# Applied/ACMS/absS15: Difference between revisions

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=== Murad Banaji (Portsmouth) === | === Murad Banaji (Portsmouth) === | ||

''Nonexpansivity in chemical reaction networks'' | |||

This work is motivated by the observation that quite often systems of differential equations describing chemical reaction networks (CRNs) display simple global behaviour such as convergence of all orbits to a unique equilibrium under only weak and physically reasonable assumptions on the reaction rates (kinetics). We are led to wonder if the structure of a CRN may sometimes force some distance between solutions to decrease (or at least not increase) with time. If so, how can we find this nonincreasing quantity? We explore different ways in which CRNs can define nonexpansive semiflows (recall that a semiflow <math>(\phi_t)_{t \geq 0}</math> on some Banach space <math>(X, |\cdot|)</math> is nonexpansive if <math>|\phi_t(x)-\phi_t(y)| \leq |x-y|</math> for all <math>x,y \in X</math> and all <math>t \geq 0</math>). It turns out that in CRNs the natural evolution of chemical concentrations may be nonexpansive; or a nonexpansive semiflow may be obtained from the evolution of the so-called "extents" of reactions. In both cases we may be able to draw global conclusions about convergence of chemical concentrations. In each case the challenge is to find the correct norm to get nonexpansivity for arbitrary kinetics. To construct such norms and show nonexpansivity we appeal to the theory of monotone dynamical systems. Families of CRNs which can be analysed in this way are presented; however characterising fully the class of CRNs to which this theory applies remains an open - and undoubtedly difficult - task. | This work is motivated by the observation that quite often systems of differential equations describing chemical reaction networks (CRNs) display simple global behaviour such as convergence of all orbits to a unique equilibrium under only weak and physically reasonable assumptions on the reaction rates (kinetics). We are led to wonder if the structure of a CRN may sometimes force some distance between solutions to decrease (or at least not increase) with time. If so, how can we find this nonincreasing quantity? We explore different ways in which CRNs can define nonexpansive semiflows (recall that a semiflow <math>(\phi_t)_{t \geq 0}</math> on some Banach space <math>(X, |\cdot|)</math> is nonexpansive if <math>|\phi_t(x)-\phi_t(y)| \leq |x-y|</math> for all <math>x,y \in X</math> and all <math>t \geq 0</math>). It turns out that in CRNs the natural evolution of chemical concentrations may be nonexpansive; or a nonexpansive semiflow may be obtained from the evolution of the so-called "extents" of reactions. In both cases we may be able to draw global conclusions about convergence of chemical concentrations. In each case the challenge is to find the correct norm to get nonexpansivity for arbitrary kinetics. To construct such norms and show nonexpansivity we appeal to the theory of monotone dynamical systems. Families of CRNs which can be analysed in this way are presented; however characterising fully the class of CRNs to which this theory applies remains an open - and undoubtedly difficult - task. | ||

This is joint work with Bas Lemmens (University of Kent) and Pete Donnell (University of Portsmouth). | This is joint work with Bas Lemmens (University of Kent) and Pete Donnell (University of Portsmouth). |

## Revision as of 17:59, 5 March 2015

# ACMS Abstracts: Spring 2015

### Irene Kyza (U Dundee)

*Adaptivity and blowup detection for semilinear evolution convection-diffusion equations based on a posteriori error control*

We discuss recent results on the a posteriori error control and adaptivity for an evolution semilinear convection-diffusion model problem with possible blowup in finite time. This belongs to the broad class of partial differential equations describing e.g., tumor growth,chemotaxis and cell modelling. In particular, we derive a posteriori error estimates that are conditional (estimates which are valid under conditions of a posteriori type) for an interior penalty discontinuous Galerkin (dG) implicit-explicit (IMEX) method using a continuation argument. Compared to a previous work, the obtained conditions are more localised and allow the efficient error control near the blowup time. Utilising the conditional a posteriori estimator we are able to propose an adaptive algorithm that appears to perform satisfactorily. In particular, it leads to good approximation of the blowup time and of the exact solution close to the blowup. Numerical experiments illustrate and complement our theoretical results. This is joint work with A. Cangiani, E.H. Georgoulis, and S. Metcalfe from the University of Leicester.

### Daniel Vimont (UW)

*Linear Inverse Modeling of Central and East Pacific El Niño / Southern Oscillation (ENSO) Events*

Research on the structure and evolution of individual El Niño / Southern Oscillation (ENSO) events has identified two categories of ENSO event characteristics that can be defined by maximum equatorial SST anomalies centered in the Central Pacific (around the dateline to 150 deg. W; CP events) or in the Eastern Pacific (east of about 150 deg. W; EP events). The distinction between these two events is not just academic: both types of event evolve differently, implying different predictability; the events tend to have different maximum amplitude; and the global teleconnection differs between each type of event.

In this presentation I will (i) describe the Linear Inverse Modeling (LIM) technique, (ii) apply LIM to determine an empirical dynamical operator that governs the evolution of tropical Pacific climate variability, (iii) define norms under which initial conditions can be derived that optimally lead to growth of CP or EP ENSO events, and (iv) identify patterns of stochastic forcing that are responsible for exciting each type of event.

### Saverio Spagnolie (UW)

*Sedimentation in viscous fluids: flexible filaments and boundary effects*

The deformation and transport of elastic filaments in viscous fluids play central roles in many biological and technological processes. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance may cause a significant alteration in the long-time sedimentation orientation and filament geometry. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes which match excellently with full numerical simulations. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. Embedding the analytical results for a single filament into a mean-field theory, we show how flexibility affects a well established concentration instability in a sedimenting suspension.

Another problem of classical interest in fluid mechanics involves the sedimentation of a rigid particle near a wall, but most studies have been numerical or experimental in nature. We have derived ordinary differential equations describing the sedimentation of arbitrarily oriented prolate and oblate spheroids near a vertical or inclined plane wall which may be solved analytically for many important special cases. Full trajectories are predicted which compare favorably with complete numerical simulations performed using a novel double layer boundary integral formulation, a Method of Stresslet Images. Several trajectory-types emerge, termed tumbling, glancing, reversing, and sliding, along with their fully three-dimensional analogues.

### Jonathan Freund (UIUC)

*Adjoint-based optimization for understanding and reducing flow noise*

Advanced simulation tools, particularly large-eddy simulation techniques, are becoming capable of making quality predictions of jet noise for realistic nozzle geometries and at engineering relevant flow conditions. Increasing computer resources will be a key factor in improving these predictions still further. Quality prediction, however, is only a necessary condition for the use of such simulations in design optimization. Predictions do not of themselves lead to quieter designs. They must be interpreted or harnessed in some way that leads to design improvements. As yet, such simulations have not yielded any simplifying principals that offer general design guidance. The turbulence mechanisms leading to jet noise remain poorly described in their complexity. In this light, we have implemented and demonstrated an aeroacoustic adjoint-based optimization technique that automatically calculates gradients that point the direction in which to adjust controls in order to improve designs. This is done with only a single flow solutions and a solution of an adjoint system, which is solved at computational cost comparable to that for the flow. Optimization requires iterations, but having the gradient information provided via the adjoint accelerates convergence in a manner that is insensitive to the number of parameters to be optimized. The talk will review the formulation of the adjoint of the compressible flow equations for optimizing noise-reducing controls and present examples of its use. We will particularly focus on some mechanisms of flow noise that have been revealed via this approach.

### Markos Katsoulakis (U Mass Amherst)

*Information Theory methods for parameter sensitivity and coarse-graining of high-dimensional stochastic dynamics*

In this talk we discuss path-space information theory-based sensitivity analysis and parameter identification methods for complex high-dimensional dynamics, as well as information-theoretic tools for parameterized coarse-graining of non-equilibrium extended systems. Furthermore, we establish their connections with goal-oriented methods in terms of new, sharp, uncertainty quantification inequalities. The combination of proposed methodologies is capable to (a) handle molecular-level models with a very large number of parameters, (b) address and mitigate the high-variance in statistical estimators, e.g. for sensitivity analysis, in spatially distributed

Kinetic Monte Carlo (KMC), (c) tackle non-equilibrium processes, typically associated with coupled physicochemical mechanisms, boundary conditions, etc. (such as reaction-diffusion systems), and where even steady states are unknown altogether, e.g. do not have a Gibbs structure. Finally, the path-wise information theory tools, (d) yield a surprisingly simple, tractable and easy-to-implement approach to quantify and rank parameter sensitivities, as well as (e) provide reliable molecular model parameterizations for coarse-grained molecular systems and their dynamics, based on fine-scale data and rational model selection methods through suitable path-space (dynamics-based) information criteria. The proposed methods are tested against a wide range of high-dimensional stochastic processes, ranging from complex biochemical reaction networks with hundreds of parameters, to spatially extended Kinetic Monte Carlo models in catalysis and Langevin dynamics of interacting molecules with internal degrees of freedom.

### Tao Zhou (Chinese Academy of Sciences)

*The Christoffel function weighted least-squares for stochastic collocation approximations: applications to Uncertainty Quantification*

We shall consider the multivariate stochastic collocation methods on unstructured grids. The motivation for such a study is the applications in parametric Uncertainty Quantification (UQ). We will first give a general framework of stochastic collocation methods, which include approaches such as compressed sensing, least-squares, and interpolation. Particular attention will be then given to the least-squares approach, and we will review recent progresses in this topic.

### Elaine Spiller (Marquette)

TBA

### Murad Banaji (Portsmouth)

*Nonexpansivity in chemical reaction networks*

This work is motivated by the observation that quite often systems of differential equations describing chemical reaction networks (CRNs) display simple global behaviour such as convergence of all orbits to a unique equilibrium under only weak and physically reasonable assumptions on the reaction rates (kinetics). We are led to wonder if the structure of a CRN may sometimes force some distance between solutions to decrease (or at least not increase) with time. If so, how can we find this nonincreasing quantity? We explore different ways in which CRNs can define nonexpansive semiflows (recall that a semiflow [math]\displaystyle{ (\phi_t)_{t \geq 0} }[/math] on some Banach space [math]\displaystyle{ (X, |\cdot|) }[/math] is nonexpansive if [math]\displaystyle{ |\phi_t(x)-\phi_t(y)| \leq |x-y| }[/math] for all [math]\displaystyle{ x,y \in X }[/math] and all [math]\displaystyle{ t \geq 0 }[/math]). It turns out that in CRNs the natural evolution of chemical concentrations may be nonexpansive; or a nonexpansive semiflow may be obtained from the evolution of the so-called "extents" of reactions. In both cases we may be able to draw global conclusions about convergence of chemical concentrations. In each case the challenge is to find the correct norm to get nonexpansivity for arbitrary kinetics. To construct such norms and show nonexpansivity we appeal to the theory of monotone dynamical systems. Families of CRNs which can be analysed in this way are presented; however characterising fully the class of CRNs to which this theory applies remains an open - and undoubtedly difficult - task.

This is joint work with Bas Lemmens (University of Kent) and Pete Donnell (University of Portsmouth).