Stochastic gradient descent for semilinear elliptic equations with uncertainties

Randomness is ubiquitous in modern engineering. The uncertainty often modeled as random coefficients the differential equations that describe underlying physics. In this work, we a two-step framework for numerically solving semilinear elliptic partial with coefficients: 1) reformulate problem functional minimization based on direct method of calculus variation; 2) solve using stochastic gradient descent method. We provide convergence criterion resulted algorithm and discuss some useful technique to overcome issues ill-conditioning large variance. accuracy efficiency are demonstrated by numerical experiments.