On Solving a System of Volterra Integral Equations with Relaxed Monte Carlo Method

A random simulation method was used for treatment of systems of Volterra integral equations of the second kind. Firstly, a linear algebra system was obtained by discretization using quadrature formula. Secondly, this algebra system was solved by using relaxed Monte Carlo method with importance sampling and numerical approximation solutions of the integral equations system were achieved. It is theoretically proved that the validity of relaxed Monte Carlo method is based on importance sampling to solve the integral equations system. Finally, some numerical examples from literatures are given to show the efficiency of the method.


Introduction
In engineering, social and other areas, a lot of problems can be converted to Volterra integral equations to solve, such as elastic system in aviation, viscoelastic and electromagnetic material system and biological system, and some differential equations are often transformed into integral equations to solve in order to simplify the calculation.For example, the drying process in airflow, pipe heating, gas absorption and some other physical processes can be reduced to the Goursat problem.Then, some of the Goursat problem can be described by Volterra integral equations [1].Another example, when one-dimensional situations are concerned and the coolant flow is incompressible, the definite solution problem of the transpiration cooling control with surface ablation appears as Volterra integral equations of second kind [2].In practice, the analytical solutions for this kind of integral equations are difficult to obtain.Therefore, it is more practical to research the numerical method for solving this kind of integral equations.
The main aim of this paper is to propose a numerical algorithm based on Monte Carlo method for approximating solutions of the following system of Volterra integral equations where ( , ), ( , ) 0, 1, 2 p q = are known kernel functions, the functions 1 ( ) f x , 2 ( ) f x are given and defined in a x b ≤ ≤ , and 1 2 ( ), ( ) are the unknown functions to be determined.One of the earliest methods for solving integral equations using Monte Carlo method was proposed by Albert [3], and was later developed [4].Literatures [5]- [8] employed Monte Carlo method to solve numerical solutions of Fredholm integral equations of the second kind.But very few studies are devoted to employing Monte Carlo method to solve Volterra integral equations and the system of Volterra integral equations.In this paper, we present and discuss a relaxed Monte Carlo approach with importance sampling to solve numerically systems of Volterra integral equations.Due to less accuracy and lower efficiency of Monte Carlo method, in this paper, combination of Monte Carlo and quadrature formula will be used to deal with Equation (1) and importance sampling is applied to accelerate the convergence and improve the accuracy of Monte Carlo method.Some numerical examples are given to show the efficiency and the feasibility of proposed Monte Carlo method.

Discretizing System of Integral Equations
Here, Newton-Cotes quadrature formula is used to discretize Equation (1).Dividing the interval ( , ) . For convenience, denoting the notation ( ) , where , 0,1, , Thus the following linear algebra system can be obtain ( , , , , , , , ) ( ) , j ω is the weight of Newton- Cotes quadrature formula.The matrix of coefficients of Equation ( 2) is A I B = − .If we assume that there ex- ists a unique solution of (2), the solution would be a numerical approximation of (1).This process will produce an error which is determined by numerical quadrature formula and can be reduced by increasing the number of nodes for a given quadrature formula.For a large number of nodes, Equation ( 2) is too large to solve directly.It is well known that Monte Carlo technique has a unique advantage for large systems or high-dimensional problems.At the same time, this method can obtain function values at some specified points or their linear combination that is just what researchers need.But for determined numerical methods, in order to obtain function value at a certain point, it is often necessary that find function values for all nodes.Here relaxed Monte Carlo method is used to Equation (2) based on a random sample from Markov chain with discrete state.According to theory of importance sampling, probability transition kernel is selected to suggest a possible move.To obtain solution of the linear algebraic system (2), the following iterative formula is considered where is chosen such that it minimizes the norm of L for accelerating the convergence, and 1 F DF = .The iterative formula (3) can define a Neumann series, as following Set iterative initial value (0) Φ is the exact solution of Equation ( 2), the truncation error and con- vergency of the iterative formula (3) can be obtained by the following expression This conclusion can be proved by using theories in numerical analysis.Here, the iterative matrix L satisfies ) To achieve a desirable norm in each row of L , a set of relaxation parameters, { } 2( 1) will be used in place of a single γ value.According to the arguments of Faddeev and Faddeeva [9] [10], the relaxed Monte Carlo method will converge if

Relaxed Monte Carlo Method with Importance Sampling
For Neumann series (4), we have In order to obtain the approximation solution of linear system (2) and system of integral Equation ( 1), the kth iteration ( )

Φ
of i Φ be evaluated by means of computing the following series .
Construct the Markov chain on the state space { } 1, 2, , The weight function m W of Markov chain is defined as follows By expressions ( 7) and ( 8), the following conclusion can be gotten.Theorem 3.1 For the given ( 0) m m > , we have This theorem is easy to prove.In the light of the expression (7), the following estimator is defined ) .
Due to Theorem 3.1, the conclusion ( 11) is easy to prove.
< , ε is the precision of truncation error and given in ad- vance.Then one can evaluate the sample mean is bounded, according to the Central Limit Theorem, we would obtain ( ) So the precision of the estimator ( ) k ξ in the sense of probability can be measured by its variance . Based upon the minimum variance of estimator 1 ξ , by the variance expression ( ) Var( ) , the transition probability ij p of Markov chain should be chosen in the following form .
This form of ij p leads to that more samples are taken in regions which have higher function values.This is importance sampling.

Numerical Examples
In this section, we employ the proposed relaxed Monte Carlo method with importance sampling (say RMCIS) to compute the numerical solution of some examples and compare it with their exact solutions.The numerical results are presented in Table 1 and Table 2, where AE means absolute error for ( ) ( 1, 2) p x p ϕ = . We plot the       2.

Conclusion
In this paper, a relaxed Monte Carlo numerical method is provided to solve a system of linear Volterra integral equations.The most important advantage of this method is simplicity and easy-to-apply in programming, in comparison with other methods.The implementation of current approach RMCIS is effective.The numerical examples that have been presented in the paper and the compared results support our claims.

A
denotes the row norm of the given matrix A .

Figure 2 .
Figure 2. The figure of average absolute errors (MAE) for Example 2 at eleven points 1 , , 1 .0 , 0  , (a) for ) ( 1 x ϕ numerical results are listed in Table

Table 1 .
Numerical results of Example 1 with Figure 1 and Figure 2. Below are the numerical results for some of them.