I am looking for stochastic methods for solving a very high-dimensional PDE (with one time dimension and very large number of spatial dimensions), which would reduce it to a lower-dimensional problem, probably at the cost of carrying out a Monte Carlo simulation. Any pointers?



Stochastic methods for solving very high-dimensional PDE
It seems to me this question "has not received enough attention" because of the conflation of two issues: dimensional reduction of a high-dimensional PDE and stochastic (Monte Carlo) integration of the PDE. 


The socalled ["curse of dimensionality"](http://en.wikipedia.org/wiki/Curse_of_dimensionality) refers to the fact that the computational time required for a non-stochastic solution of a PDE grows exponentially with the number of dimensions (each dimension needs to be discretized, say with $N$ points, so the total number of points in $d$ dimensions is $N^d$). Dimensional reduction is then imperative, and one method to achieve that is [principal component analysis](http://en.wikipedia.org/wiki/Principal_component_analysis).


The computational time for a solution of a PDE by Monte Carlo integration grows only linearly with dimension, so dimensional reduction is not needed. The accuracy of this approach is low, and this is why one tries to avoid resorting to a Monte Carlo method.


The two approaches to the solution of a high-dimensional PDE, dimensional reduction by principal component analysis and Monte Carlo integration, are compared in these [lecture notes.](https://www.it.uu.se/edu/course/homepage/projektTDB/vt07/Presentationer/Projekt3/Dimension_Reduction_for_the_Black-Scholes_Equation.pdf)




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Upon some more search, I found one dimensional-reduction scheme with a stochastic component. It goes by the acronym RS-HDMR = Random-Sampling-High-Dimensional-Modeling-Representation and is described [here.](https://www.ima.umn.edu/talks/workshops/3-15-19.2000/li/hdmr.pdf) (The HDMR [Wikipedia page](http://en.wikipedia.org/wiki/High-dimensional_model_representation) could use some expansion...)



