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Current contacts: Benjamin Seibold or Daniel B. Szyld
The Seminar usually takes place on Wednesdays at 4:00 PM in Room 617 on the sixth floor of Wachman Hall. Click on title for abstract.
Vincenzo Carnevale, Temple University
The response and conformational changes of voltage-gated ion channels are well understood at a single-molecule level, but there is limited knowledge about how these channels interact with the lipids and with one another in their native environment. In particular, current models cannot accommodate recent experimental observations that highlight a dramatic and so far unsuspected collective behavior: voltage gated ion channels in physiological membranes form clusters, gate cooperatively, and show pronounced hysteresis effects suspected to give rise to multistability of membrane-potentials and thus to"cellular memory". To reconcile these seemingly conflicting views and bridge these two vastly different length scales, I will present a quantitative model based on the statistical mechanics of interacting, diffusing agents with internal degrees of freedom and subject to an external field. I will thus show that channels embedded in membranes close to a miscibility transition develop attractive long-range interactions and hysteresis. This model sheds light on several poorly understood aspects of ion channels behavior, including the non-Markovian character of single channel currents.
Fabiana Russo, Temple University
The application of granular biofilms in engineered systems for wastewater treatment and valorisation has significantly increased over the past years. Granular biofilms have a regular, dense structure and allow the coexistence of a high number of microbial trophic groups. A mathematical model is presented to describing the de novo granulation, and the evolution of multispecies granular biofilms, in a continuously fed bioreactor. The granular biofilm is modeled as a spherical free boundary domain with radial symmetry and a vanishing initial value. All main phenomena involved in the process are accounted: initial attachment by pioneer planktonic cells, biomass growth and decay, substrates diffusion and conversion, invasion by planktonic cells and detachment. Specifically, non-linear hyperbolic PDEs govern the advective transport and growth of sessile biomasses which constitute the biofilm matrix, and quasi-linear parabolic PDEs model the diffusive transport and conversion of dissolved substrates and planktonic species within the biofilm granule. Non-linear ODEs describe the dynamics of substrates and planktonic biomass within the bulk liquid. The free boundary evolution is governed by an ordinary differential equation which accounts for microbial growth, attachment and detachment phenomena. The model is applied to cases of biological and engineering interest. Numerical simulations are performed to test its qualitative behavior and explore the main aspects of the de novo granulation: ecology, microbial species distribution within the granules, dimensional evolution of the granules, and dynamics of dissolved substrates and planktonic biomass within the bioreactor.
Benjamin Seibold, Temple University
Order reduction, i.e., the convergence of the solution at a lower rate than the formal order of the chosen time-stepping scheme, is a fundamental challenge in stiff problems. Runge-Kutta schemes with high stage order provide a remedy, but unfortunately high stage order is incompatible with DIRK schemes. We first highlight the spatial manifestations of order reduction in PDE IBVPs. Then we introduce the concept of weak stage order, and (a) demonstrate how it overcomes order reduction in important linear PDE problems; and (b) how high-order DIRK schemes can be constructed that are devoid of order reduction.
Nicola Guglielmi, Gran Sasso Science Institute, L'Aquila, Italy
We present a new class of contour integral methods for linear convection–diffusion
parametric PDEs and in particular those arising from modeling in finance.
These methods aim to provide a numerical approximation of the solution by computing
its inverse Laplace transform. The choice of the integration contour is determined by a
pseudospectral roaming technique, which depends on few (weighted) pseudo-spectral
level sets of the operator in the equation.
Next we discuss how to deal efficiently with parametric problems. The main advantage of
the proposed method is that, differently from time stepping methods as Runge-Kutta
integrators, the Laplace transform allows to compute the solution directly at a given instant
or in a given time window.
In terms of the reduced basis methodology, this determines a significant improvement in the
reduction phase.
Some illustrative examples arising from finance will be presented to show the effectiveness
of the method.
This talk is based on joint work with Maria Lopez Fernandez and Mattia
Michal Outrata, Virginia Tech
When using implicit Runge-Kutta methods for solving parabolic PDEs, solving the stage
equations is often the computational bottleneck, as the dimension of the stage equations Mk=b for an s-stage Runge-Kutta method becomes $sn$ where the spatial discretization dimension $n$ can be very large. Hence the solution process often requires the use of iterative solvers, whose convergence can be less than satisfactory. Moreover, due to the structure of the stage
equations, the matrix $M$ does not necessarily inherit any of the preferable properties of the spatial operator, making GMRES the go-to solver and hence there is a need for a preconditioner. Recently in [3] and also [1, 2]
a new block preconditioner was proposed and numerically tested with promising results.
Using spectral analysis and the particular structure of M , we study the prop-
erties of this class of preconditioners, focusing on the eigen properties of the preconditioned
system, and we obtain interesting results for the eigenvalues of the
preconditioned system for a general Butcher matrix. In particular, for low number
of stages, i.e., s = 2, 3, we obtain explicit formulas for the eigen properties of the
preconditioned system and for general s we can explain and predict the character-
istic features of the spectrum of the preconditioned system observed in [1]. As the
eigenvalues alone are known to not be sufficient to predict the GMRES convergence
behavior in general, we also focus on the eigenvectors, which altogether allows us to
give descriptive bounds of the GMRES convergence behavior for the preconditioned system.
We then numerically optimize the Butcher tableau for the performance of the
entire solution process, rather than only the order of convergence of the Runge-
Kutta method. To do so requires careful balancing of the numerical stability of the
Runge-Kutta method, its order of convergence and the convergence of the iterative
solver for the stage equations.
(Joint work with Martin Gander)
References
[1] M. M. Rana, V. E. Howle, K. Long, A. Meek, W. Milestone. A New Block Preconditioner
for Implicit Runge-Kutta Methods for Parabolic PDE Problems. SIAM Journal on Scientific
Computing, vol (43): S475–S495, 2021
[2] M. R. Clines, V. E. Howle, K. R. Long. Efficient order-optimal preconditioners for implicit
Runge-Kutta and Runge-Kutta-Nystr ̈om methods applicable to a large class of parabolic and
hyperbolic PDEs. arXiv: https: // arxiv. org/ abs/ 2206. 08991 , 2022
[3] M. Neytcheva, O. Axelsson. Numerical solution methods for implicit Runge-Kutta methods of
arbitrarily high order. Proceedings of ALGORITHMY 2020, ISBN : 978-80-227-5032-5, 2020
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