an other update to future work

latex/main paper/main.tex

@@ -1618,40 +1618,48 @@ \subsection{First Passage Sampling}
 
 \section{Limitations and Future Work}
 
-In this section we discuss the limitations of the presented work and interesting future directions. Many
-areas of presented work are not fully addressed, worked out or neither well integrated in
-the current literature. At time of writing this thesis we were mainly aware of the WoS literature
-and we independently rediscovered the improved random convergence of integration and IVPs,
-the use of a Poisson process to avoid recursion in \ref{jl:main poisson}
-and the recursive estimator for $e^{E[X]}$ in example \ref{ex:exp int} ,
-these discoveries guided us to the existing literature of work on it. \\
+In this section, we explore the limitations of our study and propose potential avenues for future work. 
+Several aspects of our work remain underdeveloped, inadequately explored, or insufficiently integrated
+into the existing body of literature. At the time of writing this thesis, our 
+primary reference was the WoS literature. We independently rediscovered 
+the improved random convergence of integration and IVPs, the 
+use of a Poisson process to circumvent recursion (as shown in \ref{jl:main poisson}), 
+and the recursive estimator for $e^{E[X]}$ (as demonstrated in example \ref{ex:exp int}). 
+These discoveries subsequently led us to other relevant literature. \\
 
 %maybe I should add this
 We lack a parallel complexity analysis for RMC for linear IVPs. Different to MC, the cost of an estimator in RMC
 is also a RV. Naively we would expect unlimited parallel vs wall time scaling just like in MC but the due
 to the tails of the cost distribution wall time at risk scales atleast logarithmic with splitting. The analysis
 for \ref{jl:main poisson} simplifies because the cost distribution of the estimator is Poisson distributed.\\
 
-The part on RRMC was written before we were aware of algorithm \ref{jl:main poisson} therefore we have not explored
-combining the two. The way we applied recursion in recursion to RMC for IVPs in RRMC is limited to
-achieving convergence behavior similar to classical solvers while staying unbiased.
-Recursion in recursion allows to 'freeze' terms in the outer recursion while staying unbiased,
-'freezing' slow varying terms is the next natural thing to try. \\
-
-As our techniques for linear IVPs are based on transforming to integral equations, generalizations to
-linear delay differential equation may be possible. An other direction is directly estimating the Magnus expansion
-where unbiased estimators for $e^{\int_{0}^{\Delta t} A(s)ds} y(0)$ are needed. This connects to
-work on evaluating matrix functions with MC see \cite{guidotti_fast_2023}.\\
-
-We have not figured out the best way to implement coupled splitting while avoiding recursion. In subsection
-\ref{sec:greens functions} we derived global integral equations for our test problem but ODEs are local in nature.
-We are interested in methods that can exploit the local nature of ODEs directly in the estimators based on the
-global integral equations.\\
-
-We used the adjoint main Poisson method \ref{jl:adjoint main poisson} as base for deriving our point estimators
-for the heat equation. It is still unclear what class of problems that can be handled by the adjoint main Poisson method
-similar to how the heat equation can be handled. \\
-
+The section on RRMC was composed prior to our discovery of algorithm \ref{jl:main poisson}, 
+and as such, we did not investigate potential integration of the two. Our application of 
+recursion in recursion in RRMC for IVPs is primarily aimed at achieving convergence behavior 
+akin to classical solvers, while maintaining unbiasedness. This recursive approach allows us to 
+'freeze' terms in the outer recursion without bias. The next logical step would be 
+to attempt 'freezing' terms that vary slowly. We believe that RRMC, even after optimizations, could
+prove beneficial in specific use cases. Furthermore, insights derived from unbiased algorithms could
+enhance the performance of biased algorithms. \\
+
+Our techniques for linear IVPs, which are based on transforming to integral equations, may be extendable 
+to linear delay differential equations. Another potential direction involves directly estimating 
+the Magnus expansion, where unbiased estimators for $e^{\int_{0}^{\Delta t} A(s)ds} y(0)$ are required. 
+This is related to work on evaluating matrix functions with Monte Carlo methods, as seen in \cite{guidotti_fast_2023}. \\
+
+In subsection \ref{sec:greens functions}, we developed global integral equations for our test problem. 
+Despite the inherent local nature of ODEs, we are curious whether this locality is retained in the global integral equations. 
+More specifically, we are interested in whether this local nature can be leveraged to enhance the efficiency of 
+estimators solving these global integral equations, even without knowledge of the original ODEs. \\
+
+Coupled splitting's convergence behavior and unbiasedness at each iteration are key attributes. 
+However, the use of recursion and Russian roulette in its implementation poses practical challenges. 
+Despite these challenges, numerous opportunities exist to further optimize the performance of coupled splitting
+with typical MC techniques.
+
+We employed the adjoint main Poisson method \ref{jl:adjoint main poisson} as the foundation for deriving our 
+point estimators for the heat equation. However, it remains uncertain which class of problems can be addressed 
+by the adjoint main Poisson method in a manner similar to the heat equation. 
 % maybe I should add this
 We have not fully discussed how to combine presampling with path stitching and when it is possible to do so.
 It has a nice interepertation with a U-statistic. \\