143,126 research outputs found
Theoretical spectroscopy: Full quantum dynamical study of the vibrational structure of hydrated hydroxide
In many processes, the impact of nuclear quantum phenomena cannot be neglected if a physically correct simulation is aimed for [1]. This quantum description of the nuclear motion is usually obtained by mapping the system onto a grid, thus implying a discretisation of the configuration space. As a consequence, all quantities take the form of high-order tensors. Unfortunately, multidimensional representations imply an exponential scaling of data and the concomitant number of operations with system size as well as the difficulty of computing multidimensional integrals [2].
Solutions to these issues, in particular when considering the representation of the potential energy surface, exist in the form of tensor-decomposition schemes. We present the recently developed the Multigrid POTFIT (MGPF) algorithm [3] which alleviates the exponential scaling by avoiding the calculations on the full grid. Moreover, we introduceour latest improvements to the algorithm which provide numerical stability and higher accuracy through the use of non-product grids for combined modes [4].
We illustrate the power of the MGPF algorithm in conjunction with the Multiconfiguration Time-Dependent Hartree (MCTDH) method in the case of the full-dimensional (9D) study of the vibrational structure [5] and the computation of the infrared spectrum [6] of the hydrated hydroxide complex (H3O2-).
[1] Fabien Gatti (Ed.) in Molecular Quantum Dynamics From Theory to Applications. Springer (2014).
[2] H.-D. Meyer, F. Gatti, G. A. Worth (Eds.) Multidimensional Quantum Dynamics: MCTDH Theory and Applications, Wiley (2009).
[3] D. Peláez, H.D. Meyer, The multigrid POTFIT (MGPF) method: Grid representations of potentials for quantum dynamics of large systems, J. Chem. Phys., 138, 014108 (2013).
[4] D. Peláez, H.D. Meyer (in preparation)
[5] D. Peláez, K. Sadri, H.-D. Meyer, Full-dimensional MCTDH/MGPF study of the ground and lowest lying vibrational states of the bihydroxide complex, Spectrochimica Acta Part A,119, 42 (2014).
[6] D. Peláez, H.D. Meyer, On the infrared absorption spectrum of the hydrated hydroxide (H3O2-) cluster anion, Chem. Phys., (in press) (2016).Universidad de Málaga. Campus de Excelencia Internacional AndalucĂa Tech
Multidimensional persistent homology is stable
Multidimensional persistence studies topological features of shapes by
analyzing the lower level sets of vector-valued functions. The rank invariant
completely determines the multidimensional analogue of persistent homology
groups. We prove that multidimensional rank invariants are stable with respect
to function perturbations. More precisely, we construct a distance between rank
invariants such that small changes of the function imply only small changes of
the rank invariant. This result can be obtained by assuming the function to be
just continuous. Multidimensional stability opens the way to a stable shape
comparison methodology based on multidimensional persistence.Comment: 14 pages, 3 figure
Stable comparison of multidimensional persistent homology groups with torsion
The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance d_T that represents a possible solution to this problem. Indeed, d_T is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with vector-valued filtering functions. Furthermore, we prove a result showing the relationship between d_T and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made
Stable comparison of multidimensional persistent homology groups with torsion
The present lack of a stable method to compare persistent homology groups
with torsion is a relevant problem in current research about Persistent
Homology and its applications in Pattern Recognition. In this paper we
introduce a pseudo-distance d_T that represents a possible solution to this
problem. Indeed, d_T is a pseudo-distance between multidimensional persistent
homology groups with coefficients in an Abelian group, hence possibly having
torsion. Our main theorem proves the stability of the new pseudo-distance with
respect to the change of the filtering function, expressed both with respect to
the max-norm and to the natural pseudo-distance between topological spaces
endowed with vector-valued filtering functions. Furthermore, we prove a result
showing the relationship between d_T and the matching distance in the
1-dimensional case, when the homology coefficients are taken in a field and
hence the comparison can be made.Comment: 10 pages, 3 figure
Invariance properties of the multidimensional matching distance in Persistent Topology and Homology
Persistent Topology studies topological features of shapes by analyzing the
lower level sets of suitable functions, called filtering functions, and
encoding the arising information in a parameterized version of the Betti
numbers, i.e. the ranks of persistent homology groups. Initially introduced by
considering real-valued filtering functions, Persistent Topology has been
subsequently generalized to a multidimensional setting, i.e. to the case of
-valued filtering functions, leading to studying the ranks of
multidimensional homology groups. In particular, a multidimensional matching
distance has been defined, in order to compare these ranks. The definition of
the multidimensional matching distance is based on foliating the domain of the
ranks of multidimensional homology groups by a collection of half-planes, and
hence it formally depends on a subset of inducing a
parameterization of these half-planes. It happens that it is possible to choose
this subset in an infinite number of different ways. In this paper we show that
the multidimensional matching distance is actually invariant with respect to
such a choice.Comment: 14 pages, 2 figure
The Theory of the Interleaving Distance on Multidimensional Persistence Modules
In 2009, Chazal et al. introduced -interleavings of persistence
modules. -interleavings induce a pseudometric on (isomorphism
classes of) persistence modules, the interleaving distance. The definitions of
-interleavings and generalize readily to multidimensional
persistence modules. In this paper, we develop the theory of multidimensional
interleavings, with a view towards applications to topological data analysis.
We present four main results. First, we show that on 1-D persistence modules,
is equal to the bottleneck distance . This result, which first
appeared in an earlier preprint of this paper, has since appeared in several
other places, and is now known as the isometry theorem. Second, we present a
characterization of the -interleaving relation on multidimensional
persistence modules. This expresses transparently the sense in which two
-interleaved modules are algebraically similar. Third, using this
characterization, we show that when we define our persistence modules over a
prime field, satisfies a universality property. This universality result
is the central result of the paper. It says that satisfies a stability
property generalizing one which is known to satisfy, and that in
addition, if is any other pseudometric on multidimensional persistence
modules satisfying the same stability property, then . We also show
that a variant of this universality result holds for , over arbitrary
fields. Finally, we show that restricts to a metric on isomorphism
classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in
Foundations of Computational Mathematics. 36 page
- …