CAM Seminar

Computational and Applied Mathematics Seminar

Spring 2021

Mondays at 4:10 p.m. via ZOOM talks

(Note the starting time maybe a few minutes later following the adjusted class times on MWF required by the University)

The CAM Seminar is organized in the ISU Mathematics Department. It brings speakers from inside and outside of ISU, raising issues and exchanging ideas on topics of current interest in the are of computational and applied mathematics.

 

  • April 26

       Title: Phase-encoded Linear Algebra with the Quantum Fourier Transform

       Robert Basili

       Iowa State University 

Abstract: 

Since the discovery of Shor's algorithm in 1994 for performing integer factorization in polynomial time, the possibility of computational speed-up through quantum algorithms has received considerable attention by the scientific community. Recent rapid advancement in quantum-capable hardware has largely eliminated lingering doubts as to the future feasibility of quantum computing, and enormous efforts are being made to develop this new technology, identify strategies for its effective utilization, and prepare for its incorporation in various areas of academia and industry.

In practice, quantum computing applications often gain advantage over their classical counterparts by performing a large number of operations in superposition. A key example of this is quantum arithmetic, which allows basic numerical operations to be taken on superpositions of numbers. This may in turn be extended to superpositions of more complex structures, such as matrices found in computational linear algebra. Meanwhile, the Quantum Fourier Transform (QFT) offers a novel approach to encode numbers and perform arithmetic on a quantum computer while reducing the number of qubits necessary for addition. In this talk, I will introduce basic concepts of quantum computing, describe how one may use the QFT to digitally encode numbers and their arithmetic manipulation, and consider the challenges of extending these methods to matrices. I will discuss the query and gate complexities for these methods and provide results of a preliminary implementation.

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  • February 01

       Title: 

       Speaker

       University 

Abstract: 

ZOOM Link:

  • February 08

       Title:  Positivity-preserving, energy stable numerical schemes for the Cahn-Hilliard equation with logarithmic potential

       Cheng Wang

       University of Massachusetts Dartmouth

Abstract: 

The Cahn-Hilliard model with logarithmic potential is considered, in which the key difficulty has always been associated with the singularity of the logarithmic terms. An energy stable finite difference scheme, which implicitly treats the logarithmic terms, is proposed and analyzed in this talk. In particular, how to ensure the positivity of the logarithmic arguments, so that the numerical scheme is well-defined at a point-wise level, has been a long-standing mathematical challenge. It is proved that, given any numerical solution with a fixed bound at the previous time step, there exists a unique numerical solution that satisfies the given bound (-1,1) at a point-wise level. As a result, the numerical scheme is proven to be well-defined, and the unique solvability and energy stability could be established with the help of convexity analysis. In addition, an optimal rate convergence analysis could be appropriately established. Some numerical results are also presented in the talk.

ZOOM Link:

 

  • February 15

       Title: 

       Speaker

       University 

Abstract: 

ZOOM Link:

  • February 22

       Title:  A $C^0$ finite element method for the biharmonic problem with Navier boundary conditions in a polygonal domain

       Peimeng Yin

       Wayne State University 

Abstract: In our work, we study the biharmonic equation with the Navier boundary conditions in a polygonal domain. In particular, we propose a method that effectively decouples the 4th-order problem into a system of Poisson equations. Different from the usual mixed method that leads to two Poisson problems but only applies to convex domains, the proposed decomposition involves a third Poisson equation to confine the solution in the correct function space, and therefore can be used in both convex and non-convex domains. A $C^0$ finite element algorithm is in turn proposed to solve the resulted system. In addition, we derive the optimal error estimates for the numerical solution on both quasi-uniform meshes and graded meshes. Numerical test results are presented to justify the theoretical findings. This is a joint work with Hengguang Li and Zhimin Zhang.

ZOOM Link:

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  • March 01

       Title:  Nonlocality in nonlinear conservation model

       Anh Thuong Vo

       University of Nebraska-Lincoln

Abstract:  In this talk, we investigate the convergence of solutions of nonlocal conservation PDE to the local counterparts in one space dimension. Nonlocal operators are integral operators that mimic differential operators but account for long-range interactions over a finite horizon. Nonlocality appears in many physical phenomena (fracture, phase separation) and has a wide range of applications (image processing). In Du et al, it was shown that nonlocal operators can be reduced into local operators in a distributional sense. The solution of the nonlocal Burgers equation is also shown to converge to the local counterpart numerically. In our research, we generalize the nonlocal advection operator. In the limit when the horizon parameter approaches zero, we are able to prove nonlocal operator convergence pointwise to its local counterpart. Then, we apply the result to show the convergence of the solution of the nonlocal conservation equation to the local counterpart.

ZOOM Link:

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  • March 08

       Title:  Integration factor method for a class of high order differential equations with moving free boundaries

       Xinfeng Liu

       University of South Carolina 

Abstract:  The systems of reaction-diffusion equations coupled with moving boundaries defined by Stefan condition have been widely used to describe the dynamics of spreading population. There are several numerical difficulties to efficiently handle such systems. Firstly extremely small time steps are usually demanded due to the stiffness of the system. Secondly it is always difficult to efficiently and accurately handle the moving boundaries. In this talk, to overcome these difficulties, we first transform the one-dimensional problem with a moving boundary into a system with a fixed computational domain, and then introduce four different temporal schemes: Runge-Kutta, Crank-Nicolson, implicit integration factor (IIF) and Krylov IIF for handling such stiff systems. Numerical examples are examined to illustrate the efficiency, accuracy and consistency for different approaches, and it can be shown that Krylov IIF is superior to other three approaches in terms of stability and efficiency by direct comparison. I will also briefly review the integration factor methods and their applications in viscous fluid flows with moving interfaces.

ZOOM Link:
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  • March 15

       Title: Conservative, Positivity Preserving and Free Energy Dissipative Numerical Methods for the Poisson-Nernst-Planck Equations

       Zhongming Wang

       Florida International University

Abstract:  We design and analyze some numerical methods for solving the Poisson-Nernst-Planck (PNP) equations. The numerical schemes, including finite difference method and discontinuous Galerkin method, respect three desired properties that are possessed by the analytical solutions: I) conservation, II) positivity of solution, and III) free-energy dissipation. Advantages of different types of methods are discussed. Numerical experiments are performed to validate the numerical analysis. An application to an electrochemical charging system is also studied to demonstrate the effectiveness of our schemes in solving realistic problems. This is  joint work with  D. Jie, H. Liu, P. Yin, and S. Zhou.

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  • March 22

       Title:  Modeling and nonlinear simulation of solid tumor growth with chemotaxis 

       Shuwang Li

       Illinois Institute of Technology 

Abstract:  The ability of tumors to metastasize is manifested by morphological instabilities such as chains or fingers that invade the host environment. In this talk, we develop a computational method for computing the nonlinear dynamics of a tumor-host interface within the sharp interface framework.  We are interested in solid tumor growth with chemotaxis and cell-to-cell adhesion, together with the effect of the tumor microenvironment by the variability in spatial diffusion gradients, the uptake rate of nutrients inside/outside the tumor and the heterogeneous distribution of vasculature modeled using complex far-field geometries. We solve the nutrient field (modified Helmholtz equation) and the Stokes/Darcy flow field using a spectrally accurate boundary integral method, and update the interface using a nonstiff semi-implicit approach. Numerical results highlight the complexity of the problem, e.g. development of spreading branching-patterns and encapsulated morphologies in a long period of time. 

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  • March 29

       Title:  Geometry inspired DNNs on Manifold-structured Data

       Rongjie Lai

       Rensselaer Polytechnic Institute

Abstract:  Deep neural networks have made great success in many problems in science and engineering. In this talk, I will discuss our recent efforts on leveraging non-trivial geometry information hidden in data to design adaptive DNNs. In the first part, I will discuss our work on advocating the use of a multi-chart latent space for better data representation. Inspired by differential geometry, we propose a Chart Auto-Encoder (CAE) and prove a universal approximation theorem on its representation capability. CAE admits desirable manifold properties that auto-encoders with a flat latent space fail to obey, predominantly proximity of data. In the second part, I will discuss our work on a new way of defining convolution on manifolds via parallel transport. This geometric way of defining parallel transport convolution (PTC) provides a natural combination of modeling and learning on manifolds. PTC allows for the construction of compactly supported filters and is also robust to manifold deformations. I will demonstrate its applications to shape analysis and point clouds processing using PTC-nets. This talk is based on a series of joint work with a group of my students and collaborators.
 

ZOOM Link:

Time: Mar 29, 2021 04:00 PM Central Time (US and Canada)
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  • April 05

       Title: 

       TBD

       TBD

Abstract: 

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  • April 12

       Title:  First-order image restoration models for staircase reduction and contrast preservation

       Wei Zhu

       University of Alabama

Abstract: In this talk, we will discuss two novel first-order variational models for image restoration. In the literature, lots of higher-order models were proposed to fix the staircase effect. In our first model, we consider a first-order variational model that imposes stronger regularity than total variation on regions with small image gradients in order to achieve staircase reduction. In our second model, we further propose a novel regularizer that presents a lower growth rate than any power function with a positive exponent for regions with large image gradients. Besides removing noise and keeping edges effectively, this regularizer also helps preserve image contrasts during the image restoration process. We employ augmented Lagrangian method (ALM) to minimize both models and provide the convergence analysis. Numerical experiments will be then presented to demonstrate the features of the proposed models.

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  • April 19

       Title:  Scattering Resonances Through Subwavelength Holes and Their Applications in Imaging and Sensing

       Junshan Lin

       Auburn University 

Abstract: 

The so-called extraordinary optical transmission (EOT) through metallic nanoholes has triggered extensive research in modern plasmonics, due to its significant applications in bio-sensing, imaging, etc. The mechanisms contributing to the EOT phenomenon can be complicated due to the multiscale nature of the underlying structure. In this talk, I will focus on mechanisms induced by scattering resonances.

In the first part of the talk, based upon the layer potential technique, asymptotic analysis and the homogenization theory, I will present rigorous mathematical analysis to investigate the scattering resonances for several typical two-dimensional structures, these include Fabry-Perot resonance, Fano resonance, spoof surface plasmon, etc. In the second part of the talk, preliminary mathematical studies for their applications in sensing and super-resolution imaging will be given. I will focus on the resonance frequency sensitivity analysis and how one can achieve super-resolution by using plasmonic nanohole structures.

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