Section:
New Results
Solving Systems over the Reals and Applications
Exact algorithms for linear matrix inequalities
Let be a linear matrix, or pencil,
generated by given symmetric matrices of size
with rational entries. The set of real vectors such that the
pencil is positive semidefinite is a convex semi-algebraic set called
spectrahedron, described by a linear matrix inequality (LMI). In
[13], we design an exact algorithm that, up to
genericity assumptions on the input matrices, computes an exact
algebraic representation of at least one point in the spectrahedron,
or decides that it is empty. The algorithm does not assume the
existence of an interior point, and the computed point minimizes the
rank of the pencil on the spectrahedron. The degree of the
algebraic representation of the point coincides experimentally with
the algebraic degree of a generic semidefinite program associated to
the pencil. We provide explicit bounds for the complexity of our
algorithm, proving that the maximum number of arithmetic operations
that are performed is essentially quadratic in a multilinear Bézout
bound of . When (resp. ) is fixed, such a bound, and hence
the complexity, is polynomial in (resp. ). We conclude by
providing results of experiments showing practical improvements with
respect to state-of-the-art computer algebra algorithms.
Real root finding for determinants of linear matrices
Let be given square matrices of size with
rational coefficients. In [14], we focus on the
exact computation of one point in each connected component of the real
determinantal variety
.
Such a problem finds applications in many areas such as control
theory, computational geometry, optimization, etc. Using standard
complexity results this problem can be solved using
arithmetic operations. Under some genericity assumptions on the
coefficients of the matrices, we provide an algorithm solving this
problem whose runtime is essentially quadratic in
. We also report on experiments with a
computer implementation of this algorithm. Its practical performance
illustrates the complexity estimates. In particular, we emphasize that
for subfamilies of this problem where is fixed, the complexity is
polynomial in .
A nearly optimal algorithm for deciding connectivity queries
in smooth and bounded real algebraic sets
A roadmap for a semi-algebraic set is a curve which has a
non-empty and connected intersection with all connected components of
. Hence, this kind of object, introduced by Canny, can be used to
answer connectivity queries (with applications, for instance, to
motion planning) but has also become of central importance in
effective real algebraic geometry, since it is used in higher-level
algorithms. In [15], we provide a
probabilistic algorithm which computes roadmaps for smooth and bounded
real algebraic sets. Its output size and running time are polynomial
in , where is the maximum of the degrees of the
input polynomials, is the dimension of the set under consideration
and is the number of variables. More precisely, the running time
of the algorithm is essentially subquadratic in the output size. Even
under our assumptions, it is the first roadmap algorithm with output
size and running time polynomial in .
Determinantal sets, singularities and application to optimal
control in medical imagery
Control theory has recently been involved in the field of nuclear
magnetic resonance imagery. The goal is to control the magnetic field
optimally in order to improve the contrast between two biological
matters on the pictures. Geometric optimal control leads us here to
analyze mero-morphic vector fields depending upon physical parameters,
and having their singularities defined by a determinantal
variety. The involved matrix has polynomial entries with respect to
both the state variables and the parameters. Taking into account the
physical constraints of the problem, one needs to classify, with
respect to the parameters, the number of real singularities lying in
some prescribed semi-algebraic set. In [24], we
develop a dedicated algorithm for real root classification of the
singularities of the rank defects of a polynomial matrix, cut with a
given semi-algebraic set. The algorithm works under some genericity
assumptions which are easy to check. These assumptions are not so
restrictive and are satisfied in the aforementioned application. As
more general strategies for real root classification do, our algorithm
needs to compute the critical loci of some maps, intersections with
the boundary of the semi-algebraic domain, etc. In order to compute
these objects, the determinantal structure is exploited through a
stratification by the rank of the polynomial matrix. This speeds up
the computations by a factor 100. Furthermore, our implementation is
able to solve the application in medical imagery, which was out of
reach of more general algorithms for real root classification. For
instance, computational results show that the contrast problem where
one of the matters is water is partitioned into three distinct
classes.
Optimal Control of an Ensemble of Bloch Equations with
Applications in MRI
The optimal control of an ensemble of Bloch equations describing the
evolution of an ensemble of spins is the mathematical model used in
Nuclear Resonance Imaging and the associated costs lead to consider
Mayer optimal control problems. The Maximum Principle allows to
parameterize the optimal control and the dynamics is analyzed in the
framework of geometric optimal control. This leads to numerical
implementations or suboptimal controls using averaging principle
as presented in [25].
Critical Point Computations on Smooth Varieties: Degree and
Complexity bounds
Let be an equidimensional algebraic set and
be an -variate polynomial with rational coefficients. Computing
the critical points of the map that evaluates at the points of
is a cornerstone of several algorithms in real algebraic geometry and
optimization. Under the assumption that the critical locus is finite
and that the projective closure of is smooth, we provide in
[31] sharp upper bounds on the degree of the
critical locus which depend only on and the degrees of the
generic polar varieties associated to . Hence, in some special
cases where the degrees of the generic polar varieties do not reach
the worst-case bounds, this implies that the number of critical points
of the evaluation map of is less than the currently known degree
bounds. We show that, given a lifting fiber of , a slight variant
of an algorithm due to Bank, Giusti, Heintz, Lecerf, Matera and
Solernó computes these critical points in time which is quadratic in
this bound up to logarithmic factors, linear in the complexity of
evaluating the input system and polynomial in the number of variables
and the maximum degree of the input polynomials.