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In modeling population genetics, I derived a 1+4 dimensional parabolic equation of the form: \begin{align} \frac{\partial u}{\partial t}=&\frac{1}{2}\sum_{i=1}^4a(x_i)^2\frac{\partial^2u}{\partial x_i^2}\\ &+b_1(x_1,x_2)\frac{\partial u}{\partial x_1}+b_2(x_1,x_2)\frac{\partial u}{\partial x_2}\\ &+b_3(x_3,x_4)\frac{\partial u}{\partial x_3}+b_4(x_3,x_4)\frac{\partial u}{\partial x_4}\\ &+\epsilon \sum_{i=1}^4 h_i(x_1,x_2,x_3,x_4)\frac{\partial u}{\partial x_i} \end{align} where $\epsilon$ is a small parameter and $h_i$ are linear functions of $x_1,\cdots,x_4$. This is simply the Kolmogorov Backward Equation governing the expectation of some function $u$ of the underlying 4 random variables satisfying four stochastic differential equations.

However, it is very difficult to numerically solve this 1+4 PDE using deterministic algorithms given the high dimensions. I also don't wanna go back to do Monte Carlo simulation over the 4 SDEs because that would make deriving this PDE meaningless..

An interesting approach, which I have been thinking, but has not figured it out, is that dimensions $x_1,x_2$ are ONLY coupled to dimensions $x_3,x_4$ through the small parameter $\epsilon$. If $\epsilon=0$, solving this PDE is equivalent to solving two 1+2 dimensional PDEs in the $(x_1,x_2)$ and $(x_3,x_4)$ domains, respectively, which will greatly simplify the numerical algorithm. And the eventual solution to the original PDE can be viewed as a perturbation (or by coupling them into two systems?).

Does any one know if there is any implementation of such idea? Or is it mathematically plausible to do so?

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  • $\begingroup$ This approach called operator splitting or just splitting is widely used in mechanics to solve numerically PDEs with several terms. $\endgroup$ – Harry49 Feb 19 at 8:20
  • $\begingroup$ Thank you @Harry49! I found a similar approach called Alternating Direction Methods $\endgroup$ – yixianshuiesuan Feb 19 at 16:46

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