Conceptually, I understand how a numerical method like Monte Carlo is used to solve a definite integral. Because integral of a function is the area bounded by the curve, the ratio of random points that land inside the curve to the total number of points is the value of the integral.
Conceptually, can someone explain for a non math person, how we can solve a PDE/ODE using Monte Carlo?
Monte Carlo (MC) simulation methods solve mathematical problems by random sampling, providing approximations where analytical or numerical methods are impractical. For ordinary differential equations (ODEs) and partial differential equations (PDEs), MC uses probabilistic interpretations to approximate solutions.
ODEs describe the rate of change of a variable with respect to a single independent variable. They are used to model systems where the future state depends only on the current state, like population growth or the motion of a pendulum. Examples include Newton’s second law of motion in physics, the logistic growth model in biology, and compound interest in finance. PDEs involve rates of change with respect to multiple independent variables, often space and time, and are used to describe more complex systems. Examples include the modelling heat diffusion in physics and engineering, reaction-diffusion equations for biological pattern formation, and the Black-Scholes equation in finance for pricing options.
For ODEs, consider an equation such as $\frac{dy}{dx} = f(x, y)$ with initial condition $y(x_0) = y_0$. This can be reformulated as:
$$ y(x) = y_0 + \int_{x_0}^x f(t, y(t)) \, dt $$
MC methods approximate this integral by sampling random paths that represent possible trajectories of $y(t)$ over the interval $[x_0, x]$. These paths are typically simulated using stochastic techniques that reflect the governing behaviour of the system. The solution at a specific point, $y(x)$, is then estimated as the average behaviour across these paths.
For PDEs, the probabilistic approach becomes more complex due to multiple variables and spatial domains. Many PDEs, such as the heat equation $\frac{\partial u}{\partial t} = \Delta u$, correspond to stochastic processes like Brownian motion. Here, MC simulates particle random walks within the domain to approximate the solution. The Feynman-Kac theorem provides a rigorous foundation by linking certain PDEs to expectations over stochastic processes. This allows MC to reformulate the solution as the statistical average of simulated behaviours.
MC methods are particularly advantageous for high-dimensional PDEs, where traditional grid-based methods (eg., finite differences) face computational challenges. By avoiding explicit gridding, MC sidesteps the curse of dimensionality. However, accuracy depends on sample size, often requiring variance reduction techniques to improve computational efficiency.
In essence, MC methods solve ODEs and PDEs by transforming them into probabilistic problems, estimating solutions through simulations of dynamic systems. While conceptually similar to MC integration, these applications address evolving trajectories rather than static areas.
The trick is to convert ODE/PDE into an integral equation and then Monte Carlo comes to play. Here are some examples. http://jotterbach.github.io/2018/08/08/MonteCarloODE/
