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The Theory of the Interleaving Distance on Multidimensional Persistence Modules
Springer Science and Business Media LLC - Tập 15 - Trang 613-650 - 2015
In 2009, Chazal et al. introduced $$\epsilon $$ -interleavings of persistence
modules. $$\epsilon $$ -interleavings induce a pseudometric $$d_\mathrm{I}$$ on
(isomorphism classes of) persistence modules, the interleaving distance. The
definitions of $$\epsilon $$ -interleavings and $$d_\mathrm{I}$$ generalize
readily to multidimensional persistence modules. In this paper, we develop the
theory of ...
The Bubble Transform and the de Rham Complex
Springer Science and Business Media LLC - - Trang 1-49 - 2022
The purpose of this paper is to discuss a generalization of the bubble transform
to differential forms. The bubble transform was discussed in Falk and Winther
(Found Comput Math 16(1):297–328, 2016) for scalar valued functions, or
zero-forms, and represents a new tool for the understanding of finite element
spaces of arbitrary polynomial degree. The present paper contains a similar
study for diffe...
Constrained Diffeomorphic Shape Evolution
Springer Science and Business Media LLC - Tập 12 - Trang 295-325 - 2011
We design optimal control strategies in spaces of diffeomorphisms and shape
spaces in which the Eulerian velocities of the evolving deformations are
constrained to belong to a suitably chosen finite-dimensional space, which is
also following the motion. This results in a setting that provides a great
flexibility in the definition of Riemannian metrics, extending previous
approaches in which shape ...
Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation
Springer Science and Business Media LLC - Tập 16 - Trang 1631-1696 - 2016
This article provides a survey of recent research efforts on the application of
quasi-Monte Carlo (QMC) methods to elliptic partial differential equations
(PDEs) with random diffusion coefficients. It considers and contrasts the
uniform case versus the lognormal case, single-level algorithms versus
multi-level algorithms, first-order QMC rules versus higher-order QMC rules, and
deterministic QMC m...
Effective Models and Numerical Homogenization for Wave Propagation in Heterogeneous Media on Arbitrary Timescales
Springer Science and Business Media LLC - Tập 20 - Trang 1505-1547 - 2020
A family of effective equations for wave propagation in periodic media for
arbitrary timescales $$\mathcal {O}(\varepsilon ^{-\alpha })$$ , where
$$\varepsilon \ll 1$$ is the period of the tensor describing the medium, is
proposed. The well-posedness of the effective equations of the family is ensured
without requiring a regularization process as in previous models (Benoit and
Gloria in Long-time ...
A Necessary and Sufficient Proximity Condition for Smoothness Equivalence of Nonlinear Subdivision Schemes
Springer Science and Business Media LLC - Tập 16 - Trang 1069-1114 - 2015
In the recent literature on subdivision methods for approximation of
manifold-valued data, a certain “proximity condition” comparing a nonlinear
subdivision scheme to a linear subdivision scheme has proved to be a key
analytic tool for analyzing regularity properties of the scheme. This proximity
condition is now well known to be a sufficient condition for the nonlinear
scheme to inherit the regul...
On the Dirac–Frenkel Variational Principle on Tensor Banach Spaces
Springer Science and Business Media LLC - Tập 19 Số 1 - Trang 159-204 - 2019
The Geometry of Off-the-Grid Compressed Sensing
Springer Science and Business Media LLC - Tập 23 - Trang 241-327 - 2021
Compressed sensing (CS) ensures the recovery of sparse vectors from a number of
randomized measurements proportional to their sparsity. The initial theory
considers discretized domains, and the randomness makes the physical positions
of the grid nodes irrelevant. Most imaging devices, however, operate over some
continuous physical domain, and it makes sense to consider Dirac masses with
arbitrary ...
Computing the Homology of Semialgebraic Sets. I: Lax Formulas
Springer Science and Business Media LLC - Tập 20 - Trang 71-118 - 2019
We describe and analyze an algorithm for computing the homology (Betti numbers
and torsion coefficients) of closed semialgebraic sets given by Boolean formulas
without negations over lax polynomial inequalities. The algorithm works in weak
exponential time. This means that outside a subset of data having exponentially
small measure, the cost of the algorithm is single exponential in the size of
th...
On the Sample Complexity of Stabilizing Linear Dynamical Systems from Data
Springer Science and Business Media LLC -
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