Leszek Plaskota
University of Warsaw

Continuous problems: optimality, complexity, tractability

Since a digital computer is able to store and manipulate only with
finitely many real numbers, most computational problems of contin-
uous mathematics can only be solved approximately using partial in-
formation. A branch of computational mathematics that studies the
inherent difficulty of continuous problems, for which available informa-
tion is partial, noisy, and priced, is called information-based complex-
ity (IBC). IBC emerged as a branch of computational mathematics
some 30 years ago as a consequence of the need to study theoreti-
cal aspects of computations related to continuous problems. Since
then IBC developed in different directions, see, e.g., the monographs
[1, 2, 6, 7, 8, 9, 10, 11].
In IBC study, an important role, both theoretical and practical,
play problems that are defined on functions of many or even infinitely
many variables. In physical or chemical applications, the number of
variables can be even millions. Such problems often suffer from the
curse of dimensionality, i.e., their ε-complexity grows exponentially
fast as the number d of variables increases to ∞. How to deal with
the curse or, if possible, how to vanquish the curse, is a fundamen-
tal theoretical and practical question of contemporary computational
mathematics. The three volume monograph [3, 4, 5] is the present
state of the art in the subject.
In this talk, we first present the main ingredients and ideas behind
IBC and then show how they can be applied and what results can be
obtained for the particular problem of numerical integration of scalar
and multivariate functions.

Georg Regensburger
Johann Radon Institute for Computational and Applied Mathematics (RICAM)
Austrian Acadamy of Sciences

Generalized mass action systems and positive solutions of polynomial
systems with real exponents
The study of dynamical systems arising from chemical reaction networks
with mass action kinetics was initiated by Feinberg, Horn, and Jackson
in the 1970s. Chemical reaction network theory (CRNT) provides
statements about uniqueness, existence, and stability of positive
steady states for all rate constants and initial conditions depending
on the underlying network structure alone. In terms of the
corresponding polynomial equations, they give existence and uniqueness
of positive real solutions for all positive parameters. We survey and
illustrate these results from CRNT, emphasizing the consequences for
polynomial equations with real exponents and addressing computational
aspects.
Further, we discuss an extension of several statements to generalized
mass action systems where reaction rates are allowed to be power-laws
in the concentrations. Hence in contrast to mass action kinetics, the
exponents (kinetic orders) can differ from the corresponding
stoichiometric coefficients. In this setting, uniqueness and existence
additionally depend on sign vectors of the stoichiometric and
kinetic-order subspaces. We discuss the corresponding extension of
Birch's theorem for generalized polynomial equations, which is robust
with respect to perturbations in the exponents, as determined by sign
conditions. This is joint work with Stefan Müller.
Finally, we outline some recent results about necessary and sufficient
conditions in terms of sign vectors for the injectivity of generalized
polynomial maps. In the context of chemical reaction networks with
power-law kinetics, they can be used to preclude as well as to
guarantee multiple positive steady states. This is joint work Carsten
Conradi, Alicia Dickenstein, Elisenda Feliu, Stefan Müller, and Annen Shiu.