A Radial Basis Function Method for Solving PDE Constrained Optimization Problems

Transcription

1 Report no. /6 A Radial Basis Fnction Method for Solving PDE Constrained Optimization Problems John W. Pearson In this article, we appl the theor of meshfree methods to the problem of PDE constrained optimization. We derive new collocation-tpe methods to solve the distribted control problem with Dirichlet bondar conditions and the Nemann bondar control problem, both involving Poisson s eqation. We prove reslts concerning invertibilit of the matrix sstems we generate, and discss a modification to garantee invertibilit. We implement these methods sing Matlab, and prodce nmerical reslts to demonstrate the methods capabilit. We also comment on the methods effectiveness in comparison to the widel-sed finite element formlation of the problem, and make some recommendations as to how this work ma be extended. Ke words and phrases: Radial basis fnctions; PDE constrained optimization; Poisson control; collocation method. Oxford Universit Mathematical Institte Nmerical Analsis Grop St Giles Oxford, England OX 3LB john.pearson@maths.ox.ac.k Jl, 2

2 2 J. W. PEARSON Introdction A fast-developing area of research is the application of meshfree methods, sing radial basis fnctions (RBFs), to find the nmerical soltion of partial differential eqations (PDEs) [3, 4, 5, 2]. A starting point for man sch pieces of work is Poisson s eqation [, 6, 3]. In this article, we extend this research to an area of considerable recent attention, that of PDE constrained optimization (see for example [5, 6, 7, 8]). We investigate the applicabilit of RBF collocation methods to the problem of Poisson control (that is, optimizing a qantit with Poisson s eqation as a constraint), which is probabl the most commonl investigated PDE constrained optimization problem. When Poisson control is investigated however, the method sed is generall that of appling a finite element method (FEM) to it. To or knowledge, the application of an RBF method to this problem has not been ndertaken. We comment on the soltion of a nmber of problems sing this method with Gassian kernels, as well as the effect of altering the RBF parameter, the nmber of RBFs sed, and the penalt constant (which we will define in Section 3.) in the control problem. In Section 2, we give a brief overview of research that has been ndertaken in the field of nmerical soltion of PDEs, in particlar Poisson s eqation, sing RBF collocation methods. In Section 3, we then derive a method to solve problems involving the distribted control of Poisson s eqation with Dirichlet bondar conditions and the Nemann bondar control of Poisson s eqation. This involves sing a Lagrange mltiplier approach with a similar collocation strateg as one cold se to solve the PDE itself. In Section 5, we test or method on a nmber of problems to illstrate its effectiveness. We will also comment on the method s performance when parameters and the penalt constant are varied. 2 Meshfree soltion of Poisson s eqation Recentl, the application of meshfree methods to solve PDE problems has become a topic of mch research. Literatre sch as [, 4, 5, 6] motivate and derive sch methods, and appl them to Poisson s eqation. In [3], three collocation strategies are discssed for this problem. The first is straight collocation, where the soltion is taken to be a straightforward sm of RBFs mltiplied b coefficients, which are fond b solving a matrix sstem. The centres of the RBFs are taken to be inside the domain where the soltion is soght, with some on the bondar. The second method is smmetric collocation, where one modifies the basis fnction in the interpolant b inclding the operator involved in the PDE being solved. The third method, that of direct collocation, is similar to the first, bt also ses the PDE on the bondar, and takes the centres of some of the RBFs to be otside the domain. In [9], convergence proofs and error estimates are shown for meshless Galerkin methods for a class of PDEs of which Poisson s eqation is an example, and it is shown that the error bonds obtained in the energ norm are the same as for a finite element approach. Another sefl contribtion is the discssion of Domain Decomposition Methods

3 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION 3 for preconditioning sch problems, as discssed in [4]. Meshfree methods have also been researched for solving other PDEs; for example in [3], a meshless method for solving the stead state Navier-Stokes eqations has been investigated. In [2], amongst a nmber of other isses, a soltion procedre for hperbolic and parabolic PDEs is also discssed. If one wishes to solve Poisson s eqation within a domain Ω sing straight collocation, one takes n RBFs with centres at the interior points ξ j Ω interior, j =,..., n and n RBFs with centres at ξ j Ω, j = n +,..., n + n. We denote the RBF centred at ξ j as φ j. Then we approximate or soltion variable b n+n i= Y i φ i, where Y i are coefficients to be fond. Each RBF takes the form φ = φ(r), with r = x ξ, where x denotes the position vector and ξ denotes the centre of the RBF. Now sppose that we wish to solve Poisson s eqation with Dirichlet bondar conditions, that is 2 = g in Ω, D = f on Ω, in this wa. Then the matrix sstem that we obtain from straight collocation is [ ] [ ] A g [] =, (2.) f where denotes the vector of coefficients Y i, i =,..., n + n, and A, D, g and f are given b A = {a ij } i=,...,n, j=,...,n+n, a ij = 2 φ j x=ξi, D = {d,ij } i=,...,n, j=,...,n+n, d,ij = φ j x=ξn+i, g = {g i } i=,...,n, g i = g(ξ i ) f = {f i } i=,...,n, f i = f(ξ n+i ), with 2, as in the rest of this paper, denoting the Laplacian with respect to the position vector x. On the other hand, if we are solving Poisson s eqation with Nemann bondar conditions, that is 2 = g in Ω, = h on Ω, n in this wa, then the reslting matrix sstem that is [ ] [ ] A g [] =, (2.2) h where A and g are as above, and N and h are given b N N = {n,ij } i=,...,n, j=,...,n+n, n,ij = φ j n h = {h i } i=,...,n, h i = h(ξ n+i )., x=ξn+i

4 4 J. W. PEARSON Using direct collocation to solve these sstems will ield similar sstems for the Dirichlet and Nemann problems. When sing smmetric collocation, we instead approximate as n i= Y i 2 ξ φ i+ n+n i=n+ Y iφ i (where ξ denotes the Laplacian with respect to ξ). When solving the Dirichlet problem for instance, this will ield the following matrix sstem: where h and f are as above, and with [ ] [ à h [] = D f à = [ à à 2 ], D = [ D D2 ], ], (2.3) à = {ã,ij } i=,...,n, j=,...,n, ã,ij = 2 ξ( 2 φ j ) x=ξi, à 2 = {ã 2,ij } i=,...,n, j=,...,n, ã 2,ij = 2 φ j x=ξn+i, D = { d,ij } i=,...,n, j=,...,n, d,ij = 2 ξφ j x=ξi, D 2 = { d 2,ij } i=,...,n, j=,...,n, d2,ij = φ j x=ξn+i. It shold be noted that in general, more mesh points are needed when sing smmetric collocation to obtain the same accrac as when sing straight collocation [8, 9]. However, when sing the smmetric collocation method, invertibilit is garanteed for all configrations of (distinct) centres of RBFs [3, 2]. We will find that man of the matrices that appear when deriving the model for solving Poisson s eqation sing an RBF method will also appear when solving the Poisson control problems discssed in Section 3. The methods we derive in Section 3 will be based on the straight collocation and smmetric collocation methods. 3 Extension to Poisson control In this Section, we explain how we extend the theor of solving Poisson s eqation sing a collocation method to solving the problem of Poisson control, that is minimizing a fnctional sch that Poisson s eqation holds (an example of a PDE constrained optimization problem). 3. Formlation of the distribted control problem The problem that we examine is of the form min, 2 ŷ 2 L 2 (Ω) + β 2 2 L 2 (Ω) (3.)

5 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION 5 sch that 2 = in Ω, = f on Ω, for some penalt constant β >. Throghot this work, we take Ω = [, ] 2. The vale of β is important; if β is small, then being far awa from ŷ is penalized heavil, whereas if β is large, it is more important that has small L 2 (Ω)-norm. In this Section, we consider a method based on straight collocation as defined in Section 2; a method based on smmetric collocation is given in Section 3.2. We define here to be the state and to be the control. The fact that we are controlling inside the entire domain Ω makes this a distribted control problem. Or theor will be extended in Section 3.3 to a bondar control problem, that is a problem where we cold control onl over the bondar of or domain Ω. We first trn to interpreting this problem as a matrix sstem of eqations. To do this, we discretize, as h, h, and introdce Lagrange mltipliers λ, µ, χ, which we discretize as λ h, µ h and χ h. We collocate or variables as follows: λ h = h = n+n i= n Λ i φ i, µ h = i= Y i φ i, h = n+n i=n+ n+n i= U i φ i, M i φ i, χ h = n+n i=n+ X i φ i. where φ,..., φ n denote trial fnctions whose centres are located inside the domain, and φ n+,..., φ n+n denote trial fnctions whose centres are on the bondar. These φ i are defined in terms of radial basis fnctions. We now define the vectors,, λ, µ and χ as = { i } i=,...,n+n, = { i } i=,...,n+n, λ = {λ i } i=,...,n, µ = {µ i } i=,...,n, χ = {χ i } i=,...,n, at which point we can write or discretized problem as sch that min h, h 2 h ŷ 2 L 2 (Ω) + β 2 h 2 L 2 (Ω) 2 h = h in Ω, with h attaining the same vales as f at all the radial basis fnction centres on the bondar.

6 6 J. W. PEARSON as Now, in the L 2 -norm, the qantities 2 h ŷ 2 L 2 (Ω) and β 2 h 2 L 2 (Ω) 2 h ŷ 2 L 2 (Ω) = 2 = 2 Ω Ω ( h ŷ) 2 dω hdω 2 h ŷdω + Ω 2 = 2 T M d T + C, β 2 h 2 L 2 (Ω) = β 2 2 hdω Ω = β 2 T M, Ω ŷ 2 dω can be written where M and M are mass matrices (which are eqal de to the fact that and are discretized in the same wa in or formlation) defined as M = {m ij } i,j=,...,n+n, m ij = φ i φ j dω, d is defined as d = {d i } i=,...,n+n, d i = Ω Ω ŷφ i dω, and C is a constant, the vale of which we will see to be nimportant. We se a straight collocation method similar to the method sed for solving the PDE itself (as discssed in Section 2), to obtain three eqations as shown below. The first garantees that Poisson s eqation is solved at ever point corresponding to a centre of an RBF, the second ensres that the Dirichlet bondar conditions are satisfied, and the third ensres that the control is set to zero on the bondar (as for these problems we onl wish to control the interior of the domain Ω). where A + D 2 =, D = f, D =, A = {a ij } i=,...,n, j=,...,n+n, a ij = 2 φ j x=ξi, D = {d,ij } i=,...,n, j=,...,n+n, d,ij = φ j x=ξn+i, D 2 = {d 2,ij } i=,...,n, j=,...,n+n, d 2,ij = φ j x=ξi, f = {f i } i=,...,n, f i = f(x n+i ). As before, ξ i denotes the centre of the i-th RBF we are considering, and f is the vector containing the bondar vales (i.e. the specified vales at nodes x n+,..., x n+n.

7 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION 7 We are therefore now left with the following Lagrangian L = L(,, λ, µ, χ), of which we wish to find the stationar point: L = 2 T M d T + C + β 2 T M + λ T (A + D 2 ) + µ T (D g) + χ T (D ), where λ, µ and χ denote the vectors of the relevant coefficients for or three Lagrange mltipliers. Now, differentiating L with respect to,, λ, µ and χ and setting the partial derivatives to be zero gives s the following eqations which mst be satisfied: : M + A T λ + D T µ = d, : βm + D2 T λ + D T χ =, λ : A + D 2 =, µ : D = f, χ : D =. Finall, combining these 5 eqations gives the sstem that we wish to solve: M A D T βm D2 T D T A D 2 D D λ µ χ = d f. (3.2) The sstem that we obtain is smmetric, and frthermore is of saddle point strctre, which is garanteed if we se the above discretize-then-optimize formlation. This is sefl for proving the conditions for invertibilit of this sstem, as in Section An alternative expansion in terms of basis fnctions In this Section, we discss an alternative method for solving (3.), motivated b the fact that when solving Poisson s eqation sing smmetric collocation, one is garanteed to be solving an invertible matrix sstem, whatever the configration of RBF centres. The conseqence of this with regard to the distribted control problem is given in Theorem 3, thogh this methodolog cold also be extended to Nemann bondar control problems.

8 8 J. W. PEARSON To carr ot this method, we take n n+n h = Y i 2 ξφ i + Y i φ i, h = i= λ h = i=n+ n Λ i 2 ξφ i, µ h = i= n+n i=n+ n U i 2 ξφ i + i= M i φ i, χ h = n+n i=n+ n+n i=n+ X i φ i. U i φ i, Using the same method as in Section 3., we obtain the matrix sstem M Ã T DT β M d DT 2 DT Ã D2 λ D µ = g, (3.3) D χ where d and f are as defined in Section 3., and [ ] M = M M M2 =, M 2 M22 Ã = [ Ã Ã 2 ], D = [ D D2 ], D 2 = [ D2 D22 ], with M = { m,ij } i=,...,n, j=,...,n, m,ij = M 2 = { m 2,ij } i=,...,n, j=,...,n, m 2,ij = M 2 = { m 2,ij } i=,...,n, j=,...,n, m 2,ij = M 22 = { m 22,ij } i=,...,n, j=,...,n, m 22,ij = Ω Ω Ω Ω 2 φ i 2 φ j dω, 2 φ i φ n+j dω, φ n+i 2 φ j dω, φ n+i φ n+j dω, Ã = {ã,ij } i=,...,n, j=,...,n, ã,ij = 2 ( 2 ξφ j ) x=ξi, Ã 2 = {ã 2,ij } i=,...,n, j=,...,n, ã 2,ij = 2 φ j x=ξn+i, D = { d,ij } i=,...,n, j=,...,n, d,ij = 2 ξφ j x=ξi, D 2 = { d 2,ij } i=,...,n, j=,...,n, d2,ij = φ j x=ξn+i, D 2 = { d 2,ij } i=,...,n, j=,...,n, d2,ij = 2 ξφ j x=ξi, D 22 = { d 22,ij } i=,...,n, j=,...,n, d22,ij = φ j x=ξn+i. For the remainder of this paper, we will be working with the simpler formlation derived in Section 3., along with eqall-spaced RBF centres. However, the sstem derived in this Section is sefl, as we prove in Section 4 that (3.3) is invertible for an configration of RBF centres (provided the locations of the centres are distinct).

9 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION RBF soltion of Nemann bondar control problem In this Section, we describe a method (based on the straight collocation method for solving Poisson s eqation with Nemann bondar conditions) to solve the Nemann bondar control problem sch that min, 2 ŷ 2 L 2 (Ω) + β 2 2 L 2 ( Ω) 2 = g in Ω, n = on Ω. As well as sing φ i to denote a basis of RBFs defined within Ω, we now take ψ i, i =,..., n, to be a new basis of RBFs defined solel on the bondar Ω. Then writing h = n+n i= n Y i φ i, h = U i ψ i, λ h = i= n n Λ i φ i, µ h = M i ψ i, i= i= and proceeding b the same method as in Section 3.2 ields the following matrix sstem: M A T N T d β ˆM N2 T A λ = h, (3.4) N N 2 µ where M, A and d are as defined in Section 3., and ˆM = { ˆm ij} i=,...,n, j=,...,n, ˆm ij = N = {n,ij } i=,...,n, j=,...,n+n, Ω n,ij = φ j n ψ i ψ j ds,, x=ξn+i N 2 = {n 2,ij } i,j=,...,n, n 2,ij = ψ j x=ξn+i = d,ij, g = {g i } i=,...,n, g i = g(ξ n+j ). Theorem 4 in Section 4 gives a reslt on the invertibilit of this matrix sstem. 3.4 Implementation aspects When implementing the methods we have derived in Sections 3. and 3.3, we need to consider a nmber of things. First of all, we need to choose what radial basis fnctions we will se in or methods. A nmber of widel sed RBFs are discssed in literatre sch as [4, 7, 3]. Throghot this work, we will se Gassians (which are basis fnctions

10 J. W. PEARSON of the form φ = φ(r) = e cr2 ), althogh or theor cold be extended to an positive definite RBF. We will var the vale of the coefficient c in the Gassians; as we will demonstrate in Section 5, it is not a trivial problem to choose the appropriate vale of c to obtain an accrate soltion to the matrix sstems previosl derived. The choice of the so called shape parameter c, when it appears, is likel to be important, as this featre is observed when solving Poisson s eqation itself [, 3]. Large vales of c reslt in a well-conditioned problem bt poor convergence, whereas small vales of c reslt in a good approximation, bt bad conditioning. We will therefore var the vales of c in or nmerical tests; all plots shown in Section 5 are prodced with vales of c seen to be appropriate for the particlar problem. We mst also choose the centres of the RBFs (which will be needed to constrct the matrices and vectors in (3.2)), and note their locations. We will proceed sing randoml generated centres inside the domain and along the bondar for simplicit. When constrcting the matrices as defined in Sections 3. and 3.3, we note a cople of things. First, the matrices M (and hence M if we se the same nmber of basis fnctions to represent and ) and ˆM are clearl smmetric. This will save some comptational time when constrcting the matrix sstem (3.2). When constrcting the mass matrices, we will need an effective qadratre rle: we find that sing Gass- Legendre qadratre with arond 5 qadratre points in each direction is sfficient when Gassians are sed (note that in this case, we ma separate the terms along the individal axes, leaving effectivel two 5 point qadratre rles, rather than one 25 point rle, for a 2D problem). Finall, we note that d will clearl depend on or choice of ŷ, and f or g will depend on what we choose the bondar terms to be this will need to be taken into accont when implementing or method. We solve the matrix sstem we have constrcted sing backslash in Matlab, and perform standard a posteriori tests to verif the accrac of or soltions. To test or method, we consider for specific problems, listed below. Problems, 2 and 3 are all distribted control problems with Dirichlet bondar conditions, which we will solve sing the method derived in Section 3., and Problem 4 is a Nemann bondar control problem, which we will solve sing the method discssed in Section 3.3. The problems are all in two dimensions on a domain Ω = [, ] 2 for illstrative prposes, bt or theor can be easil extended to solving three dimensional problems. In the problems as stated, x = [x, ] T denote the coordinates we are working with. Problem : We wish to solve the distribted Poisson control problem min, 2 ŷ 2 L 2 (Ω) + β 2 2 L 2 (Ω) sch that 2 =, x Ω, = f, x Ω,

11 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION where Ω = [, ] 2, and ŷ = x ( x ) ( ), x Ω, f =. Problem 2: We consider the same formlation as Problem, except that { if x Ω, ŷ = if x Ω 2, { if x Ω := Ω Ω f =, if x Ω 2 := Ω Ω 2, Ω = [, 2] 2, Ω2 = Ω\Ω. This problem was previosl considered sing an FEM method in [7]. Problem 3: We again consider the same formlation as Problem, bt with { if x Ω, ŷ = if x Ω 2, f =, Ω = [, 2] 2, Ω2 = Ω\Ω. This problem was previosl considered sing an FEM method in [5]. Problem 4: We wish to solve the Nemann bondar control problem sch that where Ω = [, ] 2, and min, 2 ŷ 2 L 2 (Ω) + β 2 2 L 2 ( Ω) 2 = g, x Ω, n =, x Ω, ŷ = x ( x ) ( ), x Ω, g = g(x, ) = x. Problems, 2 and 3 are variants on the Poisson distribted control problem, with different bondar conditions and target fnctions, and, as stated above, Problem 4 is an example of the Nemann bondar control problem. Problems 2 and 3 are interesting ones specificall becase we cannot exactl attain or target soltion, no matter how

12 2 J. W. PEARSON x x.5 (a) State (b) Control Figre : Plot of finite element soltion (interior points onl) to optimal control Problem 3 sing Q-Q approximation, step size h = 2 5 and β = 6, and the control needed to obtain this soltion x.5.5 x.5 (a) Problems & 4 (b) Problems 2 & 3 Figre 2: Plots of target soltion = ŷ for Poisson control Problems and 4 (left) and Problems 2 and 3 (right). small we set β. The reason for this is that we have specified that mst be eqal to on all of Ω, and so the target ŷ = on [, 2] 2 cannot be reached. Conseqentl, for small β, the control ma exhibit a lack of smoothness as gets closer to ŷ. Figre illstrates a finite element soltion we obtain for the state when β = 6, and the non-smooth control that we reqire for this to happen. Figre 2 shows the targets ŷ that we hope the state to be as close to as possible, for each of or 4 problems. 4 Solvabilit of matrix sstems In order to garantee the existence of soltions to PDE constrained optimization problems sing or methods, it is necessar to obtain reslts on the invertibilit of the matrix sstems derived in Section 3. We therefore prove a nmber of sch reslts in this Section. Theorem is a general reslt from saddle point theor, see [2]. We will se this reslt to prove conditions for invertibilit of the matrix sstems that we have derived in Section 3. Theorem 2 gives a reslt concerning the invertibilit of the matrix sstem (3.2) for the soltion of distribted control problems in terms of the invertibilit of the

13 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION 3 matrix sstem (2.), that is the matrix sstem generated b solving Poisson s eqation with Dirichlet bondar conditions sing straight collocation. Theorem 3 proves the nconditional solvabilit of the alternative matrix sstem (3.3) for distribted control problems, sing the fact that the matrix sstem (2.3) is invertible, provided that all RBF centres are distinct [3, 2]. Theorem 4 gives a similar reslt as Theorem 2, this time concerning the invertibilit of (3.4), the matrix sstem for Nemann bondar control problems, in terms of the invertibilit of (2.2), the matrix sstem generated b solving Poisson s eqation with Nemann bondar conditions. [ ] Φ Ψ T Theorem An saddle point sstem of the form, where Φ is an m m Ψ matrix and Ψ is an n m matrix with m n, is invertible provided that Φ is invertible and the Schr complement ΨΦ Ψ T is invertible. Theorem 2 The matrix A given b M A T D T βm D2 T D T A = A D 2 D D (that is the matrix corresponding to the [ Poisson ] distribted control problem as described A in Section 3.) is invertible, provided is invertible. D [ ] C E T Proof. The matrix A ma be written as a saddle point sstem A =, where E [ ] A D M 2 C = and E = D βm. Now, becase of Theorem, we simpl D need to show that C and (EC E T ) exist. [ The fact that ] mass matrices are M invertible means that we ma write C = M to prove the first part. β We prove that EC E T is invertible b showing that it is positive definite. To do this, we note that, for an v R n+2n, v T EC E T v = ( C /2 E T v ) T ( C /2 E T v ), and so v T EC E T v > nless C /2 E T v =, which occrs if and onl if E T v = as C /2 is invertible. Now, writing v = [v v 2 v 3 ] T, where v R n, v 2 R n, v3 R n, we obtain E T v = [ A D ] T [ v v 2 ] = [ ], D T 2 v + D T v 3 =.

14 4 J. W. PEARSON [ ] A B the invertibilit of (and hence also the fact that D D has fll rank), we have [ ] [ ] v from the first of the above eqations that =, from which it follows from v 2 the second eqation that v 3 =. We therefore obtain that E T v = if and onl if v =, and so EC E T is invertible. Theorem 3 The matrix is invertible. M à T DT β M DT 2 DT à D2 D D Proof. This can be proved [ in the ] same wa as Theorem 2, sing Theorem, as well à the fact that the matrix is alwas invertible in the case of smmetric D collocation provided that all RBF centres are distinct. Theorem 4 The matrix is invertible, provided [ A N M A T N T β ˆM N T 2 A N N 2 ], the (straight) collocation matrix corresponding to Poisson s eqation with Nemann bondar conditions, is invertible. Proof. This can also be proved in a similar wa as Theorem 2 sing Theorem. In this Section we have therefore shown that if one proves the invertibilit of a matrix sstem for solving Poisson s eqation (with Dirichlet or Nemann bondar conditions) sing a particlar arra of nodes, one has atomaticall proved the invertibilit of the eqivalent PDE constrained optimization problem (that is the distribted conrol problem and the Nemann bondar control problem respectivel). The invertibilit of matrix sstems for solving Poisson s eqation will hold almost alwas, bt conterexamples can be constrcted [, 3].

15 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION x.5.5 x.5 (a) State, β = 3 (b) Control, β = x.5.5 x.5 (c) State, β = 5 (d) Control, β = 5 Figre 3: Diagrams of the RBF soltions to the distribted Poisson control Problem for the state and control, for β = 3 and 5 with c = 5 and 3 respectivel. Here, we took h = Nmerical reslts In this Section we present nmerical reslts, all of which were generated sing eqallspaced points. Figres 3, 4, 5 and 6 show soltions obtained for state and control sing or RBF methods with β = 3 and β = 5, for the distribted control problems (Figres 3 5) and Nemann bondar control problem (Figre 6) stated in Section 3.4. The figres exhibit man featres which we wold expect of soltions to a PDE constrained optimization problem. The state is close to the target fnction ŷ for each problem (as shown in Figre 2), and it is even closer to the target fnction if we set β to be smaller. In Figre 5 in particlar, or control exhibits ver similar non-smoothness near Ω for small β, as we saw for the finite element soltion in Figre. We observe that the soltions we obtained seemed to depend on the vale of c we sed in or Gassians. We therefore seek to analze how the vales of c, as well as β, affected the vales of ŷ L2 (Ω), L 2 (Ω) and 2 ŷ 2 L 2 (Ω) + β 2 2 L 2 (Ω)(the fnctional or method has attempted to minimize). As or method merel generates the vales of the state and control at each RBF centre, it is difficlt to evalate an of these fnctionals exactl, as the represent integrals. We find that if we approximate ŷ L2 (Ω) b the l 2 -norm of the vector of the difference between the RBF soltion for and the target

16 6 J. W. PEARSON x x.5 (a) State, β = 3 (b) Control, β = x x.5 (c) State, β = 5 (d) Control, β = 5 Figre 4: Diagrams of the RBF soltions to the distribted Poisson control Problem 2 for the state and control, for β = 3 and 5 with c = 25 in each case. Here, we took h = 2 5. soltion ŷ at each point (denoted b ŷ l2 (Ω) ), and approximate L 2 (Ω) b the l 2 -norm of the vector of the RBF soltion of the control (denoted b l2 (Ω)), these give an illstration of or sccess in minimizing 2 ŷ 2 L 2 (Ω) + β 2 2 L 2 (Ω). Note that these vales will clearl depend on h, bt we will keep this vale constant when we carr ot this analsis. In Table, we analze the vales of ŷ l2 (Ω), l 2 (Ω) and 2 ŷ 2 l 2 (Ω) + β 2 2 l 2 (Ω) for varing β. As we wold expect, at first, as β gets smaller, the state gets rapidl closer to the target fnction ŷ, and althogh l2 (Ω) does become mch larger, or approximation of the fnctional we wish to minimize, 2 ŷ 2 L 2 (Ω) + β 2 2 L 2 (Ω), gets smaller and smaller. In Table 2, we considered a specific case (namel Problem 3 with h =.4 and β = 4 ), and again analzed the vales of ŷ l2 (Ω), l 2 (Ω) and 2 ŷ 2 l 2 (Ω) + β 2 2 l 2 (Ω), bt this time varing the vale of the parameter c. The effects seemed to be similar as for Poisson s eqation itself; taking a vale of c that is too large reslts in convergence being a problem, bt a vale of c that is too small reslts in ill-conditioning. In Table 2, we observe that the middle grond of vales of c (for this case roghl c [2, 5]) generate ver similar reslts the reslts still var slightl as h is still relativel large, and the soltion ma not have converged to man digits. We recommend

17 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION x.5.5 x.5 (a) State, β = 3 (b) Control, β = x x.5 (c) State, β = 5 (d) Control, β = 5 Figre 5: Diagrams of the RBF soltions to the distribted Poisson control Problem 3 for the state and control, for β = 3 and 5 with c = 25 and 3 respectivel. Here, we took h = 2 5. taking c to be roghl within this interval, althogh the ideal vales ma var if h is changed, for example when considering the problems for which the soltion is plotted in Figres 3, 4, 5 and 6, or if β is varied, as in Table. We conclde that choosing the vale of c appropriatel is important when considering or methods for solving Poisson control problems. In Table 3, we displa the condition nmbers for the matrix in (3.2) for h =.4 and a variet of vales of c and β; the reslts demonstrate the ill-conditioning for the problem when sing small vales of c. The case h =.4 is an interesting one, as it is a case for which the condition nmber can be above or below 6 ; if the condition nmber is above this nmber, one cannot expect to obtain an digits of accrac when implementing the method in doble precision arithmetic. The conditioning also becomes more of a problem as h is decreased, so it wold be desirable to find optimal choices of c for an h and β. Providing the vale of c is chosen appropriatel, we can conclde that or methods are powerfl ones for solving the Poisson control problems that we have considered. [For frther details abot Poisson control problems and the featres of FEM soltions, see for example [5, 6, 7].] We compare the FEM method and the RBF collocation method we have presented here in Section 6.

18 8 J. W. PEARSON x x.5 (a) State, β = 3 (b) Control, β = x x.5 (c) State, β = 5 (d) Control, β = 5 Figre 6: Diagrams of the RBF soltions to the Nemann bondar control Problem 4 for the state and control, for β = 3 and 5 with c = 5 in each case. Here, we took h = Conclding remarks In this article, we first reviewed some research that has been ndertaken to solve Poisson s eqation sing meshfree methods, and derived sstems of eqations that are commonl sed for Dirichlet and Nemann bondar vale problems. We then motivated, derived, implemented and tested a new radial basis fnction method for solving optimal control problems involving Poisson s eqation, and proved a nmber of solvabilit reslts relating to the reslting matrix sstems. The problems considered were the distribted control problem with Dirichlet bondar conditions and the Nemann bondar control problem. We fond that this method was a powerfl one for each of these formlations. There are clear advantages to sing an RBF method for sch problems over the more commonl sed finite element method. Most notabl, it is not necessar to constrct a mesh for this method, which saves a considerable amont of comptational work and memor. If one did wish to select centres for the RBFs, there wold be no restriction on where these centres cold be (provided the did not coincide). There are also good direct methods for small matrix sstems bilt into programs sch as Matlab, so if we wished to compte an RBF soltion to this problem sing a relativel small nmber of fnctions, this method wold be ideal. In addition, the soltions obtained are infinitel smooth, provided appropriate RBFs are sed in the method, so if a high degree of

19 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION 9 smoothness were reqired, this wold be a method of choice. However, problems arise when we wish to compte a (highl accrate) soltion to the problem sing man RBFs, reslting in a large matrix sstem. As shown in Figre 7, the reslting matrix sstem for sch a meshfree method is extremel dense, which makes the sparse matrix sstem obtained when an FEM is sed preferable, as there have been powerfl iterative methods developed for sch sstems [5, 6, 7]. Compting the mass matrices in (3.2) is also an expensive procedre when creating the global matrix sstem, althogh we wold not expect the comptation time involved to be greater than the time taken to solve the reslting sstem. Frthermore, ill-conditioning of the matrix sstem is a significant drawback for large sstems in particlar. It is possible however that this isse, as well as the denseness of the sstem, cold be resolved b sing compactl spported radial basis fnctions nz=94 (a) FEM nz= (b) RBF Figre 7: sp plots in Matlab for the matrix sstem needed to be solved for the distribted Poisson control problem sing a Q-Q mixed FEM, and for the matrix sstem (exclding entries which have absolte vale less than 5 ) sing the RBF method of Section 3. sing Gassians with c = 25 to solve the same problem. In each case h = 2 4 and β = 4. There is mch more investigation that cold be ndertaken based on this work. For example, other tpes of RBFs cold be tested on these problems, in particlar compactl spported RBFs. This methodolog cold also be extended to other optimal control problems: convection-diffsion control, Stokes control and Navier-Stokes control to name bt a few. It is also desirable to provide a theoretical framework for this method, proving, for example, nmerical stabilit and error bonds, as well as theoretical reslts concerning the optimal choices of parameters involved in the method (in particlar the optimal choice of c for an given vales of h and β to take when basing the method on 2 Gassians of the form e cr ). Frthermore, it wold be helpfl to find effective solvers to the sstem of eqations that this problem leads to, inclding efficient preconditioning techniqes. Soltions to an of these problems wold be highl desirable in order to develop this new and potentiall ver sefl application of radial basis fnction theor. Acknowledgements. The athor wold like to express his gratitde to Holger Wendland for invalable help and advice while prodcing this manscript. He wold also like to thank And Wathen and Mark Richardson for helpfl discssions abot this

20 2 J. W. PEARSON work. The athor acknowledges financial spport from the Engineering and Phsical Sciences Research Concil (EPSRC). References [] Aminataei, A., Mazarei, M.M.: Nmerical Soltion of Poisson s Eqation Using Radial Basis Fnction Networks on the Polar Coordinate. Compters and Mathematics with Applications 56, (28) [2] Benzi, M., Golb, G.H., Liesen, J.: Nmerical Soltion of Saddle Point Problems. Acta Nmerica 4, 37 (25) [3] Chinchapatnam, P.P., Djidjeli, K., Nair, P.B.: Radial Basis Fnction Meshless Method for the Stead Incompressible Navier-Stokes Eqations. International Jornal of Compter Mathematics 84, (27) [4] Dan, Y.: A Note on the Meshless Method Using Radial Basis Fnctions. Compters and Mathematics with Applications 55, (28) [5] Dan, Y., Tan, Y.: A Meshless Galerkin Method for Dirichlet Problems Using Radial Basis Fnctions. Jornal of Comptational and Applied Mathematics 96, (26) [6] Elansari, M., Oazar, D., Cheng, A.H.: Bondar Soltion of Poisson s Eqation Using Radial Basis Fnction Collocated on Gassian Qadratre Nodes. Commnications in Nmerical Methods in Engineering 7, (2) [7] Fasshaer, G.E.: Meshfree Approximation Methods with MATLAB. Interdisciplinar Mathematical Series Vol. 6, World Scientific (27) [8] Franke, C., Schaback, R.: Convergence Order Estimates of Meshless Collocation Methods Using Radial Basis Fnctions. Advances in Comptational Mathematics 8, (998) [9] Giesl, P., Wendland, H.: Meshless Collocation: Error Estimates with Application to Dnamical Sstems. SIAM Jornal on Nmerical Analsis 45, (27) [] Hon, Y.C., Schaback, R.: On Unsmmetric Collocation b Radial Basis Fnctions. Applied Mathematics and Comptation 9, (2) [] Hang, C.-S., Yen, H.-D., Cheng, A.H.-D.: On the Increasingl Flat Radial Basis Fnction and Optimal Shape Parameter for the Soltion of Elliptic PDEs. Engineering Analsis with Bondar Elements 34, (2) [2] Kansa, E.J.: Motivation for Using Radial Basis Fnctions to Solve PDEs. Technical Report, available at (999)

21 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION 2 [3] Larsson, E., Fornberg, B.: A Nmerical Std of Some Radial Basis Fnction Based Soltion Methods for Elliptic PDEs. Compters and Mathematics with Applications 46, (23) [4] Ling, L., Kansa, E.J.: Preconditioning for Radial Basis Fnctions with Domain Decomposition Methods. International Workshop on MeshFree Methods (23) [5] Pearson, J.W., Wathen, A.J.: A New Approximation of the Schr Complement in Preconditioners for PDE Constrained Optimization. Sbmitted to Nmerical Linear Algebra with Applications (2) [6] Rees, T.: Preconditioning Iterative Methods for PDE Constrained Optimization. DPhil Thesis, Universit of Oxford (2) [7] Rees, T., Dollar, H.S., Wathen, A.J.: Optimal solvers for PDE-Constrained Optimization. SIAM Jornal of Scientific Compting 32, (2) [8] Tröltzsch, F.: Optimal Control of Partial Differential Eqations: Theor, Methods and Applications. Gradate Stdies in Mathematics Vol. 2, American Mathematical Societ (2) [9] Wendland, H.: Meshless Galerkin Methods Using Radial Basis Fnctions. Mathematics of Comptation 68, (999) [2] W, Z.M.: Hermite-Birkhoff Interpolation of Scattered Data b Radial Basis Fnctions. Approximation Theor and its Applications 8, (992)

22 22 J. W. PEARSON Problem ŷ l2 l2 2 ŷ 2 l 2 + β 2 2 l β Problem 2 ŷ l2 l2 2 ŷ 2 l 2 + β 2 2 l β Problem 3 ŷ l2 l2 2 ŷ 2 l 2 + β 2 2 l β Problem 4 ŷ l2 l2 2 ŷ 2 l 2 + β 2 2 l β Table : Table of soltions to Poisson control problem obtained for Problems, 2, 3 and 4 as detailed in Section 3.4, with RBFs evenl distribted with spacing h = 2 4. Reslts are given for a variet of β, and we took c = 6 for each Problem except for Problem 4, for which we took c = 2.

23 RADIAL BASIS FUNCTIONS IN PDE CONSTRAINED OPTIMIZATION 23 Problem 3 ŷ l2 l2 2 ŷ 2 l 2 + β 2 2 l c Table 2: Table of soltions to Poisson control problem obtained for Problem 3, with RBFs evenl distribted with spacing h =.4, and β = 4. Reslts are given for a variet of vales of c. β c Table 3: Approximate condition nmber of the matrix in (3.2) for the distribted Poisson control problem on [, ] 2, with RBFs evenl distribted with spacing h =.4, for a variet of vales of c and β. Approximation is obtained b se of the Matlab fnction condest.