Abstract
Based on recent progress on moment problems, semidefinite optimization approach is proposed for estimating upper and lower bounds on linear functionals defined on solutions of linear integral equations with smooth kernels. The approach is also suitable for linear integrodifferential equations with smooth kernels. Firstly, the primal problem with smooth kernel is converted to a series of approximative problems with Taylor polynomials obtained by expanding the smooth kernel. Secondly, two semidefinite programs (SDPs) are constructed for every approximative problem. Thirdly, upper and lower bounds on related functionals are gotten by applying SeDuMi 1.1R3 to solve the two SDPs. Finally, upper and lower bounds series obtained by solving two SDPs, respectively infinitely approach the exact value of discussed functional as approximative order of the smooth kernel increases. Numerical results show that the proposed approach is effective for the discussed problems.
1. Introduction
Semidefinite optimization has been successfully applied to deal with many important problems [1–10] since it was proposed in 1963 [11]. Recently, the authors in [10] presented the semidefinite optimization method for obtaining guaranteed bounds on linear functionals defined on solutions of linear differential equations with polynomial coefficients. Instead of directly handling linear differential equations, the approach gets the discussed bounds by solving SDPs based on these equations and related functionals. Their numerical results are very encouraging. The authors in [12] proposed the semidefinite optimization method for estimating bounds on linear functionals of solutions of linear integral and integrodifferential equations with polynomial kernels. The method does not directly solve these equations as successive approximations method, Runge-Kuta one, direct computation one, the Adomian decomposition one, the modified Adomian decomposition one, the variational iterative one, and so forth in the references [13–21]. Numerical results show that the proposed method can get guaranteed bounds on the discussed functionals.
In this paper, we will extend the polynomial kernels in [12] to the generally smooth kernels and propose semidefinite optimization method for providing guaranteed bounds on linear functionals defined on solutions of Fredholm and Volterra integral equations with generally smooth kernels. Firstly, we expand smooth kernel in Fredholm or Volterra integral equation as a series of Taylor polynomials, and then the Fredholm or Volterra integral equation with the smooth kernel is converted to a series of integral equations with polynomial kernels. Secondly, semidefinite programs (SDPs) are constructed based on these approximative equations and discussed functionals. Thirdly, we apply SeDuMi 1.1R3 [22] to solve these SDPs and get upper and lower bounds sequences which all converge to the exact value of related functional. Finally, we illustrate the effectiveness of the method by carrying out some numerical experiments.
The rest of this paper is organized as follows. In Section 2, we propose semidefinite optimization method for estimating guaranteed bounds on the linear functionals defined on solution of Volterra integral equation of the second kernel with smooth kernel. In Section 3, four numerical examples are tested. We end the paper with some conclusions and discussions in the final section.
2. Semidefinite Optimization Method
In this section, we propose semidefinite optimization method for estimating guaranteed bounds on linear functionals defined on solutions of Volterra integral equation of the second kind with smooth kernel.
Throughout the work, we suppose that related integral equations make unique solutions exist in the distribution space , in which the polynomial ring is dense.
Primal Problem. Computing in which satisfies where and the kernel are given in advance, and the former is an integrable function on the interval , and the latter is an infinitely smooth function in variables and .
Equation (2) is called Volterra integral equation of the second kind [17].
We expand the kernel as the following Taylor polynomial with orders at and : where
Therefore, (2) can be approximated by the following integral equation: in which is of the form (4).
For simplicity, we rewrite in (4) as
Further, Primal Problem can be written as the approximative form.
Approximative Problem. Compute where , satisfies (7), in which is just the one in (2) and the kernel is defined by (8).
In [12], semidefinite optimization method has been proposed for estimating bounds on linear functionals defined on solution of linear integral equation with polynomial kernel. Now we present semidefinite optimization method for providing guaranteed bounds on the linear functional (1) in Primal Problem, which is a generalized form of the method proposed in [12].
Algorithm 1.
Step 0. Let , and give a tolerance .
Step 1. Convert Primal Problem to Approximative Problem according to the above analysis.
Step 2. Generate linear equality constraints.
Suppose that the solution of (7) is bounded from below; that is, there exists such that
We define
which may be called moments even though may not be a probability distribution.
Multiplying (2) by the testing functions , and integrating it over the interval , we can get
By (11), we obtain
Substituting (13) into (12), we can get
in which
Because the solution of (2) is in the distribution space , in which is dense, (7) can be transformed into the system which consists of the equations as described by (14) where .
Step 3. Generate semidefinite constraints.
It is obvious that is equivalent to .
Denote
where is a nonnegative integer.
By the method in [10], for , we have
We can obtain the following positive semidefinite matrices:
in which
where in the three matrices , , and , by replacing in (17) with and setting the four subsets of the set to in (17), respectively.
Step 4. Construct two SDPs.
Assuming that the testing function with the highest degree is , we get the following two SDPs:
where decision variables are , .
Step 5. Apply SeDuMi 1.1R3 to solve the two SDPs.
Denote by the decision variable obtained by solving the above maximizing programming. has similar meaning.
Step 6. Define whether the highest degree of the testing function increases or not.
When , go to Step 7; or if
all hold, go to Step 7, or let and go to Step 4.
Step 7. Judge whether iteration goes on or not.
When , go to Step 2, or if
all hold, stop the iteration and output and , which are upper and lower bounds of , respectively; or let and go to Step 1.
Remark 2. Obviously, in Algorithm 1 can extend to with . Of course, some necessary modifications must be done.
Remark 3. The proposed method is also suitable for other linear integral and integrodifferential equations with smooth kernels.
Remark 4. In general, in (5) is unknown. But we do not need infimum of over , so we can set a small value to .
Remark 5. The semidefinite constraints (18) in Algorithm 1 only depend on the integral interval . The moments in the semidefinite constraints (18) do not appear in the linear constraints. They are extension moments (see [9] or [10] for details).
Remark 6. In practical applications, for reducing computation amounts of Algorithm 1, we usually set two suitable positive integers to and , respectively.
Remark 7. In some cases, in (2) can also be expanded as Taylor polynomials in variables and with different orders, respectively.
3. Numerical Experiments
In this section, we give four examples to illustrate the effectiveness of Algorithm 1. For simplicity, the interval is always taken as the integral interval in these examples.
3.1. Volterra Integral Equation of the First Kind with Smooth Kernel
Example 1. Computing
where , satisfies
The exact solution of (24) is .
Multiplying (24) by and integrating it over the interval , we can get
In this example, for all . Zero is set to in (11). Then we define
Taylor polynomial with degree 6 of at and Taylor polynomial with degree 7 of at are as follows:
where , , , , , , , , , , , and , respectively.
Substituting (27) into (25) and replacing with , we can obtain
For , we know that the semidefinite constraints are the same as (18).
We construct the following two SDPs:
Letting , we apply SeDuMi 1.1R3 to solve the SDPs. The partial numerical results are reported in Table 1.
In Table 1, UN-Ob, LLFoS, ULFoS, and ELFoS mean decision variables or objective functions in the above two SDPs, lower bounds, upper bounds, and exact values on linear functionals defined on solution of (24), respectively. Error means . These signs in Tables 2, 3, 4, and 5 have the same meanings. From Table 1, we can see that for every , is more than , and these errors do not decrease as increases. Maybe the case is resulted in by accumulative error.
In order to increase the precision of numerical results of Example 1, we first convert (24) to the following equivalent integral equation (30) and then apply Algorithm 1 to estimate (23), where satisfies (30).
Differentiating both sides of (24) with respect to gives
Use the two polynomials where , , , , , , , and , to approximate and , respectively.
Substituting (31) into (30), multiplying its both sides by , and integrating it over the interval , we have
Let
Substituting (33) into (32) and simplifying it, we get
According to , we have semidefinite optimization as stated by (18).
We construct the SDPs:
Letting and , we apply SeDuMi 1.1R3 to solve the above SDPs. Some numerical results are listed in Table 2.
From Table 2 we can see that the numerical results are accurate to four decimal points of the related exact values. Numerical results in Table 2 show that the proposed approach can efficiently estimate upper and lower bounds on the linear functionals defined on solution of Volterra integral equation of the first kind. If we want to increase precision of the numerical results, we can reach the goal by expanding and in the kernel of (24) as Taylor polynomials with higher degrees. We also solved the max/min programs when . Numerical results show that and , , are all less than . So the examples are as follows.
3.2. Volterra Integral Equation of the Second Kind with Smooth Kernel
Example 2. Computing
where , satisfies
The exact solution of the equation is . Define as stated by (26). We expand as follows:
where , , , , and .
Multiplying (37) in which is replaced with by and integrating it over the interval , we can get
Because , we obtain semidefinite constraints as stated by (18).
We construct the following SDPs:
Letting and , we apply SeDuMi 1.1R3 to solve the above SDPs. The partial numerical upper and lower bounds are listed in Table 3.
Numerical results in Table 3 show that the proposed method is very effective for obtaining guaranteed upper and lower bounds whose precision can reach .
3.3. Fredholm Integral Equation of the Second Kind with Smooth Kernel
Example 3. Computing
where , satisfies
The exact solution of (42) is .
Chebyshev polynomial with degree 3 of at and Chebyshev polynomial with degree 4 of at are as follows:
Define as (26). Substituting (43) into (42), multiplying (42) by , and integrating it over the interval , we have
For , we get semidefinite constraints as stated by (18).
We construct the following SDPs:
Letting and , we apply SeDuMi 1.1R3 to solve the above SDPs and get numerical upper and lower bounds of related functionals which are partially listed in Table 4.
Obviously, the numerical results in Table 4 are accurate to four decimal points of the exact functional values.
3.4. Volterra Integrodifferential Equation with Smooth Kernel
Example 4. Computing
where , satisfies
with boundary condition .
The exact solution of (47) is .
Integrating (47) over the interval , we get
Replace in the kernel of (48) with (38), and do the similar replacement for in the kernel. Applying SeDuMi 1.1R3 to solve the corresponding approximative problem which is just Volterra integral equation of the second kind, we get numerical results partially reported in Table 5.
Numerical results in Table 5 show that the semidefinite optimization method can provide guaranteed bounds on linear functionals defined on solution of (47). If we hope to obtain more accurate numerical results, we can expand and in the kernel of (47) as Taylor polynomials with higher degrees.
4. Conclusions and Discussions
In this paper, we have presented the semidefinite optimization method for providing guaranteed bounds on linear functionals defined on solutions of linear integral equations with smooth kernels. Four examples show that the proposed approach is effective for estimating bounds on linear integral and integrodifferential equations with smooth kernels. The proposed approach requires that the related integral equation is linear. It cannot be directly applied to solve the nonlinear integral equation. So next work is to improve the proposed method, so that it can handle nonlinear problems.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the anonymous reviewers for their helpful comments. This work is partially supported by the National Natural Science Foundation, Grants nos. 61202397 and 11301445, the Research Foundation of Education Bureau of Hunan Province, Grant no. 13B121, the Research Foundation for Doctoral Program of Higher Education of China Grant no. 20114301120001, and the Science and Technology Project of Hunan Province.