Applications of Tikhonov regularization to inverse problems using reproducing kernels

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1 Journal of Physics: Conference Series Applications of Tikhonov regularization to inverse problems using reproducing kernels Related content - Reproducing kernels in forms on toric varieties A V Shuplev volume To cite this article: Saburou Saitoh 2007 J. Phys.: Conf. Ser View the article online for updates enhancements. - On the choice of the regularization parameter for ordinary iterated Tikhonov regularization of nonlinear illposed problems Qi-Nian Jin Zong-Yi Hou - Filtering techniques for efficient inversion of two-dimensional Nuclear Magnetic Resonance data V. Bortolotti, L. Brizi, P. Fantazzini et al. Recent citations - Toeplitz Operators for Wavelet Transform Related to the Spherical Mean Operator Besma Amri - A Discretized Tikhonov Regularization Method for a Fractional Backward Heat Conduction Problem Zhi-Liang Deng Xiao-Mei Yang - Best approximation for weierstrass transform connected with spherical mean operator L.T. Rachdi N. Msehli This content was downloaded from IP address on 07/03/2019 at 15:26

2 Applications of Tikhonov regularization to inverse problems using reproducing kernels Saburou Saitoh Department of Mathematics, Faculty of Engineering, Gunma University, Kiryu JAPAN Abstract. This paper is a survey article based on our recent papers on inverse problems using Tikhonov regularization reproducing kernels. Tikhonov regularization is a basically important idea method in numerical analysis, however, the extremal functions of the Tikhonov functional are commonly represented by using the associated singular values singular functions for the case of compact operators. So, their representations are restrictive abstract in a sense. We shall give new representations for the case of general bounded linear operators by using the theory of reproducing kernels we shall introduce various concrete representations for typical cases. Our representations are both analytical numerical. We introduce a general theory of the Tikhonov regularization using the theory of reproducing kernels including error estimates convergence rates. We shall present the general theory its concrete results for the typical inverse problem for heat conduction with computer graphs (in the cited references) as evidence of the power of our inverse formulas. 1. Reproducing kernels First we recall a basic relation between linear mappings in the framework of Hilbert spaces reproducing kernels. In particular, we can see here why we meet ill-posed problems, indeed, we can see the idea method for the avoidance of the ill-posed problems in the framework of Hilbert spaces. However, this will be a mathematical theory for the purpose of developing numerical methods, we will need the idea of Tikhonov regularization discussed in Section 5. However we will need essentially the applications of the theory of reproducing kernels to both mathematical numerical theories for bounded linear operators in the framework of Hilbert spaces. See [5] for a general elementary theory of inverse problems ill-posed problems. We consider any positive matrix K(p, q) on a fixed set E; that is, for an abstract set E the complex valued function K(p, q) on E E satisfies, for any finite points {p j } of E for any complex numbers {C j }, C j C j K(p j, p j ) 0. j j Then, by the fundamental theorem by Moore Aronszajn, we have: Proposition 1.1 ([1]) For any positive matrix K(p, q) on E, there exists a uniquely determined functional Hilbert space (abbreviated RKHS) H K comprising functions {f} on E admitting c 2007 Ltd 1

3 the reproducing kernel K(p, q) satisfying characterized by K(, q) H K for any q E (1), for any q E for any f H K f(q) = (f( ), K(, q)) HK. (2) For some general properties of reproducing kernel Hilbert spaces for various constructions of the RKHS H K from a positive matrix K(p, q), see the recent book [23] its Chapter 2, Section 5, respectively. 2. Connections with linear mappings Let us connect linear mappings in the framework of Hilbert spaces with reproducing kernels ([19]). For an abstract set E for any Hilbert (possibly finite dimensional) space H, we shall consider an H valued function h on E h : E H (3) the linear mapping from H into a linear space comprising functions on E, given by f f, where f(p) = (f, h(p)) H for f H. (4) This represents, in particular, the Fredholm integral equations of the first kind in the framework of Hilbert spaces. For this linear mapping (4), we form the positive matrix K(p, q) on E defined by which is, by Proposition 1.1, a reproducing kernel. Then, we have the following fundamental results: K(p, q) = (h(q), h(p)) H on E E, (5) (I) For the RKHS H K admitting the reproducing kernel K(p, q) defined by (5), the images {f(p)} by (4) for H are characterized as the members of the RKHS H K. (II) In general, we have the inequality in (4) f HK f H, (6) however, for any f H K there exists a uniquely determined f H satisfying f(p) = (f, h(p)) H on E (7) f HK = f H. (8) In (6), the isometry holds if only if {h(p); p E} is complete in H. (III) We can obtain the inversion formula for (4) in the form f f, (9) 2

4 by using the RKHS H K. However, this inversion formula will depend on, case by case, the realizations of the RKHS H K. (IV) Conversely, if we have an isometric mapping L from the RKHS H K admitting a reproducing kernel K(p, q) on E onto a Hilbert space H, then the mapping is linear its isometric inversion L 1 is represented in the form (4). Here, the Hilbert space H valued function h satisfying (3) (4) is given by h(p) = LK(, p) on E (10), {h(p) : p E} is complete in H. When (4) is isometrical, sometimes we can use the isometric mapping for a realization of the RKHS H K, conversely that is, if the inverse L 1 of the linear mapping (4) is known, then we have f HK = L 1 f H. 3. General applications We shall state some general applications of the results (I) (IV) to several wide subjects their basic references: (1) Linear mappings ([21],[23]). The fact that the image spaces of linear mappings in the framework of Hilbert spaces are characterized as reproducing kernel Hilbert spaces defined by (5) is the most important one in the general theory of reproducing kernels. This means that the theory of reproducing kernels is fundamental a general concept in mathematics. To look for the characterization of the image space is a starting point when we consider the linear equation (4). (II) gives a generalization of the Pythagorean theorem (see also [18]) means that in the general linear mapping (4) there exists essentially an isometric identity between the input the output. (III) gives a generalized (natural) inverse (solution) of the linear mapping (equation) (4). (IV) gives a general method determining constructing the linear system from an isometric relation between outputs inputs by using the reproducing kernel in the output space. (2) Linear mappings among smooth functions ([28]). We considered linear mappings in the framework of Hilbert spaces, however, we can consider linear mappings in the framework of Hilbert spaces comprising smooth functions, similarly. Conversely, reproducing kernel Hilbert spaces are considered as the images of some Hilbert spaces by considering some decomposed representations (5) of the reproducing kernels. Such decomposition is, in general, possible. This idea is important in [29] also in the following items (6) (7). (3) Nonharmonic linear mappings ([21]). If the linear system vectors h(p) move in a small way (perturbation of the linear systems) in the Hilbert space H, then we can not calculate the related positive matrix (5), however, we can discuss the inversion formula an isometric identity of the linear mapping. The prototype result is the Paley-Wiener theorem on nonharmonic Fourier series. (4) Various norm inequalities ([24],[26]). Relations among positive matrices correspond to those of the associated reproducing kernel Hilbert spaces, by the minimum principle. So, we can derive various norm inequalities among reproducing kernel Hilbert spaces from relations among positive matrices. We were able to derive many beautiful norm inequalities. 3

5 (5) Nonlinear mappings ([24],[26]). In a very general nonlinear mapping of a reproducing kernel Hilbert space, we can look for a natural reproducing kernel Hilbert space containing the image space furthermore, we can derive a natural norm inequality in the nonlinear mapping. What is a basic relation between linear mappings non-linear mappins in the framework of reproducing kernel Hilbert spaces? It seems that the theory of reproducing kernels gives a fundamental interesting answer for this question. (6) Linear (singular) integral equations ([9],[29]). We can apply our general theory to various integral equations containing Volterra type in the framework of Hilbert spaces. (7) Linear differential equations with variable coefficients ([29],[32]). In linear integro-differential equations with general variable coefficients, we can discuss the existence construction of the solutions, if the solutions exist. This background method is called a backward transformation method by reducing the equations to Fredholm integral equations of the first kind we can discuss the classical solutions, in very general linear equations. (8) Approximation theory ([3],[23]). Reproducing kernel Hilbert spaces are very nice function spaces, because the point evaluations are continuous. Therefore, reproducing kernels are a fundamental tool in the related approximation theory. See also Section 4. (9) Representations of inverse functions ([23],[25]). For any mapping, we discussed the problem of representing its inverse in terms of the direct mapping we derived a unified method for this problem. As a simple example, we can represent the Taylor coefficients of the inverse of the Riemann mapping function on the unit disc on the complex plane in terms of the Riemann mapping function. This fact was important in the representation of analytic functions in terms of local data in ([30]). (10) Various operators among Hilbert spaces ([27]). Among various abstract Hilbert spaces, we can introduce various operators of sum, product, integral derivative by using the linear mapping or very general nonlinear mappings. The prototype operator is convolution we discussed it from a wide general viewpoint with concrete examples. (11) Sampling theorems ([23], Chapter 4, Section 2). The Whittaker-Kotel nikov-shannon sampling theorem may be interpretated by (I) (II) very well, we can discuss the truncation errors in the sampling theory. J. R. Higgins [7] established a fully general theory for [23]. See also, Section 7, as its essence. (12) Interpolation problems of Pick-Nevanlinna type ([21],[22]). General abstract theory of interpolation problems of Pick-Nevanlinna type may be discussed by using the general theory of reproducing kernels. (13) Analytic extension formulas their applications ([6],[30]). We were able to obtain various analytic extension formulas their applications from various isometric identities (II). For their applications to nonlinear partial differential equations, see the survey article by N. Hayashi [6]. (14) Inversions of a family of bounded linear operators on a Hilbert space into various Hilbert spaces ([31]). From fully general theorems for the Pythagorean theorem, we derived the concept of inversions of a family of bounded linear operators on a Hilbert space into various Hilbert spaces. 4

6 4. Best approximation problems We shall consider the fundamental well-known best approximation problem inf f H K Lf d H (11) for a vector d in H where L : H K H is a bounded linear operator. Then, we have, by our theory in Section 2 Proposition 4.1 ([3],[23]) For a vector d in H, there exists a function f in H K such that inf f H K Lf d H = L f d H (12) if only if, for the RKHS H k admitting the reproducing kernel defined by k(p, q) = (L LK(, q), L LK(, p)) HK, (13) L d H k. (14) Furthermore, if the best approximation f satisfying (12) exists, then there exists a unique extremal function f d with the minimum norm in H K, the function f d is expressible in the form f d (p) = (L d, L LK(, p)) Hk on E. (15) In Proposition 4.1, note that (L d)(p) = (L d, K(, p)) HK = (d, LK(, p)) H ; (16) that is, L d is expressible in terms of the known d, L, K(p, q) H. This representation will, in particular, show that for bounded linear operators on a Hilbert space, if the Hilbert space admits a reproducing kernel, then the representations of their adjoint operators are concretely given. This fact will be important for general bounded linear operator theory. f d in (15) is the Moore-Penrose generalized inverse solution L d of the equation Lf = d. Therefore, Proposition 4.1 gives a necessary sufficient condition for the existence of the Moore-Penrose generalized inverse. Proposition 4.1 is rigid is not useful in practical applications, because, practical data contain noise or errors the criteria (14) is not suitable. So, we shall consider the Tikhonov regularization we shall establish a good relation between the Tikhonov regularization the theory of reproducing kernels. 5. Spectral theory In order to discuss operator equations for general bounded linear operators L, following [4] we shall fix the well-established theory among spectral theory, the Moore-Penrose generalized inverse the Tikhonov regularization. See [5] for the corresponding results for compact operators L. Let {E λ } be a spectral family for the self-adjoint operator L L. If L L is continuously invertible, then 1 (L L) 1 = λ de λ. In this case, the Moore-Penrose generalized inverse (15) can be represented by the Gaussian normal equation 1 f d (p) = λ de λl d. (17) 5

7 If R(L) is non-closed d / D(L ), i.e. if the equation Lf = d is ill-posed, then the integral in (17) does not exist. Then, we shall define 1 f d,α (p) = λ + α de λl d. (18) By construction, the operator on the right-h side of (18) acting on d is continuous, so that, for noisy data d δ with d d δ H δ, we can bound the error between f d,α f δ d,α (p) = 1 λ + α de λl d δ (19) as follows: Proposition 5.1 ([4], pages 71-73) For any d D(L ), Furthermore, lim α 0 (L L + αi) 1 L d = lim f α 0 d,α = f d. (20) Lf d,α Lf δ d,α H δ (21) f d,α f δ d,α H K δ α. (22) Proposition 5.2 ([4], pages ) For any d δ D(L ) with d d δ H δ, the function f δ defined by (19) is the unique minimizer of the Tikhonov functional d,α then If α = α(δ) is such that inf {α f 2 H f H K + d δ Lf 2 H}. (23) K lim α(δ) = 0 δ 0 lim δ 0 δ 2 α(δ) = 0, lim f δ δ 0 d,α = f d = L (d). (24) These results are important, because real physical data contain error noises. 6. Representation of the extremal functions in Tikhonov regularization Our main purpose here is to give an effective representation of the extremal function f d,α or f δ in the Tikhonov regularization, since the representation by spectral theory is abstract, in d,α many practical problems. We set K L (, p; α) = (L L + αi) 1 K(, p). Then, by introducing the inner product, for any fixed positive α > 0 (f, g) HK (L;α) = α(f, g) HK + (Lf, Lg) H, (25) we shall construct the Hilbert space H K (L; α) comprising functions of H K. This space, of course, admits a reproducing kernel. Furthermore, we obtain 6

8 Proposition 6.1 ([33]) The extremal function f d,α (p) in the Tikhonov regularization is represented in terms of the kernel K L (p, q; α) as follows: inf f H K {α f 2 H K + d Lf 2 H} (26) f d,α (p) = (d, LK L (, p; α)) H (27) where the kernel K L (p, q; α) is the reproducing kernel for the Hilbert space H K (L; α) it is determined as the unique solution K(p, q; α) of the equation: K(p, q; α) + 1 α (L K q, LK p ) H = 1 K(p, q) (28) α with K q = K(, q; α) H K for q E, (29) K p = K(, p) H K for p E. In (27), when d contains error or noises, we need its error estimate. For this, we can obtain the general result: Proposition 6.2 ([9],[17]). In (27), we obtain the estimate f d,α (p) 1 α K(p, p) d H. Propositions will show the importance to consider bounded linear operators on a Hilbert space which admitts a reproducing kernel. For many concrete applications of these general theorems, see, for example, [2,9,11-17,32-35]. 7. A typical example for the inversion of the heat conduction We shall give simple approximate real inversion formulas for the Gaussian convolution (the Weierstrass transform) u F (x, t) = (L t F )(x) = { } 1 ξ x 2 (4πt) n/2 F (ξ) exp dξ (30) n R 4t for the functions of L 2 ( ). This integral transform which represents the solution u(x, t) of the heat equation u t (x, t) = u xx (x, t) on {t > 0} satisfying the initial condition u(x, 0) = F (x) on, is very fundamental has many applications to mathematical sciences. See the recent article [39] its references for its many significant applications to medical science physics. Over twenty years ago, in the one dimensional case n = 1, the author [20] gave a surprise characterization of the image u F (x, t) of (30) for L 2 (R) = L 2 (R, dx) functions in terms of an analytic function established a very simple complex inversion formula following the idea (I). 7

9 The paper created a new method many applications to general integral transforms in the framework of Hilbert spaces analytic extension formulas. See, for example [2] [23], their many references. However, in particular, its real inversion formulas are very involved, for example, recall that: For a bounded continuous function F (x) for t = 1, for the differential operator D = d dx e D2 [(L 1 F )(x)] = F (x) pointwisely on R ([8], p. 182). So, one might think that its real inversion formulas will be essentially involved for catching analyticity in terms of the data on the real line as in the real inversion formulas of the Laplace transform. See also [10] for a recent related article. Indeed, this inverse problem is very famous as a typical ill-posed problem that is very difficult. In those papers [33,15], however we were able to obtain simple practical approximate real inversion formulas by the method in Section 6 using the Sobolev reproducing Hilbert spaces. Furthermore, we illustrated their numerical experiments by using computers we can realize that we were able to obtain practical real inversion formulas. In [17], we applied the Paley-Wiener spaces as the reproducing kernel Hilbert spaces in the above theory we got an improved numerical inversion. At first we shall fix notations basic results in the Paley-Wiener spaces following the book by F. Stenger[38] at the same time we shall show the basic relation of the sampling theory the theory of reproducing kernels. We shall consider the integral transform, for L 2 (, ( π/h, +π/h) n ), (h > 0) functions g f(z) = 1 (2π) n χ h (t)g(t)e iz t dt. (31) Here, z = (z 1, z 2,..., z n ), t = (t 1, t 2,..., t n ), dt = dt 1 dt 2 dt n, z t = z 1 t z n t n χ h (t) = Π n ν=1χ(t ν, ( π/h, +π/h)), the characteristic function χ of ( π/h, +π/h). In order to identify the image space following the idea (I), we form the reproducing kernel K h (z, u) = 1 (2π) n χ h (t)e iz t e iu t dt (32) = Π n ν 1 π(z ν u ν ) sin π h (z ν u ν ). The image space of (31) is called the Paley-Wiener space W ( π h ) (:= Wh ) comprised of all analytic functions of exponential type satisfying, for each ν, for some constant C ν as z ν ( ) π zν f(z 1,..., z ν, z ν+1,..., z n ) C ν exp h f(x) 2 dx <. From the identity, for multi-index j = (j 1, j 2,..., j n ) Z n K h (jh, j h) = Π n 1 ν=1 h δ(j ν, j ν) 8

10 (the Kronecker s δ), for each ν, since δ(j ν, j ν) is the reproducing kernel for the Hilbert space l 2, from (II) the Parseval s identity we have the isometric identities in (31) 1 (2π) n g(t) 2 dt = h n f(jh) 2 j = f(x) 2 dx. That is, the reproducing kernel Hilbert space H Kh with K h (z, u) is characterized as a space comprising the Paley-Wiener space W h with the norms above in the both senses of discrete continuous versions. Here we used the well-known result that {jh} j is a uniqueness set for the Paley-Wiener space W h ; that is, f(jh) = 0 for all j implies f 0. Then, the reproducing property of K h (z, u) states that f(x) = (f( ), K h (, x)) HKh = h n j f(jh)k h (jh, x) = f(ξ)k h (ξ, x)dξ, in particular, on the real space x. This representation is the sampling theorem which represents the whole data f(x) in terms of the discrete data {f(jh)} j. For a general theory of sampling error estimates for some finite points {hj} j, see [7] [23]. Following our general theory, we can obtain the concrete results: Proposition 7.1 ([17]) For any function g L 2 ( ) for any λ > 0, the best approximate function Ft,λ,h,g in the sense } inf {λ F 2 H F H Kh + g u F (, t) 2 L 2 ( ) Kh = λ F t,λ,h,g 2 H Kh + g u F t,λ,h,g (, t) 2 L 2 ( ) (33) exists uniquely F t,λ,h,g is represented by Ft,λ,h,g(x) = g(ξ)q t,λ,h (ξ x)dξ (34) for If, for F H Kh result: as λ 0 uniformly. Q t,λ,h (ξ x) = 1 χ h (p)e ip (ξ x) dp (2π) n λe p 2t. + e p 2 t we consider the output u F (x, t) we take u F (ξ, t) as g, then we have the F t,λ,h,g F, (35) Here we note the fact that for the Sobolev space case, for λ = 0 the corresponding representation (34) does not exist ([33],[15]), meanwhile for the Paley-Wiener space W ( ) π h case of (34), for λ = 0 the representation (34) is still valid; that is, in Proposition 7.1, the result is valid for even λ = 0. Hence, we can consider the results for λ = 0 in the spirit of Tikhonov 9

11 regularization in which we are interested in a small λ or λ tending to zero. That is, when we use the Paley-Wiener space W ( π h ), we need not to consider the Tikhonov regularization. Then, (L t F t,0,h,g)(x) = (g( ), K h (, x)) L2 ( ) as we see from (32). Since the output is the orthogonal projection of g onto the Paley-Wiener space W ( ) π h, we can estimate the difference of the output of our inverse F t,0,h,g g, clearly as L t Ft,0,h,g g L2 ( ) which is the distance from g onto the Paley-Wiener space W ( ) π h. Of course, F t,0,h,g is the Moore-Penrose generalized inverse of the operator equation, for any g L 2 (R) F W ( ) π h, L t F = g. For the Paley-Wiener space W ( ) π h, we need not use Tikhonov regularization we can look for the Moore-Penrose generalized inverse Ft,0,h,g by using the theory of reproducing kernels ([23], pp ). However, we had better to calculate the extremal functions Ft,λ,h,g in the Tikhonov regularization to set λ = 0, because the structure of the Moore-Penrose generalized inverses is involved. We consider the heat conduction for the RKHS H K, however, our inverstion formula in the sense (33) will show that for a very general function containing the delta function, our inversion formula is valid, because we are considering the approximate inversion by the functions H K. Meanwhile, we are interested in the convergence property of the extremal functions (34) when h tends to zero for a general g. However, we can see directly that in (34) lim (L tft,0,h,g)(x) = g(x) h 0 at the points x where g(x) is continuous. We are interested in numerical experiments for the both cases of the Sobolev space H S the Paley-Wiener space W ( ) π h for their convergences as λ 0 h 0. For the Sobolev spaces λ 0, see [15]. The real inversion formula (34) will give a practical formula for the Gaussian convolution. In [17], we have shown experimental results by using computers in Figures. In [15], we stated that when we use the Sobolev spaces as the reproducing kernels, we will see that in order to overcome the high ill-posedness in the real inversion in order to catch analyticity of the image of (30) we must work hardly; that is, we must take a very small λ we must calculate the corresponding integral (34) hardly in the sense of numerical. Computers help us this hard work to calculate the integral for a very small λ. However, when we use the Paley-Wiener spaces for λ = 0 for h tending to 0, we can obtain a very simple improved inversion formula. For any λ > 0 any t > 0, we shall define a linear mapping M t,λ,h : L 2 ( ) H Kh by M t,λ,h (g) = F t,λ,h,g. Now, we consider the composite operators L tm t,λ,h M t,λ,h L t. Using Fourier s integrals it can be shown that for F H Kh, (M t,λ,h L t F )(x) = 1 (2π) n { F (ξ) χ h (p)e ip (ξ x) dp λe 2 p 2t + 1 } dξ (36) 10

12 for g L 2 ( ), (L t M t,λ,h g)(x) = 1 (2π) n { g(ξ) χ h (p)e ip (ξ x) dp λe 2 p 2t + 1 } dξ. (37) Setting in (36) (37), we have t,λ,h (x ξ) = 1 Rn χ h (p)e ip (ξ x) (2π) n λe 2 p 2t + 1 dp (M t,λ,h L t F )(x) = F (ξ) t,λ,h (x ξ)dξ, (F H Kh ) (38) (L t M t,λ,h g)(x) = g(ξ) t,λ,h (x ξ)dξ, (g L 2 ( )). (39) Then we obtain that lim t,0,h(x ξ) = δ(x ξ), h 0 (40) The precise meanings of (41) (42) are given as follows: lim M t,0,hl t = I (41) h 0 lim L tm t,0,h = I. (42) h 0 lim (M t,0,hl t F )(x) = F (x) h 0 at the points where F g are continuous. lim L tm t,0,h g = g (43) h 0 Acknowledgments Supported by the Grant-in-Aid for the Scientific Research (C)(2) (No ) from the Japan Society for the Promotion Science by the Mitsubishi Foundation, the 36th, Natural Sciences, No. 20 ( ). The author wishes to express his deep thanks to the referees who gave valuable suggestions on the representation of the paper. References [1] Aronszajn A 1950 Trans. Amer. Math. Soc [2] Asaduzzaman M, Matsuura T Saitoh S 2005 Advances in Analysis. Proceedings of the 4th International ISAAC Congress, World Scientific eds H G W Begehr, R P Gilbert, M E Muldeon M W Wong [3] Byun D W Saitoh S 1994 Proc. of the 2nd International Colloquium on Numerical Analysis. VSP-Holl eds D Bainov V Covachev [4] Engl H W, Hanke M Neubauer A 2000 Regularization of Inverse Problems. Mathematics Its Applications, 376 Kluwer Academic Publishers 11

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