Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Inverse Fast Multipole Method   This article introduces a new fast direct solver for linear systems arising
out of wide range of applications, integral equations, multivariate statistics,
radial basis interpolation, etc., to name a few. \emph{The highlight of this
new fast direct solver is that the solver scales linearly in the number of
unknowns in all dimensions.} The solver, termed as Inverse Fast Multipole
Method (abbreviated as IFMM), works on the same data-structure as the Fast
Multipole Method (abbreviated as FMM). More generally, the solver can be
immediately extended to the class of hierarchical matrices, denoted as
$\mathcal{H}^2$ matrices with strong admissibility criteria (weak low-rank
structure), i.e., \emph{the interaction between neighboring cluster of
particles is full-rank whereas the interaction between particles corresponding
to well-separated clusters can be efficiently represented as a low-rank
matrix}. The algorithm departs from existing approaches in the fact that
throughout the algorithm the interaction corresponding to neighboring clusters
are always treated as full-rank interactions. Our approach relies on two major
ideas: (i) The $N \times N$ matrix arising out of FMM (from now on termed as
FMM matrix) can be represented as an extended sparser matrix of size $M \times
M$, where $M \approx 3N$. (ii) While solving the larger extended sparser
matrix, \emph{the fill-in's that arise in the matrix blocks corresponding to
well-separated clusters are hierarchically compressed}. The ordering of the
equations and the unknowns in the extended sparser matrix is strongly related
to the local and multipole coefficients in the FMM~\cite{greengard1987fast} and
\emph{the order of elimination is different from the usual nested dissection
approach}. Numerical benchmarks on $2$D manifold confirm the linear scaling of
the algorithm.

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The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph