Appendix A
Resolvent for Integral Equations

While the computation of integrals in itself is a worthy application of Monte Carlo theory, especially in the case of high dimensionality of the integration domain, it can be employed to obtain solutions, or at least estimates, to integral equations.

Recalling that Fredholm integral equations of the second kind

ϕ (s ) = f (s) + λ  K (s,t)ϕ(t)dt                       (A.1)
have a solution of the form
          ∫                            ∑∞ ∫
ϕ(s)=f(s) + λ    R(s,t,λ)f (t)dt = f (s) + λ       Ki+1 (s,t)λif(t)dt.      (A.2)
           B                           i=0  B
as outlined in Section 4.9. This resolvent series includes repeated integrals which are accessible to Monte Carlo integration methods.

Beginning with examining the integrals in the resolvent series and rewriting them as expectation values gives

  ∫                     ∫
λ    R(s,t,λ )f (t)dt = λ    p(t)R-(s,t,λ-)f(t)dt                    (A.3)
   B                     B         p (t)
                        ∫      ∑ ∞   K   (s,t)λif(t)
                    = λ    p(t)--i=0--i+1-----------dt            (A.4)
                         B             p (t)
                        ∫      ∑∞  K    (s,t)
                    = λ    p(t)    --i+1-----λif (t)dt.             (A.5)
                         B     i=0    p(t)
Examining the terms in order of increasing summation index shows
K1(s,t)-+ K2-(s,t)λ + ...                          (A .6)
p (s,t)     p(s,t)
Recalling Equation 4.184 gives
Kn(s,t) =    Kn −1(s,t1)K (t1,t)dt,     K1(s,t) = K (s,t),          (A .7)
where further integrals appear, when expanding K2
K2 (s,t)      1
--p(t)--λ =  p(t)-   K (s,t1)K (t1,t)dt1λ,                   (A .8)
which can again be evaluated in a similar fashion as an expectation value (Definition 101) with a random variable (Definition 98).
K2-(s,t)-   -1--        K-(s,t1)K--(t1,t)
 p(s,t) =  p(t)   p(t1)     p(t )     dt1λ                 (A.9a)
           ∫    B              1
        =     p(t)K-(s,t1)K-(t1,t) dtλ                      (A.9b)
            B   1   p(s,t)p (t,t1)    1

The recursion relation A.7 allows to reuse the obtained result in the subsequent terms and further recursion reveals that the term for Kn (s,t)  evaluated using Monte Carlo methods (see Section 6.6.2) takes the shape

          ∫            (n −2          )
               K-(s,t1)-  ∏  K-(ti,ti+1)   K-(tn−-1,t)-
Kn (s,t) =  Bn p(s,t1)       p(ti,ti+1)   p(tn−1,t) dt1...dtn−1       (A.10)
Thus every term in the series requires an additional random number to be generated in order to calculate an expectation value. The notion of the procedure is depicted in Figure A.1.


Figure A.1: The general Monte Carlo scheme due to Equation A.10.