A Low Memory Solver For Integral Equations Of Chandrasekhar Type In The Radiative Transfer Problems
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This document presents a new method for solving integral equations of Chandrasekhar type that arise in radiative transfer problems. The method approximates the Jacobian inverse as a diagonal matrix using variational techniques. This avoids explicitly computing or storing the Jacobian. The method formulates an optimization problem to minimize corrections to the diagonal matrix while satisfying a weak secant condition. Solving this yields a unique solution for updating the diagonal matrix approximation at each iteration. Numerical results on benchmark problems demonstrate the reliability and efficiency of the approach compared to other Newton-like methods. The fact that it solves the integral equations without derivative computation or matrix storage is a clear advantage.
A Low Memory Solver For Integral Equations Of Chandrasekhar Type In The Radiative Transfer Problems
- 1. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2011, Article ID 467017, 12 pages doi:10.1155/2011/467017 Research Article A Low Memory Solver for Integral Equations of Chandrasekhar Type in the Radiative Transfer Problems Mohammed Yusuf Waziri, Wah June Leong, Malik Abu Hassan, and Mansor Monsi Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Malaysia Correspondence should be addressed to Mohammed Yusuf Waziri, mywaziri@gmail.com Received 4 February 2011; Revised 14 July 2011; Accepted 8 August 2011 Academic Editor: Alexei Mailybaev Copyright q 2011 Mohammed Yusuf Waziri 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. The problems of radiative transfer give rise to interesting integral equations that must be faced with efficient numerical solver. Very often the integral equations are discretized to large-scale nonlinear equations and solved by Newton’s-like methods. Generally, these kind of methods require the computation and storage of the Jacobian matrix or its approximation. In this paper, we present a new approach that was based on approximating the Jacobian inverse into a diagonal matrix by means of variational technique. Numerical results on well-known benchmarks integral equations involved in the radiative transfer authenticate the reliability and efficiency of the approach. The fact that the proposed method can solve the integral equations without function derivative and matrix storage can be considered as a clear advantage over some other variants of Newton’s method. 1. Introduction The study of Chandrasekhar’s integral equation involved in the radiative transfer prob- lem has been a foremost subject of much investigations and was first formulated by Chandrasekhar 1 in 1960. It arose originally in connection with scattering through a homogeneous semi-infinite plane atmosphere and since it has been used to model diverse forms of scattering via the H-function of Chandrasekhar 2, defined by Hx 1 Hx 1 0 x x t ψtHtdt. 1.1
- 2. 2 Mathematical Problems in Engineering Chandrasekhar H-function plays a crucial role in radiative transfer and transport theory 3, 4. Since then, there have been diverse solvers of 1.1. It is well known that the numerical solution of Chandrasekhar integral equation is difficult to be obtained 5, and thus it is con- venient to have a reliable and efficient solver. The problem of finding approximate solution of such integral equations is still popular today, and various methods of solving these integral equations have been established 5–7. The common approach for the approximate solution of 1.1 is at first discretizing 1.1 by a vector x ∈ Rn , then replacing the integrals by quadrature sums and the derivatives by difference quotients involving only the components of x ∈ Rn see 8, e.g.. By doing so, 1.1 becomes a problem of finding the solution of system of n nonlinear equations with n unknowns Fx 0, 1.2 where F : Rn → Rn is a nonlinear mapping. Often, the mapping F is assumed to satisfy the following assumptions: A1 there exists x∗ ∈ Rn s.t. Fx∗ 0, A2 F is a continuously differentiable mapping in an open neighborhood of x∗ , A3 F′ x∗ is invertible. The famous iterative method for solving 1.2 is the classical Newton’s method, where the Newtonian iteration is given by xk1 xk − F′ xk −1 Fxk, k 0, 1, 2, . . . . 1.3 The convergence rate for the Newton’s method is quadratic from any initial point x0 in the neighborhood of x∗ 9. However, an iteration of 1.3 turns to be expensive, because it re- quires to compute and store the Jacobian matrix, as well as solving Newton’s system which is a linear system in each iteration. The major difficulty of Newton’s type method is the ma- trix storage requirements especially when handling large systems of nonlinear equations 5, 6, 9. There are quite a number of revised Newton’s type methods being introduced, which include fixed Newton’s and quasi-Newton’s, to diminish the weakness of 1.3. Fixed Newton method 10 for the determination of solution x∗ is given by xk1 xk − F′ x0 −1 Fxk, k 0, 1, 2, . . . . 1.4 The method avoids computation and storing the Jacobian in each iteration except at k 0. However, it still requires solving the systems of n linear equations and may consume more CPU time as the system’s dimension increases 10. A quasi-Newton’s method is another variant of Newton’s type methods, and it replaces the Jacobian or its inverse with an approximation which can be updated at each iteration 11, and its updating scheme is given by xk1 xk − B−1 k Fxk, 1.5 where the matrix Bk is the approximation of the Jacobian at xk. The main idea behind quasi- Newton’s method is to eliminate the evaluation cost of the Jacobian matrix, in which if
- 3. Mathematical Problems in Engineering 3 function evaluations are very expensive, the cost of finding a solution by quasi-Newton’s methods could be much smaller than some other Newton’s-like methods 7, 12, 13. Various Jacobian approximations matrices such as the Broyden’s method 11, 14 are proposed. However, the most critical part of such solvers is that they need the storage of full matrix of the approximate Jacobian, which can be a very expensive task as the dimension of systems increases 15. In this paper, we propose an alternative approximation to the Jacobian inverse into a diagonal matrix by means of variational techniques. It is worth mentioning that the sug- gested method can be applied to solve 1.2 without the cost of computing or storing the true Jacobian. Hence, it can reduce computational cost, storage requirement, processing time CPU time and also eliminates the need for solving n linear equations in each iteration. The proposed method works efficiently, and the results so far are very encouraging. This paper is arranged as follows; we present our proposed method in Section 2; numerical results are reported in Section 3; finally conclusion is given in Section 4. 2. Chandrasekhar H-Equation In this section, we present the detailed process of discretizing the Chandrasekhar-type inte- gral equations in the radiative transfer problem. Chandrasekhar and Breen 16 compute H- equation as the solution of the nonlinear integral equation Hx − c x 2 Hx 1 0 H y x y dy 1, 2.1 where c ∈ 0, 1 and H : 0, 1 → R is an unknown continuous function. From 2.1, we obtain Hx 1 − c 2 1 0 xH y x y dy 1. 2.2 Let us partition 0, 1 into n subinterval, 0 x1 · · · xj j/n · · · 1. Denote Hk as Hxk, then the evaluation of 2.1 at every xi yields the equation Hi 1 − c 2 1 0 xiH y xi y dy 1, i 1, 2, . . . , n. 2.3 After multiplying both sides of 2.2 by 1 − c/2 1 0 xHy/x ydy −1 and performing some algebra, we arrive at 2.4, which is known as the Chandrasekhar H-equation 15 FHx Hx − 1 − c 2 1 0 xH y dy x y −1 0. 2.4
- 4. 4 Mathematical Problems in Engineering If 2.4 is discretized by using the midpoint quadrature formula 1 0 ftdt 1 n n j0 f tj , 2.5 for tj j − 0.5h, 0 ≤ j ≤ 1, i 2, . . . , n, h 1/n, c ∈ 0, 1, then we have the following: Fi xi − ⎛ ⎝1 − c 2n n j1 tixj ti tj ⎞ ⎠ −1 . 2.6 Function 2.6 is called the discretized Chandrasekhar H-equation which can be solved by some iterative methods. Nevertheless, the most difficult part in solving 2.6 arises dramatically as c approach- es 1, since its Jacobian is singular at c 1. Due to this disadvantage, we aim to derive a method that hopefully will not be affected by this difficulty. 3. Derivation of the Method (LMSI) Firstly, note that by the mean value theorem, we have F′xkxk1 − xk Fxk1 − Fxk, 3.1 where F′xk 1 0 F′ xk θxk1 − xkdθ. Let us denote Δxk xk1 − xk and ΔFk Fxk1 − Fxk, then 3.1 becomes F′xkΔxk ΔFk. 3.2 Equation 3.2 is always regarded as the secant equation. Alternatively, we can rearrange 3.2 to obtain Δxk F′xk −1 ΔFk. 3.3 Here, we propose to use a diagonal matrix, say D, to approximate F′xk −1 , that is, F′xk −1 ≈ Dk. 3.4 Let us consider an updating scheme for D, in which we should update D by adding a cor- rection M which is also a diagonal matrix at every iteration Dk1 Dk Mk. 3.5
- 5. Mathematical Problems in Engineering 5 In order to incorporate correct information of the Jacobian inverse into the updating matrix, Dk1, we require that Dk1 satisfies the secant equation 3.2, that is, Δxk Dk MkΔFk. 3.6 However, since it is difficult to have a diagonal matrix that satisfies the secant equation, in particular, because Jacobian approximations are not usually done in element wise, we con- sider the use of the weak secant condition 17 instead, ΔFT k Δxk ΔFT k Dk MkΔFk. 3.7 To encourage good condition number as well as numerical stability in the approximation, we attempt to control the growth error of the correction by minimizing its magnitude under some norms here, we consider the Frobenuis norm, such that 3.7 holds. To this end, we con- sider the following problem: min 1 2 Dk1 − Dk2 F s.t. ΔFT k Dk1ΔFk ΔFT k Δxk, 3.8 where · F is the Frobenuis norm. If we let Dk1 − Dk Mk diagβ1, β2, . . . , βn and ΔFk ΔF 1 k , ΔF 2 k , . . . , ΔF n k , the above problem can be expressed as follows: min 1 2 β2 1 β2 2 · · · β2 n s.t. n i1 ΔF i2 k βi − ΔFT k Δxk ΔFT k DkΔFk 0. 3.9 Since the objective function and the constraint are convex, we will have unique solution for 3.9. The solution can be obtained by considering the Lagrangian function of problem 3.9 L βi, λ 1 2 β2 1 β2 2 · · · β2 n λ n i1 ΔF i2 k βi − ΔFT k Δxk ΔFT k DkΔFk , 3.10 where λ is the corresponding Lagrangian multiplier. Taking the partial derivatives of 3.10 with respect to each βi and λ, respectively, and setting them equal to zero, we have ∂L ∂βi βi λ ΔF i k 2 0, i 0, 1, 2, . . . , n, 3.11 ∂L ∂λ n i1 ΔF i k 2 βi − ΔFT k Δxk ΔFT k DkΔFk 0. 3.12
- 6. 6 Mathematical Problems in Engineering Premultiplying both sides of 3.11 by ΔF i2 k and summing them all yield n i1 ΔF i k 2 βi λ n i1 ΔF i k 4 0. 3.13 It follows from 3.13 that n i1 ΔF i k 2 βi −λ n i1 ΔF i k 4 . 3.14 Invoking the constraint 3.12, we have n i1 ΔF i k 2 βi ΔFT k Δxk − ΔFT k DkΔFk. 3.15 Equating 3.14 with 3.15 gives λ − ΔFT k Δxk − ΔFT k DkΔFk n i1 ΔF i k 4 . 3.16 Substituting 3.16 into 3.14 and after some simplifications, we obtain βi ΔFT k Δxk − ΔFT k DkΔFk n i1 ΔF i k 4 ΔF i k 2 , i 1, 2, . . . , n. 3.17 Denoting Gk diagΔF 1 k 2 , ΔF 2 k 2 , . . . , ΔF n k 2 and n i1ΔF i k 4 TrG2 k where Tr is the trace operation, we obtain, therefore, Mk ΔFT k Δxk − ΔFT k DkΔFk Tr G2 k Gk. 3.18 Finally, the proposed updating formula for the approximation of the Jacobian inverse is given as follows: Dk1 Dk ΔFT k Δxk − ΔFT k DkΔFk Tr G2 k Gk. 3.19 To safeguard possibly very small ΔFk and TrG2 k , we require that ΔFk ≥ ǫ1 for some chosen small ǫ1 0. Else, we will skip the update by setting Dk1 Dk. Now, we can describe the algorithm for our proposed method LMSI as follows.
- 7. Mathematical Problems in Engineering 7 Algorithm LMSI Steps are the following. Step 1. Given x0 and D0, set k 0. Step 2. Compute Fxk and xk1 xk − DkFxk. Step 3. If Δxk2 Fxk2 ≤ 10−4 , stop. Else, go to Step 4. Step 4. If ΔFk2 ≥ ǫ1 where ǫ1 10−4 , compute Dk1, if not, Dk1 Dk. Set k : k 1 and go to Step 2. 4. Local Convergence Results In this section, we will give some convergence properties of LMSI method. Before we proceed further, we will make the following standard assumptions on nonlinear systems F. Assumption 4.1. We have the following. i F is differentiable in an open-convex set E in Rn . ii There exists x∗ ∈ E such that Fx∗ 0, and F′ x is continuous for all x. iii F′ x satisfies Lipschitz condition of order one, that is, there exists a positive constant µ such that F′ x − F′ y ≤ µ x − y , 4.1 for all x, y ∈ Rn . iv There exists constants c1 ≤ c2 such that c1ω2 ≤ ωT F′ xω ≤ c2ω2 for all x ∈ E and ω ∈ Rn . We will also need the following result which is a special case of a more general theorem of 15. Theorem 4.2. Assume that Assumption 4.1 holds. If there exists KB 0, δ 0, and δ1 0, such that for x0 ∈ Bδ and the matrix-valued function Bx satisfies I − BxF′ x∗ ρx δ1 for all x ∈ Bδ, then the iteration xk1 xk − BxkFxk 4.2 converges linearly to x∗ . For the proof of Theorem 4.2, see 15. Using Assumption 4.1 and Theorem 4.2, one has the following result. Theorem 4.3. Assume that Assumption 4.1 holds. There exist β 0, δ 0, α 0, and γ 0, such that if x0 ∈ E and D0 satisfies I − D0F′ x∗ F δ for all xk ∈ E, then for iteration xk1 xk − DkFxk, 4.3
- 8. 8 Mathematical Problems in Engineering Dk defined by 3.19, I − DkF′ x∗ F δk, 4.4 holds for some constant δk 0, k ≥ 0. Proof. Since Dk1F Dk MkF, it follows that Dk1F ≤ DkF MkF. 4.5 For k 0 and assuming D0 I, we have M i 0 ΔFT 0 Δx0 − ΔFT 0 D0ΔF0 TrG2 ΔF i2 0 ≤ ΔFT 0 Δx0 − ΔFT 0 D0ΔF0 Tr G2 0 ΔF max2 0 , 4.6 where ΔF max 0 2 is the largest element among ΔF i 0 , i 1, 2, . . . , n. After multiplying 4.6 by ΔF max 0 2 /ΔF max 0 2 and substituting TrG2 0 n i1ΔF i 0 4 , we have M i 0 ≤ ΔFT 0 Δx0 − ΔFT 0 D0ΔF0 ΔF max 0 2n i1 ΔFi 4 ΔF max 0 4 . 4.7 Since ΔF max 0 4 / n i1ΔF i 0 4 ≤ 1, then 4.7 turns into M i 0 ≤ ΔFT 0 F′ xΔF0 − ΔFT 0 D0ΔF0 ΔF max 0 2 . 4.8 From Assumption 4.1iv and D0 I, 4.8 becomes M i 0 ≤ |c − 1| ΔFT 0 ΔF0 ΔF max 0 2 , 4.9 where c max{|c1|, |c2|}. Since ΔF i 0 2 ≤ ΔF max 0 2 for i 1, . . . , n, it follows that M i 0 ≤ |c − 1|n ΔF max 0 2 ΔF max 0 2 . 4.10 Hence, we obtain M0F ≤ n3/2 |c − 1|. 4.11
- 9. Mathematical Problems in Engineering 9 Table 1: Results of Chandrasekhar H-equation number of iteration/CPU time. n NM FN BM LMSI 200 4/6.9134 7/5.1673 6/4.7112 5/0.0312 500 4/14.0945 7/11.0418 8/6.9813 5/0.0624 1000 4/64.5813 9/27.2974 8/9.7416 5/0.1716 c 0.9 2000 4/95.6180 9/118.2094 8/11.1690 5/1.6410 5000 — — — 5/2.3556 10000 — — — 5/2.8209 20000 — — — 5/3.7915 200 6/8.1834 8/6.4282 7/5.1946 6/0.0780 500 6/18.2457 8/14.6132 7/7.6137 6/0.0824 1000 6/83.0569 9/30.0542 8/11.0432 6/0.2184 c 0.99 2000 6/121.5309 9/153.0351 8/15.9724 4/1.3104 5000 — — — 4/1.4976 10000 — — — 5/1.7910 20000 — — — 5/2.0371 200 8/15.1792 — 9/5.0388 5/0.3312 500 8/36.0330 — 10/7.1412 5/0.6024 1000 8/134.9045 — 10/12.0644 6/0.7803 c 0.9999 2000 8/175.0521 — 10/14.9064 6/1.2537 5000 — — — 6/1.5132 10000 — — — 6/1.5808 20000 — — — 6/1.6301 Suppose that α n3/2 |c − 1|, then M0F ≤ α. 4.12 From the fact that D0F √ n, it follows that D1F ≤ β, 4.13 where β √ n α 0. Therefore, if we assume that I − D0F′ x∗ F δ, then I − D1F′ x∗ F I − D0 M0F′ x∗ F ≤ I − D0F′ x∗ F M0F′ x∗ F ≤ I − D0F′ x∗ F M0F F′ x∗ F , 4.14 hence I − D1F′ x∗ F δ αφ δ1. And hence, by induction, I − DkF′ x∗ F δk for all k.
- 10. 10 Mathematical Problems in Engineering 0 20 40 60 80 100 120 140 0 1000 2000 3000 4000 5000 6000 NM FN BM LMSI CPU time (seconds) System’s dimensions Figure 1: Comparison of NM, FN, BM, and LMSI methods when c 0.9, in terms of CPU time. 5. Numerical Results In this section, we compare the performance of LMSI method with that of the Newton’s meth- od NM, fixed Newton’s method FN, and Broyden’s method BM. We apply the algo- rithms to the well-known benchmarks integral equations involved in radiative transfer. The comparison is based upon the following criterion: number of iterations, CPU time in seconds, and storage requirement. The computations are done in MATLAB 7.0 using double-precision computer. The stopping criterion used is Δxk Fxk ≤ 10−4 . 5.1 The starting point x0 is given by 1, 1, . . . 1T . The symbol “−” is used to indicate a failure due to the following: 1 The number of iteration is at least 200, but no point of xk satisfying 5.1 is obtained, 2 CPU time in second reaches 200, 3 insufficient memory to initial the run. The numerical results of the methods when solving Chandrasekhar H-Equation in different parameter are reported in Table 1. The first column of the table contains the param- eter of problem. Generally, with our choice of c, the corresponding Jacobian is not diagonally dominate; however, when c → 1, the Jacobian is nearly singular. From Table 1, it was shown that only LMSI is able to solve problems where n 2000. This is due to the fact that LMSI requires very low-storage requirement in building the approximation of the Jacobian inverse. Indeed, the size of the updating matrix increases in On as the dimension of the system increases, as opposed to NM, FN, and BM methods that increase in On2 . Moreover, we observe that LMSI method has a 100% of success rate convergence to the solution when compared with NM method having 57%, FN method with 39% and BM with 71%, respectively. In addition, it is worth mentioning that the result of LMSI in solving
- 11. Mathematical Problems in Engineering 11 0 20 40 60 80 100 120 140 160 180 0 1000 2000 3000 4000 5000 6000 NM FN BM LMSI CPU time (seconds) System’s dimensions Figure 2: Comparison of NM, FN, BM, and LMSI methods when c 0.99, in terms of CPU time. 0 20 40 60 80 100 120 140 160 180 200 0 1000 2000 3000 4000 5000 6000 NM LMSI BM CPU time (seconds) System’s dimensions Figure 3: Comparison of NM, FN, BM, and LMSI methods when c 0.9999, in terms of CPU time. problem 1 when c 0.9999 shows that the method could be a good solver even when the Jacobian is nearly singular. Figures 1, 2, and 3 reveal that the CPU time of LMSI method in- creases linearly as the dimension of the systems increases, whereas for NM, FN, and BM, the rates grow exponentially. This also suggests that our solver is a good alternative when the dimension of the problem is very high. 6. Conclusion In this paper, we present a low memory solver for integral equation of Chandrasekhar type in the radiative transfer problems. Our approach is based on approximating the Jacobian inverse into a diagonal matrix. The fact that the LMSI method can solve the discretized integral
- 12. 12 Mathematical Problems in Engineering equations without computing and storing the Jacobian makes clear the advantage over NM and FN methods. It is also worth mentioning that the method is capable of significantly reducing the execution time CPU time, as compared to NM, FN, and BM methods while maintaining good accuracy of the numerical solution to some extend. Another fact that makes the LMSI method appealing is that throughout the numerical experiments it never fails to converge. Finally, we conclude that our method LMSI is a good alternative to Newton-type methods for solving large-scale nonlinear equations with nearly singular Jacobian. References 1 S. Chandrasekhar, Radiative Transfer, Oxford University Press, Londo, UK, 1950. 2 J. Caballero, A. B. Mingrarelli, and K. Sadarangani, “Existence of solution of an integral equation of Chandrasekhar type in the radiative transfer,” Electronic Journal of Differential Equations, vol. 27, pp. 1–11, 2006. 3 G. A. Hively, “On a class of nonlinear integral equations arising in transport theory,” SIAM Journal on Mathematical Analysis, vol. 9, no. 5, pp. 787–792, 1978. 4 W. Kalkofen, Numerical Radiative Transfer, Cambridge University Press, Cambridge, UK, 1987. 5 I. W. Busbridge, “On solutions of Chandrasekhar’s integral equation,” Transactions of the American Mathematical Society, vol. 105, pp. 112–117, 1962. 6 R. W. Leggett, “A new approach to the H-equation of Chandrasekhar,” SIAM Journal on Mathematical Analysis, vol. 7, no. 4, pp. 542–550, 1976. 7 C. T. Kelley and J. I. Northrup, “A pointwise quasi-Newton method for integral equations,” SIAM Journal on Numerical Analysis, vol. 25, no. 5, pp. 1138–1155, 1988. 8 M. O. James and C. R. Werner, “On discretization and differentiation of operators with application to Newton’s method,” SIAM Journal on Numerical Analysis, vol. 3, no. 1, pp. 143–156, 1966. 9 J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall Series in Computational Mathematics, Prentice Hall , Englewood Cliffs, NJ, USA, 1983. 10 K. Natasa and L. Zorna, “Newton-like method with modification of the right-hand vector,” Mathematics of Computation, vol. 71, pp. 237–250, 2001. 11 C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Mathematics of Computation, vol. 19, pp. 577–593, 1965. 12 M. Farid, W. J. Leong, and M. A. Hassan, “A new two-step gradient-type method for large-scale unconstrained optimization,” Computers Mathematics with Applications, vol. 59, no. 10, pp. 3301– 3307, 2010. 13 M. A. Hassan, W. J. Leong, and M. Farid, “A new gradient method via quasi-Cauchy relation which guarantees descent,” Journal of Computational and Applied Mathematics, vol. 230, no. 1, pp. 300–305, 2009. 14 C. G. Broyden, “Quasi-Newton methods and their application to function minimisation,” Mathematics of Computation, vol. 21, pp. 368–381, 1967. 15 C. T. Kelley, Iterative Methods for Linear and Nonlinear Equations, vol. 16 of Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics SIAM, Philadelphia, Pa, USA, 1995. 16 S. Chandrasekhar and F. H. Breen, “On the radiative equilibrium of a stellar atmosphere. XIX,” The Astrophysical Journal, vol. 106, pp. 143–144, 1947. 17 J. E. Dennis and H. Wolkowicz, “Sizing and least-change secant methods,” SIAM Journal on Numerical Analysis, vol. 30, no. 5, pp. 1291–1314, 1993.
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