Stochastic approximation applied to fixed source problem

[wronski77]

Dear forum members,


I am trying to solve the following system of equations.

ψ(x,y,z)=∫∫ψ(x',y',z)K(x',y',z)dx'dy'

z=f(ψ)

What I do is to solve the integral equation with a Monte Carlo method, evaluate "z" and do a loop until convergence.

My question to you is whether it is possible to accelerate the convergence by using stochastic approximation method such as Stochastic gradient descent (a.k.a. Robbins-Monro). I would highly appreciate any comments on the subject, including general information about the Robbins-Monro algorithm. What I know about the Robbins-Monro algorithm is that it is used to find zeros of nonlinear stochastic equations. Can It be applied to stochastic Fredholm problems like the one above?


Thank you in advance,

 

[mmwave]

[username]

Dear [username],

Thank you for your question and for sharing your approach to solving the system of equations. The Monte Carlo method is a commonly used technique for solving integrals, and it is interesting that you are using it in conjunction with the function "z=f(ψ)". As for your question about accelerating the convergence, the answer is yes. The Robbins-Monro algorithm can indeed be applied to stochastic Fredholm problems like the one you have described.

The Robbins-Monro algorithm, also known as stochastic gradient descent, is a powerful method for finding the zeros of nonlinear stochastic equations. It is a type of stochastic approximation method that uses a sequence of stochastic approximations to converge to the desired solution. In simple terms, it is an iterative algorithm that updates the solution based on the gradient of the objective function at each iteration.

In the context of your problem, the objective function would be the difference between the calculated value of "z" and the desired value of "z". By using the Robbins-Monro algorithm, you can update your solution at each iteration and hopefully reach convergence faster than with the Monte Carlo method alone.

However, it is important to note that the success of the algorithm depends on the specific characteristics of your problem. It may not always result in faster convergence and could potentially lead to instability in certain cases. Therefore, I would recommend testing the algorithm on your specific problem and comparing the results with the Monte Carlo method to determine its effectiveness.

I hope this helps answer your question. If you have any further inquiries, please do not hesitate to ask.