CMU Applied and Computational Mathematics Seminar

Fall 2023

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Schedule

Speaker: Hiruni Pallage
Title: Recovery of Initial conditions through later time samples
Abstract: Full knowledge of the initial conditions of an initial value problem (IVP) is necessary to solve said IVP but is often impossible in real-life applications due to the unavailability or inaccessibility of a sufficiently large sensor network. One way to overcome this impairing is to exploit the evolutionary nature of the sampling environment while working with a reduced number of sensors, i.e., to employ the concept of dynamical sampling. A typical dynamical sampling problem is to find sparse locations that allow one to recover an unknown function from various times samples at these locations. The classical problem of inverse heat conduction has been recently revisited by Devore and Zuazua (2014). We study conditions on an evolving system and spatial samples in a more general setup using bases. Specifically, we study when $u(x, t) = \sum_{n=1}^\infty a_nf_n(x) g_n(t)$, where $a_n \in \mathbb{R}$, $x \in [0, 1]$, $t \in [0, \infty)$ can be reasonably approximated through later-time samples at a single sampling location. The results of our research are relevant in applications, and we present examples of solving the Laplace equation and variable coefficient wave equation using our general method. Kim and Aceska (2021) retrieved the unknown initial condition function of the above systems via exponentially growing samples. However, in the approximation process, they observed exponential growth in error terms of the coefficients. Our recent research demonstrates that we can incorporate a linear growth pattern of errors in the recovered coefficients in these systems.

Speaker: Hongsong Feng
Title: Mathematics-AI for drug discovery and drug addiction studies
Abstract: Traditional drug discovery is a time-consuming and costly process, often taking over a decade to bring a new drug to market. AI has great potential to transform the entire landscape of pharmaceutical research and developments. It is promising to combine mathematics with AI to promote the drug discovery development. Due to the importance of accurate predictions of binding affinity between proteins and drug-like compounds, there is pressing need for reliable mathematical predictive models. Models based on persistent Laplacian (PL) theory were proposed in this aspect by our research group and can serve as a useful tool for binding affinity predictions. In my talk, I will present our recent machine learning studies with persistent Laplacian models and natural language processing (NLP) methods for drug discovery and drug addiction problems.

CMU Applied and Computational Mathematics Seminar

Spring 2023

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

If you would like to give a talk, please email any one of us.

Meeting Times

Fridays, 2:00pm – 3:00pm, on Webex

Schedule

Speaker: Mohsen Zayernouri
Title: Tittle: Data-Driven Fractional Modeling, Analysis, and Simulation of Anomalous Transport & Materials
Abstract: The classical calculus and integer-order differential and integral models, due to their inherently local characters in space-time, cannot fully describe/predict the realistic nonlocal and complex nature of the anomalous transport phenomena. Nature is abundant with such processes, in which for instance a cloud of particles spreads in a different manner than traditional diffusion. This emerging class of physical phenomena refers to fascinating processes that exhibit non-Markovian (long- range memory) effects, non-Fickian (nonlocal in space) interactions, non- ergodic statistics, and non-equilibrium dynamics. The phenomena of anomalous transport have been observed in a wide variety of complex, multi-scale, and multi-physics systems such as: sub-/super-diffusion in subsurface transport, kinetic plasma turbulence, aging polymers, glassy materials, in addition to amorphous semiconductors, biological cells, heterogeneous tissues, and fractal disordered media. In this talk, we present a series of recent spectral theories and global spectral methods for efficient numerical treatment of fractional ODEs/PDEs. Finally, a number of applications including fluid turbulence, power-law rheology, and material failure processes will be also presented, in which fractional modelling emerge as a natural language for high-fidelity modelling and prediction.

Speaker: Andrea Liu
Title: Machine Learning Glassy Dynamics
Abstract: The three-dimensional glass transition is an infamous example of an emergent collective phenomenon in many-body systems that is stubbornly resistant to microscopic understanding using traditional mathematical statistical physics approaches. Establishing the connection between microscopic properties and the glass transition requires reducing vast quantities of microscopic information to a few relevant microscopic variables and their distributions. I will demonstrate how machine learning, designed for dimensional analysis reduction, can provide a natural way forward when standard statistical physics tools fail. We have harnessed machine learning to identify a useful microscopic structural quantity for the glass transition, have applied it to simulation and experimental data, and have used it to build a new mathematical model for glassy dynamics.

Speaker: Kyle Harshbarger
Title: Scheduling and Planning High Uncertainty Seasonal Products at Dow
Abstract: Dow manufactures sells products with a joint problem of high seasonality and high uncertainty. Proper framing and modeling of the problem allows scheduling and planning processes to be brought under control. Planning for the whole season requires changing buffers with backwards scheduling. Scheduling requires balancing immediate high service level needs to avoid stockouts and rest-of-season analysis to minimize excess inventory. Model fitting of sales history identified a Gamma distribution as best fit to predict future demand in future periods. Monte Carlo approaches are used to accumulate periodic demand for operational requirements.

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Schedule

Speaker: Qian Zhang
Title: Gradcurl-Conforming Finite Elements Based on De Rham Complexes for the Fourth-Order Curl Problems
Abstract: The fourth-order curl operator appears in various models, such as electromagnetic interior transmission eigenvalue problems, magnetohydrodynamics in hot plasmas, and couple stress theory in linear elasticity. The key to discretizing these problems is to discretize the fourth-order curl operator. In this talk, I will present the conforming finite element method for a simplified fourth-order curl model. Discretizing the quad-curl equations using smoother elements (such as H^2-conforming elements) would lead to wrong solutions. Specific finite elements need to be designed for the fourth-order curl operator. However, constructing such elements is a challenging task because of the continuity required by the curlcurl-conformity and the naturally divergence-free property of the curl operator. In this presentation, we provide the construction of the curlcurl-conforming elements in both 2D and 3D based on the de Rham complex. In 2D, the lowest-order grad curl-conforming element has only 6 and 8 degrees of freedom on a triangle and a rectangle, respectively. In 3D, we relate the fourth-order curl problem to fluid mechanics and a de Rham complex with higher regularity. The lowest-order element has only 18 degrees of freedom on a tetrahedron. As a by-product, we construct a family of stable and mass-preserving finite element pairs for solving the Navier-Stokes equations.

Speaker: LIK-CHUAN LEE
Title: Computer Modeling of Cardiac Microstructure and its Effects in Heart Diseases
Abstract: Microstructural pathological features such as muscle fiber disarray and excessive fibrosis are hallmarks of many heart diseases. Often present together in heart diseases such as heart failure with preserved ejection fraction (HFpEF) and hypertrophic cardiomyopathy (HCM), these features are associated with the cardiac dysfunction found in these diseases. Statistical correlation analysis can provide some insights regarding the role of microstructural pathological features on cardiac dysfunction. It is, however, difficult to distinguish or isolate the contributions of each feature using only statistical studies. Computer modeling that considers the cardiac tissue microstructure can address this limitation by isolating the effects of each pathological features on heart function. Here, we present the development of such computer models to investigate the effects of microstructural pathological features on HFpEF and HCM. First, we present a computer model of the passive left ventricular (LV) mechanics using a Lanir-type microstructural constitutive model that consider dispersion of collagen fiber waviness and orientation. The model is used to investigate the load taken up by each tissue constituent during LV passive filling. Next, we present a computer model of active LV mechanics that considers muscle fiber disarray using a structural tensor approach. This model is applied to investigate the effects of myofiber disarray on LV and cardiac myocyte function in HCM patients.

Speaker: TBA
Title:
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CMU Applied and Computational Mathematics Seminar

Spring 2022

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Schedule

Speaker: Walter G. Chapman
Title: Density Gradient Theory for Interfacial Properties of Fluids: Stabilized and Mass Conserved Algorithms for Associating Solvents to Surfactants
Abstract: Density gradient theory (DGT) is a form of density functional theory (DFT) that allows prediction of the interfacial tension and density profile of molecules through a fluid-fluid interface. In DFT, the free energy is minimized as a function of the density distribution of molecules to obtain the equilibrium structure and properties of the fluid. DGT has roots that trace back to van der Waals. In this talk, we introduce several recent extensions of the theory and algorithms for practical calculations. While conventional algorithms require a reference substance of the system, we have developed a “stabilized density gradient theory” (SDGT) algorithm to solve DGT equations for multiphase pure and mixed systems that is robust and enables other generalizations. This algorithm makes it possible to calculate interfacial properties accurately at any domain size larger than the interface thickness without choosing a reference substance or assuming the functional form of the density profile. Further, we extend DGT to enable calculations for surfactant systems. For the first time, the surfactant head group and tail group are described separately in the DGT. Finally, these extensions are applied using a mass conserved DGT. This is a true mass conserved algorithm rather than being mass constrained. Applications of the approach are demonstrated using our SAFT equation of state.

Speaker: Miranda Holmes
Title: Numerically simulating particles with short-ranged interactions
Abstract: Particles with diameters of nanometres to micrometres form the building blocks of many of the materials around us, and can be designed in a multitude of ways to form new ones. Such particles commonly live in fluids, where they jiggle about randomly because of thermal fluctuations in the fluid, and interact with each other via numerous mechanisms. One challenge in simulating such particles is that the range over which they interact attractively is often much shorter than their diameters, so the equations describing the particles’ dynamics are stiff, requiring timesteps much smaller than the timescales of interest. I will introduce methods to accelerate these simulations, which instead solve the limiting equations as the range of the attractive interaction goes to zero. In this limit a system of particles is described by a diffusion process on a collection of manifolds of different dimensions, connected by “sticky” boundary conditions. I will show how to simulate low-dimensional sticky diffusion processes, and then discuss some ongoing challenges such as extending these methods to high dimensions and incorporating hydrodynamic interactions.

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• En-Bing Lin (enbing.lin AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

If you would like to give a talk, please email any one of us.

Meeting Times

Fridays, 3:00pm – 4:00pm, on Webex

Past seminars: Fall 2014, ...

Schedule

Speaker: Promislow, Keith
Title: Packing and Entropy in Structured Polymer Blends
Abstract: Packing and entropy play crucial roles in self-assembly of structured (di-block, triblock) polymers in solvent. We start with a very simple model for the packing of star polymers in a bounded domain and show that it leads to a wide range of possible patterns, both ordered and disordered. Then we revisit the random phase reductions of self-consistent mean field models of Choksi-Ren and Uneyama-Doi. By including a longer-range interaction associated to partial charges, we derive a phase-field model for the free energy. We show that reductions of this model connect to the scalar models of Gompper-Schick and Gommper-Goos for oil-water-surfactant microemulsions, and then present an analysis of a blend of long and short polymers, identifying a scaling regime in which geometric singular perturbations techniques and some simple micro-local analysis can be applied to show that interdigitations of long and short polymers, evocative of the role of sterols in phospholipid bilayers, can have a stabilizing effect.

Speaker: Ronald G. Larson
Title: Industrial Strength Rheology: Multi-Scale Modeling of Polymers, Colloids, and Surfactant Solutions
Abstract: Continuum-level thermodynamic and transport properties relevant to industrial applications can now be computed from molecular-scale interactions using multi-scale molecular dynamics (MD) simulations and Brownian dynamics (BD) simulations, along with biasing methods, such as umbrella sampling, and forward flux sampling. We demonstrate the power of these methods by computing the dynamics and rheology of surfactant solutions, polymers, and colloid-polymer mixtures used in consumer and industrial products, such as shampoos, oil dispersants, paints, and plastic films. The complex structures of these fluids require multi-scale modeling that can include atomistic and coarse-grained molecular simulations, as well as colloidal scale simulations, and model-reduction schemes to connect commercially important rheological properties to chemical composition. We also compare the predicted results to experimental data, and extract information, that is unavailable, or not easily available, from experiments.

Speaker: Norma Ortiz-Robinson
Title: Optimal Control strategies to contain Covid-19 for a data driven model for the state of Michigan
Abstract: In this talk I will present numerically obtained optimal strategies for an optimal control model formulated to account for vaccination hesitancy, social distancing, treatment and vaccination rates. The optimal control model is based on an SIR model with parameters derived from Covid-19 epidemic data in the state of Michigan.

CMU Applied and Computational Mathematics Seminar

Spring 2021

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• En-Bing Lin (enbing.lin AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Schedule

Speaker: Wendy Robertson
Title: Application of the Modified Universal Soil Loss Equation (MUSLE) in (semi-) distributed watershed models: challenges and opportunities
Abstract: The Modified Universal Soil Loss Equation (MUSLE) is an empirically-based model that is widely applied to predict sediment yield resulting from water erosion within catchments. While it offers meaningful benefits for predicting soil loss and sediment yield (flexibility, data availability, computational power, among others), there are substantial challenges to its application and limitations for its use. This talk will discuss the theoretical framework, development and application, datasets, limitations, and future avenues of research for MUSLE and USLE-type models and their incorporation into watershed models for the prediction of sediment yield.

Speaker: Hanliang Guo
Title: Modeling ciliary flow in complex geometries
Abstract: Cilia are hair-like organelles that protrude from epithelial cell- surfaces. Being one of the most conserved micro-structures in nature, cilia are critical building-blocks of life. In particular, it is known for decades that our airway systems require the periodic movements of cilia to transport mucus that carries out the dusts and toxic particles. More recently, people have realized that being the “conveyor-belt” of unwanted particles are far from the sole function of ciliary-flows. For example, cilia can create fluid-mechanical micro environments for the active recruitment of the specific microbiome of the host; ciliary-flows in the brain ventricles behave like a “switch” that reliably and periodically alters the flow pattern. Despite the fact that cilia grow in greatly complex geometries, existing numerical works have focused on simple geometries and idealized boundary conditions such as periodic, free-space or half-space flows. In this talk, we will present a recently developed hybrid numerical method for simulating ciliary flows in complex geometries. The confining geometries are treated by FMM accelerated boundary integral method while the ciliary flows are treated by the method of regularized stokeslet. We will also present an optimization approach that can find the most efficient ciliary motion of a ciliated microswimmer. The result can have great implications on the designs of micro-robots for health-related functions such as drug delivery.

Speaker: Olga Kuksenok
Title: Mesoscale modeling of controlled degradation and erosion of polymer networks
Abstract: Controlled degradation of hydrogels plays a vital role in a variety of applications ranging from regulating growth of complex tissues and neural networks to controlled drugs and biomolecules delivery. Further, of a particular interest is photo-controlled degradation of polymer networks, which permits spatially-resolved dynamic control of physical and chemical properties of the materials. Notably, in a number of practical applications, either the characteristic features of degradable gels or the dimensions of the entire degradable gel particles range between nanometers to microns scales, the length scales referred to as mesoscopic. We develop a Dissipative Particle Dynamics (DPD) approach to capture degradation of polymer networks at the mesoscale. DPD is a mesoscale approach utilizing soft repulsive interactions between beads representing collections of atoms, thereby allowing low computational cost of simulations. To overcome unphysical topological crossings of bonded polymer chains (a known limitation of DPD), we adapted a modified Segmental Repulsive Potential (mSRP) formulation to model gels with degradable crosslinks. We track the progress of the degradation process via measuring the fraction of degradable bonds intact. Further, we track mass loss from the hydrogel films, along with the number, sizes, and spatial distributions of clusters (bonded beads) formed during the degradation process. The figure below shows representative snapshots of degradation from the three-dimensional film (side view); for clarity, solvent beads are not shown. The cluster size distribution enables us to calculate the evolution of the weight average Materials Science and Engineering College of Engineering, Computing and Applied Sciences Clemson University degree of polymerization, DPw, during degradation. The evolution of DPw depends on the crosslink density, polymer volume fraction, and solvent quality. As degradation proceeds, the hydrogel film undergoes reverse gelation and the percolating network disappears. We quantify the point at which this reverse gelation occurs (reverse gel point) from the maximum of the reduced DPw, which excludes the largest cluster in the system, and compare our measured value with previous analytical theories and experimental results. Our measured value agrees well with the value obtained from the bond percolation theory on a diamond lattice.
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Research Interest: Computational design of biomimetic materials; Theory and computer simulations of multi-component polymeric systems; Modeling pattern formation in non-equilibrium systems

Speaker: Anne V. Ginzburg
Title: Modeling the Dynamics of the Coronavirus SARS-CoV-2 Pandemic and the Role of Vaccination in Stopping its Spread
Abstract: A modeling study of the outbreak of the Coronavirus SARS-CoV-2 and the implication of a global vaccination on the spread of the virus is presented. The global outbreak of the coronavirus began in the early months of 2020 and has continued for over 11 months at the time of this presentation. By early 2021, a number of vaccines have been developed in various countries, and the administration of these vaccines is now well underway. It is important to use modeling to investigate the implications of the vaccination on the spread of the coronavirus. Our simulation utilized the “Susceptible Exposed Infected Recovered” (SEIR) model in order to recreate the likely infection and exposure rates based on the known death and net hospitalization rates. The work of You Yang Gu (MIT) and other authors were used to parameterize the virus reproduction number (R(t)) and other model inputs, and to predict the death and hospitalization rates. These predictions were then compared with the real-life data. Because of the incredibly wide scope of this pandemic, our investigation was limited to 5 U.S. states with varied geographies, population sizes and vaccine distribution plans. The next step of the investigation would include broadening the investigation to the countries of the world and predicting the general trend of the pandemic.

Speaker: Jackson A. Criswell
Title: Wavelet Neural Networks and Applications in River Flow
Abstract: River flow is a chaotic natural phenomenon characterized by extreme events causing an inherent aperiodicity across multiple time scales. This dynamic nature makes development of accurate simulations and predictive models for fluvial discharge an onerous task in geophysics and hydrology. Human development often creates a large investment of life and resources along riverbanks and in floodplains. These areas can be prone to unexpected disasters such as floods and droughts. By better understanding the hydrological processes through development of predictive models, improvements can be made in disaster preparedness and response. There was a terrible disaster that occurred on May 20, 2020 along the Tittabawassee River, near Central Michigan University. Communities in the floodplain of the Tittabawassee experienced a 500-year flash flooding event which threatened many lives and continues to cause loss through property damage and devaluation. The dammed river features a series of dams and this calamity occurred after the collapse of the final two in the chain. A neuro-wavelet method of predictive river modeling is developed here which combines wavelet analysis and artificial neural networks to perform river flow forecasting. Several test cases are studied, and a broad sweep of network types and design parameters is performed to strengthen the quality of the predictions. Results are presented to provide comparisons based on different techniques and training parameters. The algorithm design is based on a non-linear auto-regressive multilayer perceptron artificial neural network. In order to improve predictive ability, new methods are designed to incorporate the multi-resolution information from a discrete wavelet transform using the Daubechies mother wavelet. The new predictive network design is inspired by existing methods but adds more repeatability and stability to the result. A custom genetic algorithm for selecting trained networks and averaging the results of many trials provides control for the inherent randomness created from network training. Results include 30-day daily average discharge forecast, and a new predictor that is being proposed in this work, the total predicted discharge. Total predicted discharge volumes in acre-feet are presented for a week, a fortnight, and a month subsequent to several test dates. In this case study, artificial neural networks and neuro-wavelets are used to perform river flow forecasting of the May 2020 flood event of the Tittabawassee River. Preliminary results are promising and continuing work to improve the network design through the inclusion of additional climate data and increased neural net complexity is underway.

Speaker: Zachary Tickner
Title: Mathematical tools for the directed evolution of RNA devices
Abstract: SELEX (Systematic Evolution of Ligands by Exponential Enrichment) is an in-vitro directed evolution technique which enriches nucleic acids capable of specific, high-affinity binding to a variety of targets. Molecules isolated by SELEX (known as aptamers) have been applied as biosensors, diagnostic tools, and therapeutics. Careful experimental design is required for a successful selection: starting populations, known as libraries, are assembled combinatorially and may be designed to increase the probability of containing viable aptamers, while selections must be monitored and controlled to enable enrichment of aptamers while avoiding accumulation of parasitic artifacts. Following a successful selection, a variety of biochemical and computational techniques are used to identify enriched aptamers, and to evaluate their binding and structural properties. This talk will describe general considerations for planning and performing in-vitro directed evolution experiments, as well as the recent selection and characterization of RNA aptamers to the antibiotic doxycycline.

CMU Applied and Computational Mathematics Seminar

Fall 2020

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• En-Bing Lin (enbing.lin AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Speaker: Tianyu Zhang 
Title: Multiscale Flux-Based Modeling of Biofilm Communities: Linking Microbial Metabolism to Community Environment
Abstract: For environmental microbial communities, environment is destiny in the sense that microbial community structure and function are strongly linked to chemical and physical conditions. Moreover, most environments outside of the lab are physically and chemically heterogeneous, further shaping and complicating the metabolisms of their resident microbial communities: spatial variations introduce physics such as diffusive and advective transport of nutrients and byproducts for example. Conversely, microbial metabolic activity can strongly effect the environment in which the community must function. Hence it is important to link metabolism at the cellular level to physics and chemistry at the community level. In order to introduce metabolism to community-scale population dy- namics, many modeling methods rely on large numbers of reaction ki- netics parameters that are unmeasured and likely effectively unmeasur- able (because they are themselves coupled to environmental conditions), also making detailed metabolic information mostly unusable. The bio- engineering community has, in response to these difficulties, moved to kinetics-free formulations at the cellular level, termed flux balance anal- ysis. These cellular level models should respond to system level environ- mental conditions. To combine and connect the two scales, we propose to replace classical kinetics functions in community scale models and in- stead use cell-level metabolic models to predict metabolism and how it is influenced by and influences the environment. Further, our methodology permits assimilation of many types of measurement data. We will discuss the background and motivation, model development, and some numerical simulation results.

CMU Applied and Computational Mathematics Seminar

Spring 2020

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• En-Bing Lin (enbing.lin AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Schedule

Speaker: Peimeng Yin
Title: Efficient discontinuous Galerkin (DG) methods for time-dependent fourth order problems
Abstract: We design, analyze and implement efficient discontinuous Galerkin (DG) methods for a class of fourth order time-dependent partial differential equations (PDEs). The main advantages of such schemes are their provable unconditional stability, high order accuracy, and their easiness for generalization to multi-dimensions for arbitrarily high order schemes on structured and unstructured meshes. These schemes have been applied to two fourth order gradient flows such as the Swift-Hohenberg (SH) equation and the Cahn-Hilliard (CH) equation, which are well known nonlinear models in modern physics.

Speaker: Mohye Sweidan
Title: Analysis of Shortley-Weller scheme
Abstract: This series of talks aim to present the whole proof of the super-convergence of the classic Shortley-Weller scheme for Poisson problems with curved domains. This proof was provided by LIsl Waynans in 2018( Super-Convergence in Maximum Norm of the Gradient for the Shortley-Weller Method, Journal of Scientific Computing, 75:625-637).

CMU Applied and Computational Mathematics Seminar

Fall 2019

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• En-Bing Lin (enbing.lin AT cmich DOT edu)

• Leela Rakesh (leela.rakesh AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Speaker: Mohye Sweidan 
Title: A basic introduction to finite difference schemes for elliptic problems (part 1). 
Abstract: This talk will present the central difference scheme for two-point boundary problems and its error analysis. This is a standard material in numerical analysis. Any undergraduate and graduate students are welcome to attend.

Speaker: Xiaoming Zheng 
Title: A basic introduction to finite difference schemes for elliptic problems (part 2). 
Abstract: This talk will present some finite difference schemes for two dimensional elliptic problems and their error analysis. This is a standard material in numerical analysis. Any undergraduate and graduate students are welcome to attend.

Speaker: Shan, Chunhua 
Title: Complex Dynamics of Epidemic Models on Adaptive Networks
Abstract: There has been a substantial amount of well-mixing epidemic models devoted to characterizing the observed complex phenomena (such as bistability, hysteresis, oscillations, etc.) during the transmission of many infectious diseases. A comprehensive explanation of these phenomena by epidemic models on complex networks is still lacking. In this talk, we study epidemic dynamics in an adaptive network model proposed by Gross et al, where the susceptibles are able to avoid contact with the infected by rewiring their network connections. Such rewiring of the local connections changes the topology of network, and inevitably has a profound effect on the transmissions of infectious diseases, which in turn influences the rewiring process. We rigorously prove that such adaptive epidemic network model exhibits degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov-Takens bifurcation. Our study shows that human adaptive behaviors to the emergence of an epidemic may induce complex dynamics of diseases transmission, including bistability, transient and sustained oscillations, which contrast sharply to the dynamics of classic network models. Our results yield deeper insights into the interplay between topology of networks and the dynamics of disease transmission on networks.

Speaker: Xiaoming Zheng 
Title: Some finite difference schemes for elliptic problems with curved domain 
Abstract: This is part 3 of the introduction to finite difference schemes for elliptic problems. This part introduces three classic schemes to treat the curved domain: Collatz scheme, Ghost Fluid Method with first order extrapolation, and Shortley-Weller scheme. In this case, the Cartesian mesh points are not on the domain boundary. We will derive the truncation errors and global errors. The super convergence is observed in the Shortley-Weller scheme.

Speaker: Yeonhyang Kim 
Title: Orientation distribution function estimation in single shell q-ball imaging using predicted diffusion gradient directions 
Abstract: High Angular Resolution Diffusion Imaging (HARDI) has been proposed as a means to overcome some limitations imposed by diffusion tensor imaging (DTI), especially in complex models of fibre orientation distribution in voxels. A long acquisition time for HARDI is a major obstacle to the clinical implementation. In this paper, we propose a novel method to improve angular and radial resolution using measured apparent diffusion coefficients in given diffusion gradient (DG) directions.

Speaker: En-Bing Lin 
Title: The Interplay of Big Data Analytics and Mathematics 
Abstract: We begin with an overview of Big Data and some current trends of big data analytics. Applications of big data analytics in biology, business and industry will be mentioned. Moreover, we present the use of big data analytics in handling large amount of information via rough set theory and analyzing DNA sequences by utilizing wavelet analysis.

CMU Applied Mathematics Seminar

Fall 2018, Spring 2019

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Speaker: Yip, Nung Kwan
Title: Dynamics of a second order gradient model for phase transitions
Abstract: We prove in a radially symmetric geometry, the convergence in the sharp interfacial limit, to motion by mean curvature of a second order gradient model for phase transition. This is in spirit similar to the classical Allen-Cahn theory of phase boundary motion. However the corresponding dynamical equation is fourth order thus creating some challenging difficulties for its analysis. A characterization and stability analysis of the optimal profile are performed which are in turn used in the proof of convergence of an asymptotic expansion. (This is joint work with Drew Swartz.)

Speaker: Shixu Meng
Title :Qualitative approaches to inverse scattering and wave motion in complex media
Abstract: The mathematical theory of wave scattering describes the interaction of waves (e.g., sound or electromagnetic) with natural or manufactured perturbations of the medium through which they propagate. The goal of inverse wave scattering (or in short imaging) is to estimate the medium from observations of the wave field. It has applications in a broad spectrum of scientific and engineering disciplines, including seismic imaging, radar, astronomy, medical diagnosis, and non-destructive material testing. Qualitative approaches to inverse scattering problems have been the focus of much activity in the mathematics community. Examples are the linear sampling method, the factorization method, use of transmission eigenvalues, Stekloff eigenvalues and so on. Reverse time migration methods and the closely related matched field or matched filtering array data processing techniques are related to such qualitative approaches. In this talk I shall first present qualitative imaging methods in an acoustic waveguide with sound hard walls. The waveguide terminates at one end and contains an unknown obstacle of compact support or has deformed walls, to be determined from data gathered by an array of sensors that probe the obstacle with waves and measure the scattered response. To further shed light on qualitative approaches to imaging in complex media, I shall present higher-order wave homogenization in periodic media, where such media have been used with success to manipulate waves toward achieving super-focusing, sub-wavelength imaging, cloaking, and topological insulation.

Speaker: Lewei Zhao
Title: Finite Element Method for Laplace Equation in Two-dimensional Domains with a Singular Fracture Abstract: We study the Laplace equation in 2D with a line Dirac Delta function on the right hand side. We establish the regularity of this problem described by weighted Sobolev space near the endpoint of the singular line. Our numerical method relies on element not across the singular line and graded mesh controlled by a grading parameter. Numerical examples are shown to verify the optimal convergence rate.

Spring 2018

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Speaker: Chen Mu
Title: Robust Tomographic Reconstruction Techniques in Nanomanufacturing and Improvement by Data-dependent Sparse Filtered Backprojection
Abstract: Tomographic reconstruction is a method of reconstructing a high dimensional image with a series of its low dimensional projections. Filtered backprojection is one of the several popular analytical techniques for the reconstruction due to its computational efficiency and easy implementation. The accuracy of the filtered backprojection method deteriorates when the input data are noisy or the input data are available for only a limited number of projection angles. In these cases, some algebraic approaches perform better, but they require computationally slow iterations. We demonstrate an improvement of the filtered backprojection method which is as fast as the existing filtered backprojection method and is as accurate as the algebraic approaches under heavy observation noises and missing wedge issue. The new approach optimizes the filter of the backprojection operator to minimize a regularized reconstruction error, which results in a sparse filter. We compare the new approach with the state-of-the-art filtered backprojection and algebraic approaches using two simulated datasets and a real-world nanomanufacturing data set to show its competitive accuracy and fast computing speed.

Speaker: Fatih Celiker
Title: Novel nonlocal operators in arbitrary dimension enforcing local boundary conditions
Abstract: We present novel nonlocal governing operators in 2D/3D for wave propagation and diffusion that enforce local boundary conditions (BC). The main ingredients are periodic, antiperiodic, and mixed extensions of kernel functions together with even and odd parts of bivariate functions. We present all possible 36 different types of BC in 2D which include pure and mixed combinations of Neumann, Dirichlet, periodic, and antiperiodic BC. Our construction is systematic and easy to follow. We provide numerical experiments that validate our theoretical findings. We also compare the solutions of the classical wave and heat equations to their nonlocal counterparts.

Speaker: Michael Delaura
Title: Statistical Estimation of fibers from HARDI and DTI data
Abstract: High Angular Resolution Diffusion Imaging (HARDI) and Diffusion Tensor Imaging (DTI) are popular in vivo brain imaging techniques that allow medical researchers to access brain connectivity, which plays crucial role in identifying early stages of Alzheimer's disease and other brain disorders. In this talk we will introduce the mathematical and statistical models for both HARDI and DTI. We will discuss how noise enters into these models and how our statistical tractography methodology helps to estimate the fibers locations and access the uncertainty in the images obtained via HARDI and DTI. We will also discuss how our approach compares with other tractography methods. This talk is based on the joint work with Dr. Sakhanenko and Dr. Zhu.

Fall 2017

Organizers

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Abstracts

Speaker: Xiaoming Zheng
Title: A viscoelastic model of capillary growth: derivation, analysis, and simulation
Abstract: We derive a  one-dimensional viscoelastic model of blood vessel capillary growth under nonlinear friction with surroundings, analyze its solution properties, and simulate various growth patterns in angiogenesis. The mathematical model treats the cell density as the growth pressure eliciting viscoelastic response of cells, thus extension or regression of the capillary. Nonlinear analysis provides some conditions to guarantee the global existence of biologically meaningful solutions, while linear analysis and numerical simulations predict the global biological solutions exist as long as the cell density change is sufficiently slow in time. Examples with blow-ups are captured by numerical approximations and the global solutions are recovered by slow growth processes. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under several biological conditions including blood vessel extension without proliferation and blood vessel regression.

Speaker: Yeonhyang Kim
Title: The spherical harmonic basis for a HARDI signal
Abstract: High AngularResolutionDiffusionImaging(HARDI)has been proposed as a means to overcome some limitations imposed by diffusion tensor imaging (DTI), especially in complex models of fibre orientation distribution in voxels. The signal generated by the 3D diffusion measurement of each voxel is called the q-space signal. Q-ball imaging (QBI), one of the HARDI methods, directly derives the diffusion orientation distribution function (ODF) of water molecules and this ODF can be estimated directly from the raw HARDI signal S on a single sphere of q-space. Originally the Q-ball representation utilized spherical radial bases. Later, spherical harmonic bases were adopted in many applications because they provide an analytic solution for the reconstruction of ODFs. In this talk, we study some properties of the spherical harmonic basis and the representation of the ODF of a HARDI signal.

Speaker: Yingda Cheng
Title: A Sparse Grid Discontinuous Galerkin Method for High-Dimensional Transport Equations
Abstract: In this talk, we present sparse grid discontinuous Galerkin schemes for solving high-dimensional PDEs. We will discuss the construction of the scheme based on hierarchical tensor product finite element spaces, its properties and applications in kinetic transport equations.

Speaker: Roza Aceska
Title: Approximation of solutions of certain classes of PDEs via dynamical sampling
Abstract: The concept of dynamical sampling is introduced in setups where the sensing devices available are limited due to some constraints. We show how to reconstruct optimally the solution of certain (non-)linear PDEs in a suitable Sobolev class using a single (location-fixed) sensor over time. We show that the optimal sampling does not depend on the spectrum of the operators involved, but just on the order of the PDE. We generalize the problem by working with time-variant coefficients in the PDEs. We adapt our approach to solve certain nonlinear integro-differential equations and non-linear PDEs. (Preliminary research report - joint work with Alessandro Arsie and Ramesh Karki )

Speaker: Zhengfu Xu
Title: Bound preserving flux limiters and total variation stability for computation of scalar conservation laws
Abstract: Provable total variation bounded high order (at least third order) method based on variation measured on grid values will be discussed in this talk. Most of the conventional design of TVB methods is based on Harten's criteria. However, to strictly follow Harten's TVD criteria, one can only provide methods of at most second order. Popular ENO/WENO methods are very successful in producing robust numerical results with great performance of suppressing oscillations around discontinuities. However, it is still elusive to prove ENO/WENO methods are TVB. As one of the most important properties we desire for numerical methods solving conservation laws, provable TVB property is at the center of this talk. A new criteria will be provided to design TVB high order finite difference scheme for one-dimensional problems.

Spring 2016

Organizers

• Debraj Chakrabarti (chakr2d AT cmich DOT edu)

• En-bing Lin (enbing.lin@cmich.edu)

• Yeonhyang Kim (kim4y AT cmich DOT edu)

• Xiaoming Zheng (zheng1x AT cmich DOT edu)

Schedule

Abstracts

Speaker: Derek Thompson
Title: Spectra of Some Weighted Composition Operators on H^2
Abstract: We completely characterize the spectra of weighted composition operators T_\psi C_\phi on H^2 when the weight \psi is in H^\infty and continuous at the Denjoy-Wolff point of the compositional symbol \phi, and \phi converges uniformly under iteration to its Denjoy Wolff point. Though this is a rather strong condition on \phi, several well-known symbols exhibit this behavior and applications to weak normality conditions for weighted composition operators are given.

Speaker: Debraj Chakrabarti
Title: A counterexample of L. Nirenberg on embeddability of CR structures
Abstract: The aim of these talks is to discuss an early and famous contribution of this year's Fleming lecturer Louis Nirenberg to complex analysis and partial differential equations. We will try to make the talks as nontechnical and example-oriented as possible. An outline is as follows. (1) CR structures and the CR embedding problem. We discuss some classical embedding problems of geometry and analysis, such as the Whitney and Kodaira embedding theorems. We motivate the definition of CR structures, discuss some elementary properties of CR structures, and introduce the notion of pseudoconvexity. We then state the global and local versions of the CR embedding problems. (2) Two discoveries of Hans Lewy We discuss basic facts about partial differential equations, such as the Cauchy-Kowalevsky theorem and the Malgrange-Ehrenpreis Theorem. We then discuss an example of a nonsolvable first order PDE due to Lewy and also a related result on the holomorphic extension of CR functions. (3) Nirenberg's example We discuss a counterexample due to Louis Nirenberg (1974) which shows that the local CR problem does not have a solution in three dimensions.

Fall 2015

Schedule

Abstracts

Speaker: Andrew Zimmer
Title: Characterizing polynomial domains by their biholomorphism group
Abstract: In this talk we will discuss the biholomorphism group of bounded domains in $\mathbb{C}^n$. Every bounded domain has several intrinsic metrics: for instance the Kobayashi, Carath{\'e}odory, and Bergman metric. The biholomorphism group acts by isometries on each of these metrics and in particular the geometry of these metrics controls the behavior of the group of biholomorphisms. I will discuss how ideas from the theory of non-positively curved metric spaces can be used to prove new results in several complex variables. The main result I will discuss is a characterization of certain polynomial domains in terms of the asymptotic behavior of the biholomorphism group.

Speaker: Emil Straube
Title: Compactness of the $\overline{\partial}$-Neumann Operator: An Example
Abstract: We will introduce the $\overline{\partial}$-Neumann operator, explain why whether or not it is compact is important, and discuss an example. More precisely, we will discuss compactness of the $\overline{\partial}$-Neumann operator on the intersection of two (pseudoconvex) domains when the respective operators on the two domains are compact.

Speaker: Byeongseon Jeong
Title: Subdivision Schemes in Geometric Design
Abstract: Subdivision scheme is a method to obtain smooth curves/surfaces from a given set of discrete points by recursively generating denser sets. We will introduce the basic notions of subdivision schemes and discuss the properties of a scheme which enable us to reproduce certain classes of curves and surfaces. Numerical examples will be presented for the verification of design capabilities.

Speaker: Changchuan Yin
Title: Whole Genome Phylogenetic Analysis by Fourier Transform
Abstract: DNA sequence similarity comparison is a major step in computational phylogenetic analysis of genomes. The sequence comparison of closely related DNA sequences is usually performed by multiple sequence alignments (MSA); however, MSA may produce incorrect results when DNA sequences undergo rearrangements, as in many bacterial and viral genomes. It is also limited by high computational complexity when omparing large volumes of data. We present a new method for the similarity comparison of DNA sequences by Fourier transform . In this method, we map DNA sequences into 2D numerical sequences and then apply Fourier transform to convert the numerical sequences into frequency domain. In 2D mapping, the nucleotide composition of a DNA sequence is a determinant factor. The 2D mapping reduces the nucleotide composition bias in distance measuring. The method can be applicable to any DNA sequences of arbitrary length. The similarity measurement in frequency domain is successfully applied on phylogenetic analysis for large, whole bacterial genomes.

Speaker: Leela Rakesh
Title: Navier Stokes equation and polymer fluid dynamics
Abstract: The Navier-Stokes equation (NSE) is named after Claude-Louis Navier and George Gabriel Stokes. NSE with appropriate initial and boundary conditions delivers mathematical model of the motion of liquids and gases (fluid). It is the momentous equation in computational and experimental fluid dynamics by making use of various fundamental principles of vector calculus. Vector fields are convenient to study fluid dynamics and make it possible to detect the path of a fluid at any given point. Various modifications of NSEs are used in polymer and biomedical industries to study the optimality under various operating conditions. In this talk I will discuss some of the fundamental aspects of NSEs and few applications.

Speaker: Divakar Viswanath
Title: Intermittency at fine scales and complex singularities of turbulent flow
Abstract: Intermittency is a property of the finest scales of turbulent flow. If one looks at a fine scale either in time or in space, it will be quiescent much of the time (or in much of the place) except for occasional bursts. We compute complex plane singularities of turbulent flow and show that intermittency is nothing but a manifestation of complex singularities, thus numerically verifying a conjecture of Frisch and Morf (1981). This talk is joint work with Andre Souza.

Spring 2015

Schedule

Abstracts

Speaker: Kun Gou
Title: Modeling of human airway swelling by biomechanics
Abstract: The human airway, also called the trachea in a professional way, is an organ by which we breath air into the lungs. When swelling (angioedema) occurs in the airway, it can rapidly narrow the airway, reduce air transportation capability and thus lead to a life threatening condition. The symptom of swelling in the airway is studied by means of biomechanics. First we consider the airway to be an idealized cylindrical shape, and study an 1-D problem for convenience of mathematical analysis. Then we consider a practical 3-D airway geometry extracted from biomedical images, where finite element formulation is used to obtain the solution. Airway constriction and the internal stress distribution are tracked as functions of swelling effect. This modeling provides a sound continuum mechanical foundation that facilitates our understanding of airway swelling for a better cure of the disease.

Speaker: Hana Cho TBA
Title: Stable and Efficient Schemes for Parabolic Problems using the Method of Lines Transpose
Abstract: As followed up to [1] , we present a novel numerical scheme suitable for solving parabolic differential equation model using the Method of Lines Transpose (MOL^T) combined with the successive convolution operators. The primary advantage is that the operators can be computed quickly in O(N) work, to high precision; and a multi dimensional solution is formed by dimensional sweeps. We demonstrate our solver on the Allen-Cahn and Cahn-Hilliard equation.

Speaker: Michael Bolt
Title: Szeg\H{o} kernel transformation law
Abstract: Let $\Omega_1, \Omega_2$ be smoothly bounded doubly connected regions in the complex plane. We establish a transformation law for the Szeg\H{o} kernel under proper holomorphic mappings. This extends known results concerning biholomorphic mappings between multiply connected regions as well as proper holomorphic mappings from multiply connected regions to simply connected regions.

Speaker: Xiaodong Wang
Title: An integral formula in Kahler geometry with applications
Abstract: I will discuss an integral formula on a smooth, precompact domain in a Kahler manifold and some of its applications. As the 1st application I will discuss holomorphic extension of CR functions. Then I will present an isoperimetric inequality in terms of a positive lower bound for the Hermitian mean curvature of the boundary. Combining with a Minkowski type formula on the complex hyperbolic space it implies that any closed, embedded hypersurface of constant mean curvature must be a geodesic sphere, provided the hypersurface is Hopf. A similar result is valid for the complex projective space.

Speaker: Sonmez Sahutoglu
Title: Essential norm estimates for Hankel operators on convex domains in $\mathbb{C}^2$
Abstract: Let $\Omega$ be a bounded convex domain in $\mathbb{C}^2$ with $C^1$-smooth boundary and $\varphi\in C^1(\overline{\Omega})$ such that $\varphi$ is harmonic on the nontrivial analytic disks in the boundary. We estimate the essential norm of the Hankel operator $H_{\varphi}$ in terms of the $\overline{\partial}$ derivatives of $\varphi$ ``along'' the nontrivial disks in the boundary. This is joint work with Zeljko Cuckovic.

Speaker: Shravan
Title: Vesicle flows: simulations, dynamics and rheology
Abstract: In this talk, we will present recent progress in our group on numerical algorithms for simulating dense vesicle suspensions in viscous fluids. Capturing the close two-body interactions of vesicles (or other soft-particles) poses significant numerical challenges owing to the near-singularity in the hydrodynamic interaction forces. We present a new spectrally-accurate algorithm for computing such forces. A novel fast algorithm for simulating multiphase through periodic geometries of arbitrary shape will be presented. Finally, we will present new results on the dynamics and rheology of dense suspensions obtained using these computational algorithms.

Speaker: Dr. Valeriy V. Ginzburg
Title: Field-theoretic simulations and self-consistent field theory (SCFT) for studying block copolymer directed self-assembly
Abstract: In recent years, block copolymer directed self-assembly (DSA) has become a promising new approach to printing sub-40 nm features. In DSA, nanoscale patterns are obtained as a result of thermodynamic microphase separation between two or more chemically distinct blocks covalently bonded into block copolymers. By manipulating the chemical nature of each of the individual blocks and molecular weight of the block copolymer, one can vary the symmetry of the equilibrium structure (lamellar, hexagonal, body-centered-cubic, gyroid, etc.) and its equilibrium period (often referred to as “pitch”). Computational modeling is increasingly becoming part of the DSA development in industry. Modeling is used to optimize block copolymer formulations, estimate process windows, predict defect density, or just visualize the polymer morphology inside a specific feature. I will discuss the use of mesoscale field-theoretic block copolymer models, and specifically self-consistent field theory (SCFT) in the context of block copolymer DSA. In recent years, SCFT has been widely used as a tool to predict equilibrium block copolymer morphology both in the bulk and in confined geometries, and its predictions are shown to agree well with experiments. Specific examples include chemoepitaxy (lamellar PS-PMMA on brushed surfaces) and graphoepitaxy (contact hole shrink and line-space applications).

Fall 2014

Schedule

Abstracts

Speaker: Purvi Gupta
Title: A link between Fefferman's hypersurface measure and polyhedral approximation
Abstract: In convex geometry, the affine surface area measure --- studied first by Blaschke --- is a measure on convex boundaries that is preserved by the group of equi-affine transformations. It occurs, among other things, in the asymptotics of polyhedral approximations of convex bodies. In multivariate complex analysis, a similar measure, due to Fefferman, exists on boundaries of certain convex-like domains, where the relevant group is that of volume-preserving biholomorphisms. It is natural to ask whether this measure enjoys any connection with approximation problems as in the affine situation. In this talk, I will motivate and formulate this question and discuss a positive result in this direction.

Speaker: Arundhati Bagchi Misra
Title: Modified Chambolle algorithm for speckle image denoising
Abstract: In this paper, we introduce a new algorithm based on total variation for denoising speckle noise images. The total variation was introduced by Rudin, Osher, and Fatemi in 1992 for regularizing images. Chambolle proposed a faster algorithm based on duality of convex functions for minimizing the total variation. His algorithm was built for Gaussian noise removal. We modify this algorithm for speckle noise images. The first noise equation for speckle denoising was proposed by Krissian, Kikinis, Westin and Vosburgh in 2005. We apply Chambolle algorithm to the Krissian et al. noise equation to develop a faster algorithm for speckle noise images.

Speaker: Ilya Kossovskiy
Title: Dynamical Approach in CR-geometry and Applications
Abstract: Study of equivalences and symmetries of real submanifolds in complex space goes back to the classical work of Poincar\'e and Cartan and was deeply developed in later work of Tanaka and Chern and Moser. This work initiated far going research in the area (since 1970's till present), which is dedicated to questions of regularity of mappings between real submanifolds in complex space, unique jet determination of mappings, solution of the equivalence problem, and study of automorphism groups of real submanifolds. Current state of the art and methods involved provide satisfactory (and sometimes complete) solution for the above mentioned problems in nondegenerate settings. However, very little is known for more degenerate situations, i.e., when real submanifolds under consideration admit certain singularities of the CR-structure (such as non-constancy of the CR-dimension or that of the CR-orbit dimension). The recent CR (Cauchey-Riemann Manifolds) -- DS (Dynamical Systems) technique, developed in our joint work with Shafikov and Lamel, suggests to replace a real submanifold with a CR-singularity by appropriate complex dynamical systems. This technique has recently hepled to solve a number of long-standing problems in CR-geometry, related to regularity of CR-mappings. In this talk, we give an overview of the technique and the results obtained recently by using it. We also discuss a possible development in this direction, in particular, new sectorial extension phenomena for CR-mappings.

Speaker: Pin-Hung Ka
Title: An Application of the Maynard--Tao Sieve
Abstract: Goldston, Pintz, and Y\i ld\i r\i m made a breakthrough in the study of bounded gaps between primes in the recent years. They used a modified Selberg sieve to achieved bounded gaps between primes under the assumption of the Elliott--Halberstam Conjecture. Maynard and Tao expanded the idea of GPY and used a multidimensional Selberg sieve to obtain bounded gaps between primes unconditionally. In this talk, the speaker will discuss an ongoing investigation of his work in the application of the Maynard--Tao sieve to the study of gaps between $E_2$ numbers. That is, the gaps between integers with exactly two distinct prime factors.

Speaker: Liz Vivas
Title: Parabolic domains associated to formal invariant curves
Abstract: We investigate the existence of parabolic attracting domains for germs tangent to the identity, when there is a formal invariant curve associated to the given germ. Formal curves are algebraic objects that might have geometrical meaning. However this is not always the case. We review some classical results for holomorphic germs in one dimension and explain the corresponding results for holomorphic germs in several dimensions. This is joint work with Lorena Lopez-Hernanz.

Speaker: Xinghui Zhong
Title: Discontinuous Galerkin Methods: algorithm design and applications
Abstract: In this talk, we discuss discontinuous Galerkin (DG) methods with emphasis on their algorithm design targeted towards applications for shock calculation and plasma physics. DG method is a class finite element methods that has gained popularity in recent years due to its flexibility for arbitrarily unstructured meshes, with a compact stencil, and with the ability to easily accommodate arbitrary h-p adaptivity. However, some challenges still remain in specific application problems. In the first part of my talk, we design a new limiter using weighted essentially non-oscillatory (WENO) methodology for DG methods solving conservation laws, with the goal of obtaining a robust and high order limiting procedure to simultaneously achieve uniform high order accuracy and sharp, non-oscillatory shock transitions. The main advantage of this limiter is its simplicity in implementation, especially on multi-dimensional unstructured meshes. In the second part, we propose energy-conserving numerical schemes for the Vlasov-type systems. Those equations are fundamental models in the simulation of plasma physics. The total energy is an important physical quantity that is conserved by those models. Our methods are the first Eulerian solver that can preserve fully discrete total energy conservation. The main features of our methods include energy-conservative temporal and spatial discretization. In particular, an energy-conserving operator splitting is proposed to enable efficient calculation of fully implicit methods. We validate our schemes by rigorous derivations and benchmark numerical examples.

Speaker: James Angelos
Title: Best Approximation of Vector Valued Functions
Abstract: Let X be a metric space. The set C(X,R^k) denotes the set of continuous functions from X to R^k with the norm
||f|| = sup {||f(x)||_2 : x ∈ X}.
Let M be a finite dimensional subspace of C(X,R^k). p ∈ M is a best approximation to f from M if
|| f-p|| ≤ ||f-q||, ∀ q ∈ M.
We consider the charactization of p, whether or not it is unique, and properties of the best approximation operator: B : C(X,Rk) → M where B(f) = p, p the best approximation, for a particular class of subspaces known as generalized Haar spaces of tensor product type.