Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Condition

Elliptic interface problems arise in numerous scientific and engineering applications, modeling heterogeneous materials in which physical properties change discontinuously across interfaces. In this paper, we present \textit{Walk-on-Interfaces} (WoI), a grid-free Monte Carlo estimator for a class of Neumann elliptic interface problems with nonhomogeneous flux jump conditions. Our Monte Carlo estimators maintain consistent accuracy throughout the domain and, thus, do not suffer from the well-known close-to-source evaluation issue near the interfaces. We also presented a simple modification with reduced variance. Estimation of the gradient of the solution can be performed, with almost no additional cost, by simply computing the gradient of the Green's function in WoI. Taking a scientific machine learning approach, we use our estimators to provide training data for a deep neural network that outputs a continuous representation of the solution. This regularizes our solution estimates by removing the high-frequency Monte Carlo error. All of our estimators are highly parallelizable, have a $\mathcal{O}(1 / \sqrt{\mathcal{W}})$ convergence rate in the number of samples, and generalize naturally to higher dimensions. We solve problems with many interfaces that have irregular geometry and in up to dimension six. Numerical experiments demonstrate the effectiveness of the approach and to highlight its potential in solving problems motivated by real-world applications.