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

9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
35
The categories of this paper are: mtrl-th
30
The categories of this paper are: q-alg
17
The categories of this paper are: q-fin
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
3
The categories of this paper are: physics
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
32
The categories of this paper are: atom-ph
15
The categories of this paper are: nucl-ex