
Paul I. Barton, Massachusetts Institute of Technology
Title Automatic Symbolic Manipulations of Factorable Functions and Applications to Differential Equations
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Factorable functions correspond to a quite broad class of functions that can be
implemented as computer programs. Automatic differentiation (AD) provides a number
of attractive automatic methods for evaluating derivatives of factorable functions.
Moving beyond classical AD, we present automatic methods for evaluating generalized
derivatives and convex/concave relaxations of factorable functions. These symbolic
manipulations are combined with novel theory to develop numerical approaches for
evaluating generalized derivatives and convex/concave relaxations of the solutions of
parametric ODEs and DAEs. Applications include deterministic global optimization for
parameter estimation in chemical kinetics models, dynamic flux balance analysis with
genomescale metabolic models and campaign continuous manufacturing of
pharmaceuticals.
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Florent Bréhard, ENS de Lyon
Title Rigorous Polynomial Approximations for Aerospace Problems
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A wide range of numerical routines exist for solving function space problems (ODEs, PDEs, optimization, etc.). But in most cases, one lacks information about results reliability e.g., how many of the returned digits are correct. While most applications focus on efficiency, some safetycritical tasks, as well as computer assisted mathematics, need rigorous mathematical statements about the computed result such as automatic tight error bounds. A relevant practical example occurs in the spacecraft guidance and control procedures, for which it is critical to obtain rigorous trajectories approximations.
In this talk we discuss a fully automated algorithm which computes validated solutions of linear ordinary differential equations, specifically, approximate truncated Chebyshev series together with a rigorous uniform error bound. The method relies on an a posteriori validation based on a Newtonlike fixedpoint operator, which also efficiently exploits the almostbanded structure of the problem. We provide an opensource C implementation and present validated results for the aforementioned space application.
In a second time, we present some further improvements of rigorous numerics via computer algebra techniques like creative telescoping, which computes linear differential equations satisfied by multiple integrals with parameters. This is illustrated in an efficient computing method of the collision probability between two space objects, or in the evaluation of Abelian integrals arising in Hilbert's 16th problem.
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Mikhail Gilman, North Carolina State University
Title Polarimetry of Homogeneous Halfspaces (joint work with Erick Smith and Semyon Tsynkov)
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We consider reconstruction of dielectric properties of a homogeneous halfspace with plane boundary from scattered waves in different polarizations. This work is motivated by the shortcomings of existing scattering models used in radar imaging. In our pursuit of a model that can be derived from the first principles, we build on a textbook problem of calculation of the Fresnel reflection coefficients by allowing a uniaxial dielectric tensor. The corresponding inverse problem is the reconstruction of the refractive indices and orientation of the principal dielectric axis. In Born approximation, this inverse problem reduces to a set of multivariate polynomials, and the quantifier elimination procedure proved effective in finding the condition for existence of a solution. Further modifications of this model should provide backscattering; for this purpose, we consider horizontally inhomogeneous scatterer, Leontovich boundary conditions, or nonplane boundary of the scatterer.
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JrShin Li, Washington University in St. Louis
Title Computational Ensemble Control: Formulations, Techniques, and Applications
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Natural and engineered systems that consist of populations of isolated or interacting dynamical components exhibit levels of complexity that are beyond human comprehension. These complex systems often require an appropriate excitation, an optimal hierarchical organization, or a periodic dynamical structure, such as synchrony, to function as desired or operate optimally. This talk will address current theoretical and computational challenges for analysis and control of complex population systems arising in diverse areas at different scales. Approaches to characterizing controllability and computing optimal control laws for establishing dynamic structures and patterns in ensemble systems will be presented. Practical control designs, such as synchronization waveforms for pattern formation in nonlinear oscillatory networks, will be illustrated along with their experimental realizations.
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Chenqi Mou, Beihang University
Title Bifurcation Analysis of Dynamic Systems using Symbolic Methods (joint work with Wei Niu, Dongming Wang, and Xiaoliang Li)
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In this talk we first discuss, for firstorder automonous dynamic systems, how the analysis of bifurcations of codimensioan 1 like NS ones and those of codimenstion 2 like FoldNS ones can be reduced into algebraic problems and further solved with symbolic methods. The proposed symbolic approaches are illustrated by bifurcation analysis of systems arising from biology and control theory modeled as such continuous or discrete dynamic systems.
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Charles Peskin, New York University
Title Spontaneous Oscillation and FluidStructure Interaction of Cilia (joint work with Jihun Han)
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Cilia are membranebound protrusions from eukaryotic cells (which are
cells with a nucleus). They are supported internally by a
characteristic arrangement of microtubules. Motile cilia beat
spontaneously in a coordinated manner in order to move fluid parallel
to the cellular surface. Within each cilium, the beat is generated by
hundreds of dynein molecular motors that pull on the microtubules.
Since the dynein motors presumably act independently, a natural
question is how their combined action results in a spontaneous
oscillation that is the ciliary beat. On a higher level, the
coordinated action of multiple cilia is also mysterious. In this talk
I will describe a mathematical model that addresses both of these
phenomena. The model involves a geometric constraint that governs the
the spatial configurations of a cilium, a differential equation for
the tension generated by each dynein motor, and also the
fluidstructure interaction of multiple cilia. The behavior of the
model cilia involves spontaneous symmetry breaking both within a
single cilium and also in arrays of cilia immersed in fluid.
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Satya Swarup Samal, RWTH Aachen University
Title Analysis of Biochemical Reaction Network Systems Using Tropical Geometry
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We discuss a novel analysis method for reaction network systems with polynomial or rational rate functions. This method is based on computing tropical equilibrations defined by the equality of at least two dominant monomials of opposite signs in the differential equations of each dynamic variable. In algebraic geometry, the tropical equilibration problem is tantamount to finding tropical prevarieties, that are finite intersections of tropical hypersurfaces. Tropical equilibrations with the same set of dominant monomials define a branch or equivalence class. Minimal branches are particularly interesting as they describe the simplest states of the reaction network. We provide a method to compute the number of minimal branches and to find representative tropical equilibrations for each branch. Furthermore, we demonstrate the applicability of minimal branches in model reduction, robustness analysis and in symbolic dynamics of biochemical networks with separate timescales.
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Xiaoxian Tang, Texas A&M University
Title Investigating Multistationarity in Structured Reaction Networks (joint work with Alicia Dickstein, Mercedes Perez Millan, Anne Shiu)
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Many dynamical systems arising in applications exhibit multistationarity (two or more positive steady states), but it is often difficult to determine whether a given system is multistationary, and if so to identify a witness to multistationarity, that is, specific parameter values for which the system exhibits multiple steady states. In this talk we introduce a procedure to investigate multistationarity and to find a witness. In practice, the procedure is much less expensive than traditional quantifier elimination. Our method is based on two new sufficient conditions for multistationarity. First, when there are no boundary steady states and a positive steadystate parametrization exists, one can conclude multistationarity if a certain critical function changes sign. Particularly, if the steady states are defined by binomials, we have multistationarity if a certain critical function contains terms with different signs. Second, when the steadystate equations can be replaced by equivalent triangularform equations, we have multistationarity if a positive degenerate steady state exists. We also investigate the mathematical structure of this critical function, and give conditions that guarantee that triangularform equations exist by studying the specialization of Grobner bases.
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