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" 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.

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.

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" 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.

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.

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. (The HDMR Wikipedia page could use some expansion...)

It seems this question remains unanswered because you're conflating two issues: dimensional reduction of a high-dimensional PDE and stochastic (Monte Carlo) integration of the PDE.

The socalled "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.

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.

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" 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.

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.