numerical solution is dependent on the grid resolutions, and its convergence is dependent on the value of q In Figs 3–6, the first and second columns show the graph and its contours of the regularized solutions of filter (u1;a ) and filter (u2;a ), respectively As shown, once q tends to zero, we can see that if q < e, then uj;a (j ¼ 1; 2) start fluctuating which is shown by oscillating contours, afterward the solution begins to diverge, because the regularization strategies were broken down Conclusion The study on inverse Helmholtz-type problem with inhomogeneous source in 3D is still limited Recently, Nguyen et al [15] suggested two regularization methods for the modified Helmhotlz equation with inhomogeneous source and Dirichlet boundary condition in 2D In this paper, we solved inverse Helmholtz-type problem with inhomogeneous source in 3D including both Dirichlet and Neumann boundary conditions This work is a continuous development of our previous study [15] In theoretical results, we suggested a general method of regularization filter (Theorem 1), subsequently, we deduced to two concrete regularization filters (Theorem 2), which sets a fundamental to solve the problem (1)–(4) numerically as shown in Examples 1–3, and obtained the error estimation of logarithm type The numerical experiments cover both either Dirichlet (Example 1) or Neumann (Example 2) boundary conditions The numerical results prove the efficiency of the theoretical suggestion, i.e regularized solutions stably converge to the exact solution As shown in the numerical experiments, for the exact data, the convergence of the numerical solution is dependent on the grid resolution However, for the disturbed data, generally in all cases once q tends to zero and its value is still greater than or equal to the value e, the numerical solution is still converged, but afterward when q < e the numerical solution is starting to diverge Therefore, in order to make sure the numerical solution to be stable and converged, the value of q should be chosen as the same order of e This result has been confirmed by Theorem As a result, we can use the value of q (as qconv ) to estimate the accuracy of an observed data set (e) The numerical calculation of problems (1)–(4) in 3D is time consuming, particularly to deal with higher value of the wave number k, we need to refine a computational grid enough to resolve a short wave length, consequently a large number of grid points must be used Therefore, the code of numerical calculation has been parallelized by OpenMP [23] in Fortran90 Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 20 Q.V Tran et al / Applied Mathematical Modelling xxx (2014) xxx–xxx In fact, it should be stated that our regularized solutions methods are based on the Fourier series expression of solution, which may lead a limitation of the methods for applications in a complicated domain where a solution cannot be expressed by a certain series This issue will be surveyed in a further research Acknowledgments The authors gratefully acknowledge the financial support from the Advanced Research Center for River Operation and Management (ARCROM), Korea and the Institute for Computational Science and Technology at Ho Chi Minh City, Vietnam The authors also would like to thank the anonymous reviewers for their very valuable constructive comments to improve our manuscript References [1] H Jol, D Smith, Ground penetrating radar of Northern Lacustrine delta, Can J Earth Sci 28 (1991) 1939–1947 [2] G Hogan, Migration of ground penetrating radar 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Electron J Differ Eqs 211 (106) (2011) 10pp [20] Dietrich Braess, Finite Elements, third ed., Cambridge University Press, Cambridge, 2007 [21] W.H Press et al, Numerical Recipes in Fortran 90, second ed., Cambridge University Press, New York, 1996 [22] L.N.G Filon, On a quadrature formula for trigonometric integrals, Proc R Soc Edinburgh 49 (1929) 38–47 [23] The OpenMP, API specification for parallel programming Please cite this article in press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell (2014), http://dx.doi.org/10.1016/ j.apm.2014.03.001 ... press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math... press as: Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math... Q.V Tran et al., A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: Theory and numerical simulation, Appl Math Modell