Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Efficient optimization-based quadrature for variational discretization
  of nonlocal problems   Casting nonlocal problems in variational form and discretizing them with the
finite element (FE) method facilitates the use of nonlocal vector calculus to
prove well-posedeness, convergence, and stability of such schemes. Employing an
FE method also facilitates meshing of complicated domain geometries and
coupling with FE methods for local problems. However, nonlocal weak problems
involve the computation of a double-integral, which is computationally
expensive and presents several challenges. In particular, the inner integral of
the variational form associated with the stiffness matrix is defined over the
intersections of FE mesh elements with a ball of radius $\delta$, where
$\delta$ is the range of nonlocal interaction. Identifying and parameterizing
these intersections is a nontrivial computational geometry problem. In this
work, we propose a quadrature technique where the inner integration is
performed using quadrature points distributed over the full ball, without
regard for how it intersects elements, and weights are computed based on the
generalized moving least squares method. Thus, as opposed to all previously
employed methods, our technique does not require element-by-element integration
and fully circumvents the computation of element-ball intersections. This paper
considers one- and two-dimensional implementations of piecewise linear
continuous FE approximations, focusing on the case where the element size h and
the nonlocal radius $\delta$ are proportional, as is typical of practical
computations. When boundary conditions are treated carefully and the outer
integral of the variational form is computed accurately, the proposed method is
asymptotically compatible in the limit of $h \sim \delta \to 0$, featuring at
least first-order convergence in L^2 for all dimensions, using both uniform and
nonuniform grids.

23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
36
The categories of this paper are: cmp-lg
14
The categories of this paper are: math-ph
7
The categories of this paper are: hep-th
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom