Research Article Inverse Problems via the (Generalized Collage Theorem) for Vector-Valued Lax-Milgram-Based Variational Problems

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1 Mathematical Problems in Engineering Volume 25, Article ID , 8 pages Research Article Inverse Problems via the (Generalized Collage Theorem) for Vector-Valued Lax-Milgram-Based Variational Problems H Kunze, D La Torre, 2,3 K Levere, and M Ruiz Galán 4 Department of Mathematics and Statistics, University of Guelph, Guelph, O, Canada G 2W 2 Department of Applied Mathematics and Sciences, Khalifa University, PO Box 27788, Abu Dhabi, UAE 3 Department of Economics, Management, and Quantitative Methods, University of Milan, 222 Milan, Italy 4 Department of Applied Mathematics, University of Granada, 87 Granada, Spain Correspondence should be addressed to D La Torre; davidelatorre@kustaracae Received 3 December 24; Revised 5 June 25; Accepted 6 June 25 Academic Editor: Eric Florentin Copyright 25 H Kunze et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We present an extended version of the Generalized Collage Theorem to deal with inverse problems for vector-valued Lax-Milgram systems umerical examples show how the method works in practical cases Inverse Problems via the Collage Theorem In recent years a great deal of attention has been paid to the inverse problems in distributed systems, that is, the determination of unknown parameters in the functional form of the governing model of the phenomenon under study [ 4] The literature is rich in papers studying ad hoc methods to address ill-posed inverse problems by minimizing a suitable approximation error along with utilizing some regularization techniques [5 7] Many inverse problems may be recast as the approximation of a target element x in a complete metric space (X, d) by the fixed point x of a contraction mapping T : X X Thanks to a simple consequence of Banach s fixed point theorem known as the Collage Theorem, most practicalmethodsofsolvingtheinverseproblemforfixed point equations seek an operator T for which the collage distance d(x, Tx) is as small as possible Theorem ( Collage Theorem [8]) Let (X, d) be a complete metric space and T:X Xa contraction mapping with contraction factor c [, )Then,foranyx X, d (x, x) d (x, Tx), () c where x is the fixed point of T This vastly simplifies this type of inverse problem as it is much easier to estimate d(x, Tx) than to find the fixed point x and then compute d(x, x) One now seeks a contraction mapping T thatminimizestheso-called collage error d(x, Tx),in otherwords,amappingthatsendsthetargetx as close as possible to itself This is the essence of the method of collage coding which has been the basis of most, if not all, fractal image coding and compression methods Barnsley [8, 9] was the first to see the potential of using the Collage Theorem above for the purpose of fractal image approximation and fractal image coding In practical applications, from a family of contraction mappings T λ, λ Λ R n,onewishes to find the parameter λ for which the approximation error d(x, x λ ) is as small as possible In practice the feasible set is often taken to be Λ c = λ R n : c λ c < } which guarantees the contractivity of T λ for any λ Λ c A difference between the collage approach and the one based on Tikhonov regularization is the following: in the collage approach, the constraint λ Λ c guarantees that T λ is a contraction and, therefore, replaces the effect of the regularization term in the Tikhonov approach (see [4, 7]) The collage approach works well for low-dimensional parametrization in particular, while Tikhonov regularization is a fundamentally nonparametric methodology The collage-based inverse

2 2 Mathematical Problems in Engineering problem can be formulated as an optimization problem as follows: min λ Λ c d(x,t λ x) (2) This is a nonlinear and nonsmooth optimization model that canoftenbereducedtoaquadraticoptimizationprogram Several algorithms can be used to solve it including penalization methods and particle swarm ant colony techniques This method of collage coding may be applied in other situations where contractive mappings are encountered These ideas have been extended to inverse problems for Initial Value Problems (IVP) in [] In this setting, the contractive Picard operator plays the role of T and the space X contains continuous and appropriately bounded functions on a closed interval of observation Given a target function, perhaps the interpolation of observational data points, the Collage Theorem can be applied to find the Picard operator within a prescribed class that minimizes the collage distance We have applied this technique to inverse problems involving several families of differential equations and application to different areas (see [ 5]) Example 2 We present the results to an IVP inverse problem solved using collage coding, Tikhonov regularization, and the Landweber-Fridman method Consider the following steadystate heat equation on Ω=[, ]: d du (κ (x) (x)) = f(x), in Ω=(, ), dx dx u () =, u () =, where κ(x) isthevariablethermaldiffusivityofthemedium at x Ω, f(x) represents a heat source or sink at each x Ω,and u(x) denotes the temperature at x Ω The inverse problem we look at is to estimate the variable thermal diffusivity κ(x) given r uniformly distributed values of u(x), with low-amplitude Gaussian noise added, and the forcing function f(x) For this example, we assume that u(x) = 4x( x)and that f(x) = 96x 3 + 2x x 8, corresponding to κ(x) = κ True (x) = 3x 3 + 2x 2 + 4x + Morozov s discrepancy principle was used to find the value of the Tikhonov regularization parameter, with our discrepancy below a tolerance level of 8 Table presents the results; thesubscriptsindicatethemethodusedtosolvetheinverse problem In a manner analogous to the Collage Theorem we have also formulated a Generalized Collage Theorem for solving Boundary Value Problems (BVP), replacing the minimization of the true error by the minimization of something akin to the collage distance In place of Banach s fixed point theorem for contraction maps on a complete metric space, we have appealed to the Lax-Milgram representation theorem (3) These results have been extended to a wider class of ellipticequations problems in[6, 7], by considering not only Hilbert but also reflexive Banach spaces and even replacing the primal variational formulation of such a problem with a more general constrained variational one Let us mention that this kind of formulation arises, for instance, when the boundary constraints are weakly imposed Thepaper isorganizedas follows Section2 is concerned with an extension of the Collage Theorem stated in [6, Corollary 42] to a finite-dimensional vector-valued context, as well as a discretization scheme based on the use of suitable Schauder bases Section 3 presents three different numerical examples which show how to solve inverse problems for systems of elliptic differential equations 2 Vector-Valued Lax-Milgram and the Inverse Problem In this section we deal with a Lax-Milgram theorem stated in terms of a system of suitable variational equations and with a collage type result that follows from it We refer to [8 2]for some recent vectorial versions of the Lax-Milgram theorem andsomeapplicationstothestudyofmixedvariational equations The first result is the following vector-valued version of the Lax-Milgram theorem, which is a direct consequence of the characterization of the solvability of systems with infinitely many variational equations given in [22, Theorem 32] specifically of its finite-dimensional case [22,Corollary 46] and of the fact that if, E, F,,F are real vector spaces, a :E F R,,a :E F R are bilinear forms, and y :F R,,y :F R are linear forms such that the system x E with y =a (x, ) y =a (x, ) admits a solution, then such a solution is unique if, and only if, the corresponding homogeneous problem has one and only one solution Given a real normed space G, we write G for its topological dual space Theorem 3 Suppose that E is a real reflexive Banach space,, F,,F are real Banach spaces, and a :E F R,,a :E F R are continuous bilinear forms Then for all y F,,y F there exists a unique x Esuch that y =a (x, ) y =a (x, ) (4) (5)

3 Mathematical Problems in Engineering 3 Table : Recovered diffusivity using collage coding, Tikhonov regularization, and Landweber-Fridman iteration, with r being the number of uniformly distributed sample points in [, ], ε being the amplitude of the relative noise added, and the true value of κ(x) = +4x+2x 2 +3x 3 r ε κ collage (x) κ collage κ True L 2 (Ω) x+27627x x x+27685x x x+2823x x x+27627x x x+27777x x x+28989x x r ε κ Tikhonov (x) κ Tikhonov κ True L 2 (Ω) x 4295x x x 433x x x 43472x x x 9598x x x 95858x x x 9564x x r ε κ Landweber (x) κ Landweber κ True L 2 (Ω) x 4295x x x 42856x x x 424x x x 9598x x x 95758x x x 9449x x if, and only if, y =a (x, ) } x = y =a } (x, ) } and there exists ρ> satisfying (y,,y ) F F ρ k= y k a k (, y k ) k= (6) (7) continuous bilinear forms a λ : E F R,,a λ : E F R and ρ λ > with y =a λ (x, ) } x =, y =a } λ (x, ) } (y,,y ) F F ρ λ y k a kλ (, y k ) k= k= Let us also suppose that, for all λ Λ, x λ Eis the unique solution of the variational system (9) Moreover, if these equivalent conditions hold and x Eis the unique solution, then x ρ max k=,, y k (8) y =a λ (x, ) () The aforementioned generalization of the collage type result [6, Corollary 42] for a finite number of variational equations is stated in these terms Corollary 4 Let E be a real reflexive Banach space, let, let F,,F be real Banach spaces, let y F,,y F, and let Λ be a nonempty set such that for all λ Λthere exist y =a λ (x, ) Then for each x Eand for all λ Λthe inequality x λ x max ρ λ k=,, y k a kλ (x, ) () is valid

4 4 Mathematical Problems in Engineering Proof The unisolvency and continuous dependence on the initial data of the solution in Theorem 3 imply the announced result Indeed, let λ Λand notice that x λ x is a solution of the system of variational equations y a λ (x, )=a λ (x, ) y a λ (x, )=a λ (x, ) Then, in view of Theorem 3,weconcludethat (2) x λ x max ρ λ k=,, y k a kλ (x, ) (3) Let us observe that if one wants to approximate the solution x in the sense of the collage distance, that is, minimize x λ x :λ Λ}, according to Corollary 4, it suffices to minimize max ρ λ k=,, y k a kλ (x, ) :λ Λ}, (4) although if ρ:=inf ρ λ >, (5) λ Λ then we only need to minimize max k=,, y k a kλ (x, ) :λ Λ} (6) Under such an assumption, ρ >, and suppose that each space F k (k =,,)has a Schauder basis Υ ki } i,insuch awaythatifυ ki } i denotes its sequence of biorthogonal functionals, then the nonrestrictive condition M:= max Υ ki < (7) sup k=,, i holds In order to discretize our optimization problem, let us also assume that E admits a Schauder basis Θ i } i and define for each n andk=,, E n := span Θ,,Θ n }, (8) F kn := span Υ k,,υ kn } and let P n be the nth-projection of E onto E n ;thatis,forall P n x:= n i= Θ i (x) Θ i (9) We also suppose that for all λ Λ, k=,,,andn x E n, =a λ (x, ) =a λ (x, ) x =, (2) and there exists ρ n λ > suchthat (y,,y ) F n F n ρ n λ k= y k a kλ (, y k ) k= (2) Then, Theorem 3 guarantees the existence of unique x n j E n such that (y,y ) F n F n y (y )=a (x n j,y ) y (y )=a (x n j,y ) (22) Corollary 4, when applied to this vector-valued variational problem, implies xn λ P nx M ρ n λ max k=,, i= n y (Υ ki) a kλ (P n x,υ ki ), (23) and if γ:= sup ρ n λ >, (24) λ Λ,n then it suffices to minimize φ n (j) := max n k=,, i= y (Υ ki) a kλ (P n x,υ ki ) (25) or equivalently the discrete objective function n Φ n (j) := (y (Υ ki) a kλ (P n x,υ ki )) 2, (26) k= i= which is easier to minimize 3 umerical Examples In this section we present three different numerical examples Example We consider the linear system d du (κ (x) dx dx )+Au=f(x), <x<, u () =, with u () =, κ (x) = 5 + 2x, u=[ u u 2 ], A=3, f (x) =[ f (x) f 2 (x) ]=[8x ] (27) (28)

5 Mathematical Problems in Engineering u ũ 2 u x (a) x (b) 2 ũ x (c) Figure : For Example : (a) the numerical solution u,(b)thenoisedtarget u, and (c) the noised target u 2,with5%relativenoiseadded We solve the linear system, sample each solution component at M uniformly distributed data points in [, ], addrelative noise of ε, and fit 6th-degree polynomials to the resulting data to produce a target solution u Figure shows the numerical solution u and the target functions u i with M=2 and ε = 5 We consider the inverse problem: Given u and f(x), approximateκ(x) and the coefficient A such that the resulting system admits u as an approximate solution We set κ (x) = i= λ i x i (29) Example 2 We consider the coupled system d du (κ (x) dx dx )+Lu=f(x) <x<, u () =, with u () =, κ (x) = 5 + 2x, L=[ 2 3 ], (3) (3) With ξ i } n i= equal to the n-dimensional hat basis of [, ], for various values of M, noiseε, κ degree, andbasis dimension n, we construct the objective function in (26) and minimize it to find λ=(a,λ,,λ )Theresultsare presented in Table 2 x 3/5 f (x) =[ f (x) f 2 (x) ]=[ ( x) 3/5] ote that the forcing functions f (x) and f 2 (x) are not Hilbertian, living instead in W,3/2 (, ), for example We

6 6 Mathematical Problems in Engineering Table 2: Recovered parameter values for Example Truevaluesareλ=(3,5,2,) M n ε Recovered λ = =2 2 (3, 49999, 22) (3, 49999, 2, ) 2 (29628, 494, 223) (3259, 4977, 766, 3278) 2 3 (298, 4874, 23647) (3879, 4932, 296, 992) 2 5 (28898, 46858, 2678) (3637, 4852, 89, 664) 3 4 (3, 49998, 23) (3, 49998, 22, ) 3 4 (2925, 49758, 286) (3265, 523, 5585, 4778) (28888, 48936, 2259) (3537, 543, 6498, 46) (383, 47664, 2429) (3378, 545, 2832, 2467) u 3 u Figure 2: The numerical solution at 5 points for Example 2 work in the reflexive framework of Section 2 (In[6], we observed that the Hilbert space solution framework failed to work for a single equation with forcing function f (x)) Following [23,Proposition48], we can constructa Schauder basis e n } n in the Sobolev space W,p (, ) by integrating the Haar system of L p (, ) The BVP has singularities at x=andx=, since at each endpoint one of f (x) or f 2 (x) is undefined The forward problem can be solved numerically by using collocation techniquesweusecomsoltosolveitfigure 2 presents the numerical solution ext, we represent the solution in the subspace E 5 generated by the first 5 terms of the Schauder basis; call these representations u (x) and u 2 (x) We consider the inverse problem: find K(x) such that u is admitted as an approximate solution to (3) with κ(x) replaced by K(x) We solve the inverse problem by constructing the objective function in (26) with n=5, ξ i equal to the ith element of the Schauder basis, and K(x) = i= λ ix i Withthese choices, we have that (a ) λ ( u, ξ i )= (K (x) ξ i (x) u (x))dx + 2 u 2 (x) ξ i (x) dx, (a 2 ) λ ( u, ξ i )= (K (x) ξ i (x) u 2 (x))dx 3 u (x) ξ i (x) dx, ((T ) ) λ (ξ i )= f i (x) ξ i (x) dx (32) When = upon minimizing the objective function we obtain, to four decimal places, λ = 5893 and λ = 792 When = 2, we find λ = 5698, λ = 984, and λ 2 = 929 Example 3 We consider the 2D linear system (κ (x, y) u) + Au = f (x, y), with u(,y)=, u(,y)=, u (x, ) =, u (x, ) =, <x<, <y<, + 2x+3y κ (x) =[ + 2x+3y ], u=[ u u 2 ], A=[ ], f (x) =[ f (x) f 2 (x) ], (33) (34) where f (x, y) and f 2 (x, y) have been chosen so that the actual solution to the system is xy ( x) ( y) 4 u= (35) [ ] xy ( x) ( y) [ 8 ] Analogous to the process followed in Example, we sample each solution component at M Muniformly distributed data points in [, ] [, ] and add relative noise of ε to each of these data points A target solution u is constructed

7 Mathematical Problems in Engineering 7 Table 3: Recovered parameter values for Example 3Truevaluesareλ = γ = (, 2, 3, ) M ε Recovered λ Recovered γ (99282, 9728, 2934, 68) (99242, 9264, 29625, 68) (99268, 964, 2996, 97) (99233, 9257, 29594, 87) 3 (99246, 9653, 2965, 57) (9929, 928, 29573, 27) 5 (99232, 9685, 27664, 22) (9923, 782,29567, 7) 2 (9958, 9899, 2992, 8) (99574, 999, 29878, ) 2 (99575, 9886, 2978, 2) (99565, 9827, 29867, 8) 2 3 (9948, 984, 29786, 65) (99456, 924, 298, 63) 2 5 (99378, 9682, 294, 56) (99228, 952, 2968, 53) 3 (4, 9954, 2994, ) (99954, 9984, 29957, ) 3 (99796, 9965, 2989, 8) (9976, 983, 348, 8) 3 3 (99663, 246, 326, 59) (99656, 986, 29944, 59) 3 5 (99643, 2533, 29785, 22) (9964, 232, 3554, 29) using these data points together with our basis functions, ξ ij (hexagonal-based pyramids in this 2D case): M M u (x, y) = (u ) ij ξ ij, i= j= M M u 2 (x, y) = (u 2 ) ij ξ ij, i= j= for i, j =,,M (36) We consider the inverse problem: Given u, A, and f(x, y), approximate κ(x, y) such that the resulting system admits u as an approximate solution We set p q λ ij x i y j κ(x,y)= i= j= p q (37) [ γ ij x i y j ] [ i= j= ] For various values of M, noiseε, κ degree,andbasis dimension n = M, we construct the objective function in (26) and minimize it to find λ ij,γ ij,fori =,,p and j=,,qtheresultsarepresentedintable 3 Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper References [] A Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, ew York, Y, USA, 2 [2] F D M eto and A J da Silva eto, An Introduction to Inverse Problems with Applications, Springer, ew York, Y, USA, 23 [3] A Tarantola, Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, Pa, USA, 25 [4] C R Vogel, Computational Methods for Inverse Problems, vol 23 of Frontiers in Applied Mathematics, SIAM, ew York, Y, USA, 22 [5] J Milstein, The inverse problem: estimation of kinetic parameters, in Modeling of Chemical Reaction Systems, K Ebert, P Deuflhard, and W Jäger, Eds, Springer Series in Chemical Physics, pp 92, Springer, Berlin, Germany, 98 [6] A Tychonoff and V Y Arsenin, Solution of Ill-Posed Problems, Winston & Sons, Washington, DC, USA, 977 [7] A Tychonoff, Solution of incorrectly formulated problems and the regularization method, Doklady Akademii auk SSSR, vol 5, pp 5 54, 963 [8] M F Barnsley, Fractals Everywhere, Academic Press, ew York, Y, USA, 989 [9] M F Barnsley, V Ervin, D Hardin, and J Lancaster, Solution of an inverse problem for fractals and other sets, Proceedings of the ational Academy of Sciences of the United States of America, vol 83, no 7, pp , 986 [] H E Kunze and E R Vrscay, Solving inverse problems for ordinary differential equations using the Picard contraction mapping, Inverse Problems,vol5,no3,pp , 999 [] H Kunze, D la Torre, F Mendivil, and E R Vrscay, Fractal- Based Methods in Analysis, Springer, 22 [2] H Kunze, D La Torre, and E R Vrscay, Solving inverse problems for DEs using the collage theorem and entropy maximization, Applied Mathematics Letters, vol 25, no 2, pp , 22 [3] H E Kunze, D La Torre, and E R Vrscay, Random fixed point equations and inverse problems using collage method for contraction mappings, Mathematical Analysis and Applications, vol 334, no 2, pp 6 29, 27 [4] H Kunze, D La Torre, and E R Vrscay, A generalized collage method based upon the Lax-Milgram functional for solving boundary value inverse problems, onlinear Analysis Theory, Methods & Applications, vol 7, no 2, pp e337 e343, 29 [5] VCapasso,HEKunze,DLaTorre,andERVrscay, Solving inverse problems for differential equations by a generalized collage methodandapplicationtoameanfieldstochastic model, onlinear Analysis: Real World Applications,vol5,pp , 24 [6] M I Berenguer, H Kunze, D La Torre, and M Ruiz Galán, Galerkin schemes and inverse boundary value problems in reflexive Banach spaces, Computational and Applied Mathematics,vol275,pp 2,25 [7] M I Berenguer, H E Kunze, D La Torre, and M Ruiz Galán, A collage-based approach to inverse problems for constrained

8 8 Mathematical Problems in Engineering variational equations, Computational and Applied Mathematics Accepted [8] V V Semenov, Projection theorem for Banach and locally convex spaces, Cybernetics and Systems Analysis,vol44,no5, pp , 28 [9] D Boffi, F Brezzi, and M Fortin, Mixed Finite Element Methods and Applications, vol44ofspringer Series in Computational Mathematics, Springer, Heidelberg, Germany, 23 [2] G Gatica, A Simple Introduction to the Mixed Finite Element Method Theory and Applications,SpringerBriefsinMathematics, Springer, 24 [2] A I Garralda-Guillem and M Ruiz Galán, Mixed variational formulations in locally convex spaces, Mathematical Analysis and Applications,vol44,no2,pp ,24 [22] M R Galán, An intrinsic notion of convexity for minimax, Convex Analysis, vol 2, no 4, pp 5 39, 24 [23] S Fučik, Fredholm alternative for nonlinear operators in Banach spaces and its applications to differential and integral equations, Časopis pro Pěstování Matematiky,vol96,pp37 39, 97

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