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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.
Yvonne Ou, University of Delaware
In physics applications, we can encounter a Herglotz-Nevanlinna (H-N) function that does not decay fast enough along the imaginary axis to be classified as a Stieltjes function, which is well-known to have a simplified integral representation formula. However, in these applications the corresponding Nevanlinna functions might have certain nice properties along the real axis and hence can possess useful integral representation formulas. In this talk, I will present two examples of H-N functions in the context of two-phase porous/poroelastic materials. In the first example, H-N arises because of the causality encoded in the viscodynamic response of poroelastic materials; it is applied to create an effective numerical scheme for handling the memory terms in the poroelastic wave equations. In the second example, the technique of H-N functions is used to treat the effective permeability of a mixture of fluid with tiny bubbles inclusions and that of the mixture of fluid and solid inclusions in a unified framework.
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 properties of this class of preconditioners, focusing on the eigenproperties 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 eigenproperties of the preconditioned system and for general s we can explain and predict the characteristic 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
Giorgio Ascoli, George Mason University
Santiago Ramon y Cajal, widely hailed for ushering neuroscience into the modern era, described the brain jungle as impenetrable. Indeed, a century after his seminal call to arms, our understanding of neural architecture is still partial. Nevertheless, groundbreaking neuroinformatics progress in the past decades opened exciting new possibilities to test and refine the theoretical foundations of neural function with large-scale, data-driven computer simulations. This talk will highlight two open-access computational neuroanatomy resources fostering collaborative research. NeuroMorpho.Org is a database of digital reconstructions of neuronal and glial morphology from any animal species, developmental stage, and experimental methods. Hippocampome.org is a knowledge base of neuron types in the mammalian hippocampus enabling the implementation of biologically detailed spiking neural network models of associative memory and spatial navigation.
Zachary Miksis, Temple University
Fast sweeping WENO methods are a class of explicit iterative methods for solving steady-state hyperbolic PDEs. While they are able to produce high-order accurate solutions, WENO methods can be computationally expensive due to the use of multiple approximating interpolation stencils. This problem is compounded in multi-dimensional settings, and the computational cost of the iterative scheme can become enormous. In this talk, I will present the recent application of the sparse-grid combination technique, which has shown to be and effective approximation tool for high-dimensional problems, to fast sweeping WENO methods in order to reduce their computational cost.
James M. Scott, Columbia University
In recent years nonlocal models have seen a sharp increase in use across a variety of applications, such as continuum mechanics, image processing, and nonlocal diffusion. These models are often characterized by integro-differential equations, which hold on a bounded domain. An outstanding challenge in their implementation is to incorporate given boundary data into the problems.We address this challenge by considering a special class of nonlocal operators which allow us to state and analyze classical boundary value problems. The model takes its horizon parameter to be spatially dependent, vanishing near the boundary of the domain. We show the variational convergence of solutions to the nonlocal problem with mollified Poisson data to the solution of the localized classical Poisson problem with rough data as the horizon uniformly converges to zero. Several classes of boundary conditions are considered.
Noa Kraitzman, Macquarie University
Sea ice is a crucial component of the Earth’s climate system, affecting the ocean circulation, the atmospheric temperature, and the marine ecosystems. However, sea ice is not a simple solid material; it is a complex mixture of ice crystals, brine pockets, and air bubbles, that changes its structure and properties depending on the environmental conditions. In this talk, I will explore how we can model and understand the behaviour of sea ice at different scales, from the microscopic interactions of ice and salt to the macroscopic effects of heat transfer and fluid flow. I will present two mathematical models: a thermodynamically consistent model for the liquid-solid phase change in sea ice that incorporates the effects of salt, using multiscale analysis to derive a quasi-equilibrium Stefan-type problem. And a new rigorous derivation of bounds on the sea ice effective thermal conductivity obtained through Padé approximates and using Stieltjes integrals.
Malena Espanol, Arizona State University
Discrete linear and nonlinear inverse problems arise from many different imaging systems, exhibiting inherent ill-posedness wherein solution sensitivity to data perturbations prevails. This sensitivity is exacerbated by errors arising from imaging system components (e.g., cameras, sensors, etc.), necessitating the development of robust regularization methods to attain meaningful solutions. Our presentation commences with the exposition of distinct imaging systems, and their mathematical formalism, and subsequently introduces regularization techniques tailored for linear inverse problems. Then, we delve into the variable projection method, a powerful tool to address separable nonlinear least squares problems.
Benjamin Peherstorfer, Courant Institute, New York University
We introduce an online-adaptive model reduction approach
that can efficiently reduce convection-dominated problems. It exploits
that solution manifolds are low dimensional in a local sense in time
and iteratively learns and adapts reduced spaces from randomly sampled
data of the full models to locally approximate the solution manifolds.
Numerical experiments to predict pressure waves in combustion dynamics
demonstrate that our approach achieves about one order of magnitude
speedups in contrast to classical, static reduced models.
Daniel B Szyld, Temple University
We extend results known for the randomized Gauss-Seidel and the Gauss-Southwell methods for the case of a Hermitian and positive definite matrix to certain classes of non-Hermitian matrices. We obtain convergence results for a whole range of parameters describing the probabilities in the randomized method or the greedy choice strategy in the Gauss-Southwell-type methods. We identify those choices which make our convergence bounds best possible. Our main tool is to use weighted $\ell_1$-norms to measure the residuals. A major result is that the best convergence bounds that we obtain for the expected values in the randomized algorithm are as good as the best for the deterministic, but more costly algorithms of Gauss-Southwell type.
Steven Byram Roberts, Lawrence Livermore National Laboratory
Runge-Kutta methods are one of the most popular families of integrators for solving ordinary differential equations, essential in simulating dynamic systems arising in physics, engineering, biology, and various other fields.Unfortunately, classical error analysis for Runge-Kutta methods relies on assumptions that rarely hold when solving stiff ordinary differential equations (ODEs): an asymptotically small timestep and a right-hand side function with a moderate Lipschitz constant. Without idyllic assumptions, Runge-Kutta methods can experience a problematic degradation in accuracy known as order reduction.While high stage order remedies order reduction, it is only viable for expensive, fully implicit Runge-Kutta methods. In this talk, I will discuss some recent advancements in deriving practical Runge-Kutta methods that avoid order reduction. Initially, the focus will be on explicit methods applied to linear ODEs, where we have found a systematic approach to construct schemes of arbitrarily high order. Then, we will expand to classes of nonlinear ODEs where I will present new, stiff order conditions with a rich and interesting connection to rooted trees.
2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 | 2023