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BEGIN:VEVENT
SUMMARY:Axis-alignment in low rank and other structures
DTSTART:20180514T083000Z
DTEND:20180514T093000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1592@indico.math.cnrs.fr
DESCRIPTION:Speakers: Lloyd N. Trefethen (University of Oxford and LIP - E
NS de Lyon)\n\nIn two or more dimensions\, various methods have been devel
oped to represent matrices or functions more compactly. The efficacy of su
ch methods often depends on alignment of the data with the axes. We shall
discuss four cases: low-rank approximations\, sparse grids\, quasi-Monte C
arlo\, and multivariate polynomials.\n\nhttps://indico.math.cnrs.fr/event/
2828/contributions/1592/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1592/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bridging the gap between flat and hierarchical low-rank matrix for
mats
DTSTART:20180515T123000Z
DTEND:20180515T130000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1594@indico.math.cnrs.fr
DESCRIPTION:Speakers: Jean-Yves L'Excellent\, Alfredo Buttari\, Patrick R
Amestoy\, Theo Mary (University of Manchester)\n\nMatrices possessing a lo
w-rank property arise in numerous scientific\napplications. This property
can be exploited to provide a substantial reduction\nof the complexity of
their LU or LDLT factorization. Among the possible\nlow-rank matrix format
s\, the flat Block Low-Rank (BLR) format is easy to use\nbut achieves supe
rlinear complexity. Alternatively\, the hierarchical formats\nachieve lin
ear complexity at the price of a much more complex hierarchical\nmatrix re
presentation. In this talk\, we propose a new format based on\nmultilevel
BLR approximations. Contrarily to hierarchical matrices\, the number\nof l
evels in the block hierarchy is fixed to a given constant\; we prove that\
nthis provides a simple way to finely control the desired complexity of th
e\ndense multilevel BLR factorization. By striking a balance between the\n
simplicity of the BLR format and the low complexity of the hierarchical on
es\,\nthe multilevel BLR format bridges the gap between flat and hierarchi
cal\nlow-rank formats. The multilevel BLR format is of particular relevanc
e in the\ncontext of sparse (e.g. multifrontal) solvers\, for which it is
able to trade\noff the optimal dense complexity of the hierarchical format
s to benefit from\nthe simplicity of the BLR format while still achieving
O(n) sparse complexity.\n\nhttps://indico.math.cnrs.fr/event/2828/contribu
tions/1594/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1594/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Connecting geodesics for the Stiefel manifold
DTSTART:20180514T093000Z
DTEND:20180514T100000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1595@indico.math.cnrs.fr
DESCRIPTION:Speakers: Bart Vandereycken\, Marco Sutti (University of Genev
a)\n\nSeveral applications in optimization\, image and signal processing d
eal with\ndata that belong to the Stiefel manifold St(n\, p)\, that is\, t
he set of n × p\nmatrices with orthonormal columns. Some applications\, f
or example\, the\ncomputation of the Karcher mean\, require evaluating the
geodesic distance\nbetween two arbitrary points on St(n\, p). This can b
e done by explicitly\nconstructing the geodesic connecting these two point
s. An existing method for\nfinding geodesics is the leapfrog algorithm in
troduced by J. L. Noakes\, which\nenjoys global convergence properties. W
e reinterpret this algorithm as a\nnonlinear block Gauss-Seidel process an
d propose a new convergence proof based\non this framework for the case of
St(n\, p).\n\nhttps://indico.math.cnrs.fr/event/2828/contributions/1595/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1595/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Certified and Fast computation of SVD
DTSTART:20180514T120000Z
DTEND:20180514T123000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1596@indico.math.cnrs.fr
DESCRIPTION:Speakers: Joris van der Hoeven\, Jean-Claude Yakoubsohn (Insti
tut de Mathématiques de Toulouse)\n\nThis work is in progress with the co
llaboration Joris Van Der Hoeven. We\ndefine and study a new method to ap
proximate locally the SVD of a general\nmatrix: this method has a local qu
adratic convergence. We also give result of\ncomplexity of this approxima
tion using a such type homotopy method.\n\nhttps://indico.math.cnrs.fr/eve
nt/2828/contributions/1596/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1596/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Algorithms for Structured Linear Systems Solving and their Impleme
ntation
DTSTART:20180515T130000Z
DTEND:20180515T134500Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1597@indico.math.cnrs.fr
DESCRIPTION:Speakers: Seung Gyu Hyun\, Eric Schost\, Romain Lebreton (LIRM
M - Université de Montpellier)\n\nThere exists a vast literature dedicate
d to algorithms for structured matrices\,\nbut relatively few descriptions
of actual implementations and their practical\nperformance in symbolic co
mputation. In this talk\, we consider the problem of\nsolving Cauchy-like
systems\, and its application to mosaic Toeplitz systems\, in\ntwo context
s: first over finite fields where basic operations have unit cost\,\nthen
over Q. We introduce new variants of previous algorithms and describe an\n
implementation of these techniques and its practical behavior. We pay a sp
ecial\nattention to particular cases such as the computation of algebraic\
napproximants.\n\nhttps://indico.math.cnrs.fr/event/2828/contributions/159
7/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1597/
END:VEVENT
BEGIN:VEVENT
SUMMARY:A Newton-like Validation Method for Chebyshev Approximate Solution
s of Linear Ordinary Differential Equations
DTSTART:20180514T123000Z
DTEND:20180514T130000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1598@indico.math.cnrs.fr
DESCRIPTION:Speakers: Mioara Joldes\, Nicolas Brisebarre\, Florent Bréhar
d (LAAS-CNRS Toulouse and CNRS\, LIP - INRIA AriC / Plume - ENS de Lyon)\n
\nA wide range of efficient numerical routines exist for solving function
space\nproblems (ODEs\, PDEs\, optimization\, etc.) when no closed form is
known for the\nsolution. While most applications prioritize efficiency\,
some safety-critical\ntasks\, as well as computer assisted mathematics\, n
eed rigorous guarantees on\nthe computed result. For that\, rigorous numer
ics aims at providing numerical\napproximations together with rigorous mat
hematical statements about them\,\nwithout sacrificing (too much) efficien
cy and automation.\n\nIn the spirit of Newton-like validation methods (see
for example [2])\, we\npropose a fully automated algorithm which computes
both a numerical approximate\nsolution in Chebyshev basis and a rigorous
uniform error bound for a restricted\nclass of differential equations\, na
mely Linear ODEs (LODEs). Functions are\nrigorously represented using Cheb
yshev models [1]\, which are a generalization\nof Taylor models [3] with b
etter convergence properties. Broadly speaking\, the\nalgorithm works in t
wo steps: (i) After applying an integral transform on the\nLODE\, an infin
ite-dimensional linear almost-banded system is obtained. Its\ntruncation a
t a given order N is solved with the fast algorithm of [4].\n(ii) This sol
ution is validated using a specific Newton-like fixed-point\noperator. Thi
s is obtained by approximating the integral operator with a\nfinite-dimens
ional truncation\, whose inverse Jacobian is in turn approximated\nby an a
lmost-banded matrix\, obtained with a modified version of the algorithm\no
f [4].\n\nA C library implementing this validation method is freely availa
ble online at\nhttps://gforge.inria.fr/projects/tchebyapprox/\n\n[1] N. Br
isebarre and M. Joldeş. Chebyshev interpolation polynomial-based tools\nf
or rigorous computing. In Proceedings of the 2010 International Symposium
on\nSymbolic and Algebraic Computation\, pages 147-154. ACM\, 2010.\n[2] J
.-P. Lessard and C. Reinhardt. Rigorous numerics for nonlinear\ndifferenti
al equations using Chebyshev series. SIAM J. Numer. Anal.\,\n52(1):1-22\,
2014.\n[3] K. Makino and M. Berz. Taylor models and other validated functi
onal\ninclusion methods. International Journal of Pure and Applied Mathema
tics\,\n4(4):379-456\, 2003.\n[4] S. Olver and A. Townsend. A fast and wel
l-conditioned spectral method. SIAM\nReview\, 55(3):462-489\, 2013.\n\nhtt
ps://indico.math.cnrs.fr/event/2828/contributions/1598/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1598/
END:VEVENT
BEGIN:VEVENT
SUMMARY:About Diagonalisation of Para-Hermitian Matrix
DTSTART:20180515T090000Z
DTEND:20180515T093000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1599@indico.math.cnrs.fr
DESCRIPTION:Speakers: Sylvie Icart (I3S Université Côte d’Azur)\n\nIt
is well-known that a Hermitian matrix can be diagonalized by means of a\nu
nitary matrix. The aim of this talk is to present the extension of this re
sult\nto polynomial matrices\, known as PEVD (Polynomial Eigen Value Decom
position)\n[1]\, occurring e.g. in blind equalization.\n\nIn this context\
, the eigenvalues are polynomials instead of scalars. Moreover\,\npolynomi
als are Laurent polynomials\, this means with positive and negative\nexpon
ents. We will show that in this framework\, one can still define unimodula
r\nmatrices\, Smith form and invariant polynomials. We will present the di
fference\nbetween the order (or length) and the degree of a polynomial mat
rix.\n\nExtending polynomial paraconjugation to polynomial matrix\, one de
fines\npara-hermitianity. We give some properties of these matrices. Then\
, we will\nshow that diagonalization of para-hermitian matrices is not alw
ays possible.\nFurthermore we will present some results found in the liter
ature on how to\napproximate a PEVD.\n\n[1] J. G. McWhirter\, P. D. Baxter
\, T. Cooper\, S. Redif\, and J. Foster\, An\nEVD algorithm for Para-Hermi
tian polynomial matrices\, IEEE Transactions\non Signal Processing\, vol.
55\, no. 6\, June 2007\, pp. 2158-2169.\n\nhttps://indico.math.cnrs.fr/eve
nt/2828/contributions/1599/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1599/
END:VEVENT
BEGIN:VEVENT
SUMMARY:On minimal ranks and the approximate block-term tensor decompositi
on
DTSTART:20180514T143000Z
DTEND:20180514T153000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1600@indico.math.cnrs.fr
DESCRIPTION:Speakers: José Henrique DE MORAIS GOULART (GIPSA-lab\, CNRS\,
Grenoble)\n\nThe block-term tensor decomposition (BTD) is a generalizatio
n of the tensor rank (or canonical polyadic) decomposition which is well s
uited for certain source separation problems. In this talk\, I will discus
s the existence of a best approximate block-term tensor decomposition (BTD
) consisting of a sum of low-rank matrix-vector tensor products. This inve
stigation is motivated by the fact that a tensor might not admit an exact
BTD with a given structure (number of blocks and their ranks).\nAfter a br
ief introduction\, we will proceed by exploring the isomorphism between th
ird-order tensors and matrix polynomials. To every matrix polynomial one c
an associate a sequence of minimal ranks\, which is unique up to permutati
on and invariant under the action of the general linear (or tensor equival
ence) group. This notion is a key ingredient of our problem\, since it ind
uces a natural hierarchy of sets of third-order tensors corresponding to d
ifferent choices of ranks for the blocks of the BTD.\nIn the particular ca
se of matrix pencils\, I will explain how the minimal ranks of a pencil ca
n be directly determined from its Kronecker canonical form. By relying on
this result\, one can show that no real pencil of dimensions 2k x 2k havin
g full minimal ranks admits a best approximation on the set of real pencil
s whose minimal ranks are bounded by 2k-1. From the tensor viewpoint\, thi
s means that there exist third-order tensors which do not have a best appr
oximation on a certain set of two-block BTDs. These tensors form a non-emp
ty open subset of the space of 2k x 2k x 2 tensors\, which is therefore of
positive volume. I shall sketch the proof of this result and then discuss
some possible extensions of this work and open problems.\n\nhttps://indic
o.math.cnrs.fr/event/2828/contributions/1600/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1600/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Exploiting off-diagonal rank structures in the solution of linear
matrix equations
DTSTART:20180515T113000Z
DTEND:20180515T123000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1601@indico.math.cnrs.fr
DESCRIPTION:Speakers: Stefano Massei (École Polytechnique Fédérale Laus
anne)\n\nLinear matrix equations\, namely Sylvester and Lyapunov equations
\, play an important role in several applications arising in control theor
y and PDEs. In the large scale scenario\, it is crucial to exploit the str
ucture in the data in order to carry on the computations and store the fin
al solution. We focus on the case in which the coefficients have off-diago
nal blocks with low-rank and we study when this property is numerically pr
eserved in the solution. Then\, we propose a divide and conquer scheme abl
e to exploit the structure\, reaching a linear-polylogarithmic complexity
in both time and memory consumption.\n\nhttps://indico.math.cnrs.fr/event/
2828/contributions/1601/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1601/
END:VEVENT
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SUMMARY:Structured matrix polynomials and their sign characteristics: clas
sical results and recent developments
DTSTART:20180515T073000Z
DTEND:20180515T083000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1602@indico.math.cnrs.fr
DESCRIPTION:Speakers: Françoise Tisseur (University of Manchester)\n\nThe
sign characteristic is an invariant associated with particular eigenvalue
s of structured matrices\, matrix pencils\, or matrix polynomials. The con
cept of sign characteristic arises in different forms in many scientific f
ields\, and is essential for the stability analysis in Hamiltonian systems
or the perturbation behaviour of eigenvalues under structured perturbatio
ns. We will start by discussing the sign characteristics of Hermitian matr
ix polynomials\, and show how to extend its definition to eigenvalues at i
nfinity. We will discuss applications of the sign characteristic in partic
ular in control systems\, in the solution of structured inverse polynomial
eigenvalue problems and in the characterization of special structured mat
rix polynomials such as overdamped quadratics\, hyperbolic and quasidefini
te matrix polynomials.\n\nhttps://indico.math.cnrs.fr/event/2828/contribut
ions/1602/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1602/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Tensor decomposition and structured matrices
DTSTART:20180514T130000Z
DTEND:20180514T140000Z
DTSTAMP:20230930T160700Z
UID:indico-contribution-1603@indico.math.cnrs.fr
DESCRIPTION:Speakers: Bernard Mourrain (Inria Sophia-Antipolis)\n\nTensor
decomposition problems appear in many areas such as Signal Processing\, Qu
antum Information Theory\, Algebraic Statistics\, Biology\, Complexity Ana
lysis\, … as a way to recover hidden structures from data. The decomposi
tion is a representation of the tensor as a weighted sum of a minimal numb
er of terms\, which are tensor products of vectors. We present an algebrai
c approach to address this problem\, which involves duality\, Gorenstein A
rtinian algebras and Hankel operators. We show the connection with low ran
k decomposition of Hankel matrices\, discuss algebraic and optimization te
chniques to solve it and illustrate the methods on some examples.\n\nhttps
://indico.math.cnrs.fr/event/2828/contributions/1603/
LOCATION:Amphi A (ENS de Lyon)
URL:https://indico.math.cnrs.fr/event/2828/contributions/1603/
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