Applied and Computational Mathematics Seminar

• Monday, February 24, 2014 at 2:00PM in Fretwell 379
  Thanh Nguyen, UNC Charlotte
  
  Title: Imaging of buried objects using ground penetrating radar and infrared thermography
  
  Abstract: The concepts of ground penetrating radar and infrared techniques for detecting and imaging buried objects are introduced. The principle of detection and characterization of buried objects using radar is described as follows: incident electromagnetic waves are sent into the ground where we are looking for targets. Using measured scattered waves above the ground surface, it is possible to detect the presence of potential targets and classify them by estimating their physical and geometrical properties. This technique leads to a coefficient inverse problem for hyperbolic equations which determines the dielectric constant of the medium.
  
  The infrared approach is based on the measurement of the change of ground temperature and then from that determine buried objects. This method leads to several coefficient inverse problems for heat equations which aim to identifying the heat diffusivity coefficient of the medium.
  
  We will discuss how to set up the experiment and data processing steps to detect buried targets, then we will discuss how to solve the above inverse problems.

• Monday, February 17, 2014 at 2:00PM in Fretwell 379
  Thanh Nguyen, UNC Charlotte
  
  Title: Inverse obstacle scattering problems using multi-frequency measurements
  
  Abstract: We discuss the problem of reconstruction of an acoustic obstacle using far field measurements associated with incident waves sent from only one direction but at multiple frequencies. The motivation for using multi-frequency data in inverse scattering problems is explained as follows. On the one hand, the stability of the reconstruction problem is poor at low frequencies. Therefore it is not possible to reconstruct details of the obstacle. However it is not needed to choose a good initial guess. On the other hand, the problem becomes more stable (more details can be reconstructed) at high frequency but has several local minima. Therefore using multi-frequency data can help to obtain accurate reconstructions without requiring a good initial guess.
  
  The reconstruction is done as follows: We start from the lowest available frequency and obtain a approximation of the obstacle. Then the reconstruction is refined using higher frequencies. Our analysis of multi-frequency method is divided into three steps: Step 1 is devoted to the choice of an initial guess at the lowest frequency. The second step investigates the convergence of the method with respect to frequency. The third step considers the accuracy of the reconstruction in the high frequency regime.
  
  We will discuss theoretical results of the three steps and show some numerical results using computationally simulated data.

• Friday, February 7, 2014 at 2:00PM in Fretwell 379
  Prof. Hongyu Liu, UNC Charlotte
  
  Title: Locating Multiple Multiscale Scatterers by A Single Far-field Measurement
  
  Abstract: In this talk, I will describe an inverse scattering scheme of locating acoustic/electromagnetic/elastic scatterers. The proposed method can work in an extremely general setting. The number of scatterer components is not required to be known. The physical properties of each scatterer component is not required to be known either. There might be both small-size components and regular-size components (compared to the detecting EM wavelength) presented at the same time. For the regular-size components, we need know the possible shapes in advance. Our method works to locate all the scatterer components by using only a single far-field measurement. Theoretical justifications will be briefly discussed and numerical results will be demonstrated.

• Monday, November 18, 2013 at 1:00PM in Fretwell 379
  Prof. Chao Yang, Lawrence Berkeley National Lab
  
  Title: Large-scale eigenvalue calculation with applications in quantum mechanical simulations
  
  Abstract: Large-scale eigenvalue problems arise naturally from quantum mechnical simulations. Solving these problems efficiently and accurately is crucial for computational materials science and chemistry. In this talk, I will discuss the latest progress in this area. I will first review Krylov subspace methods and introduce techniques for accelerating their convergence. I will then examine methods that are based on trace minimization. The advantage of this class of methods is the possibility of using a preconditioner. One of the challenges in modern materials simulation is that one often needs to compute a relatively large number of eigenpairs (e.g. up to 1% of the dimension of the matrix). Most of the existing eigensolvers are not designed to compute that many eigenpairs. I will discuss two techniques for addressing this issue.

• Wednesday, October 2, 2013 at 1:00PM in Fretwell 319
  Prof. Zhaojun Bai, University of California, Davis (Computer Science and Mathematics)
  
  Title: Variational Principles and Steepest Descent Type Methods for Matrix Eigenvalue Problems
  
  Abstract: The variational principles, such as minimax principle and trace min principle, are of great importance in theory and computation of Hermitian eigenvalue problems. In this talk, we begin with recent results on the extension of variational principles beyond Hermitian eigenvalue problems. Then we focus on the application of newly established principles in the development of the steepest descent and conjugate gradient type methods for ill-conditioned generalized Hermitian eigenvalue problems, and linear response eigenvalue problems arising from the density functional theory (DFT) and time-dependent DFT in computational materials science. This is a joint work with Yunfeng Cai, Ren-cang Li,Dario Rocca and Giulia Galli.

• Wednesday, August 21, 2013 at 3:00PM in Fretwell 379 (Math Conference Room)
  Dr. Christopher Davis, Louisiana State University
  
  Title: A partition of unity method for an optimal control problem
  
  Abstract: In this talk, we consider an elliptic distributed optimal control problem with pointwise state constraints. We will investigate the use of a partition of unity method applied to this problem on convex polygonal domains. By using the convexity of the domain, the the optimal control problem is reformulated as a fourth order variational inequality for the state. We then solve the the variational inequality using a partition of unity method. Error estimates for the state and control are given and numerical results are shown that support these estimates.

• Monday, April 08, 2013 at 2:00PM in Fretwell 379 (Math Conference Room)
  Prof. Michael Mascagni, Florida State University
  
  Title: Novel Stochastic Methods in Biochemical Electrostatics
  
  Abstract: Electrostatic forces and the electrostatic properties of molecules in solution are among the most important issues in understanding the structure and function of large biomolecules. The use of implicit-solvent models, such as the Poisson-Boltzmann equation (PBE), have been used with great success as a way of computationally deriving electrostatics properties such molecules. We discuss how to solve an elliptic system of partial differential equations (PDEs) involving the Poisson and the PBEs using path-integral based probabilistic, Feynman-Kac, representations. This leads to a Monte Carlo method for the solution of this system which is specified with a stochastic process, and a score function. We use several techniques to simplify the Monte Carlo method and the stochastic process used in the simulation, such as the walk-on-spheres (WOS) algorithm, and an auxiliary sphere technique to handle internal boundary conditions. We then specify some optimizations using the error (bias) and variance to balance the CPU time. We show that our approach is as accurate as widely used deterministic codes, but has many desirable properties that these methods do not. In addition, the currently optimized codes consume comparable CPU times to the widely used deterministic codes. Thus, we have an very clear example where a Monte Carlo calculation of a low-dimensional PDE is as fast or faster than deterministic techniques at similar accuracy levels.

• Friday, March 29, 2013 at 3:30PM in Fretwell 379 (Math Conference Room)
  Prof. Daniele Funaro, Universit di Modena e Reggio Emilia
  
  Title: Trapping Electromagnetic Solitons in a Ring
  
  Abstract: In vacuum, the classical equations of electromagnetism naturally allow for solutions confined in ring-shaped domains. This is mainly due to the orthogonality of the electric and magnetic fields and by the enforcement of the divergence-free conditions.
  
  Explicit full solutions in terms of Bessel functions are available in the case of cylinders (see [2], [3]), where the magnetic field oscillates parallel to the axis and the electric field lyes on the circular sections. The configuration recalls that of a train of photons smoothly circulating inside a cavity. Thin rings with large diameter and circular section can be fairly well approximated by the above mentioned solutions.
  
  For more compact rings the use of numerical simulations is necessary. By the way, not all the shapes are workable. Indeed, only a restricted number of sections are compatible with all the electromagnetic constraints. Thus, the solution process must be implemented together with a sort of shape-optimization algorithm (see [1]).
  
  The search of electromagnetic waves trapped in a toroid poses interesting mathematical questions. Numerical computations show an extraordinary variety of solutions, whose dynamics depends on the section's shape. The behavior, though modeled by a different set of equations, is strikingly similar to that of a non-viscous fluid confined in a vortex ring.
  
  References
  
  [1] Chinosi C., Della Croce L., Funaro D., Rotating electromagnetic waves in toroid-shaped regions, International Journal of Modern Physics C, 21-1 (2010), pp.11-32.
  
  [2] Funaro D., Electromagnetism and the Structure of Matter, World Scientific, Singapore, 2008.
  
  [3] Funaro D., From Photons to Atoms - The Electromagnetic Nature of Matter, arXiv:1206.3110, 2012.

• Wednesday, January 23, 2013 at 3:00PM in Fretwell 379 (Math Conference Room)
  Prof. Jianfeng Lu, Duke University (Mathematics)
  
  Title: Fast Algorighms for Density Functional Theory for Metallic Systems
  
  Abstract: Electronic structure models, in particular Kohn-Sham density functional theory, are widely used in computational chemistry and material sciences nowadays. The computational cost using conventional algorithms is however expensive which limits the application to relative small systems. This calls for development of efficient algorithms to extend the first principle calculations to larger systems. In this talk, we will discuss some recent progresses in efficient algorithms for Kohn-Sham density functional theory for insulating and metallic systems.

• Friday, April 20, 2012 at 2:00PM in Fretwell 379 (Math Conference Room)
  Prof. Gunther Uhlmann, University of California at Irvine and University of Washington
  
  Title: Cloaking via Transformation Optics
  
  Abstract: We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves, acoustic waves and quantum waves. We emphasize the method of transformation optics. For the case of electromagnetic waves, Maxwell's equations have transformation laws that allow for design of electromagnetic materials that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. The object, which would have no shadow, is said to be cloaked. We recount some of the history of the subject and discuss some of the mathematical issues involved.
  
  Biograph: Gunther Uhlmann is currently a professor at both the University of California, Irvine and the University of Washington. Upon completing his Ph.D. under the direction of Professor Victor Guillemin at MIT in 1976, Prof. Uhlmann did postdoctoral work at the Courant Institute and was an assistant professor at MIT before moving to the University of Washington, where he has been from 1984-present. In 2010, he took up the Excellence in Teaching Chair in Mathematics Professorship at UC-Irvine. Prof. Uhlmann has won numerous awards for his work on inverse problems and partial differential equations using microlocal analysis, including the Kleinman Prize (2011), Bocher Prize (2011), election to the American Academy of Arts and Sciences (2009), a Guggenheim Fellowship (2001-02) and a SIAM Fellowship (2010). He has delivered many plenary talks throughout the world, among them at several ICIAM and AMS meetings. He was also an invited ICM speaker in Berlin in 1998. Among his notable works are "A global uniqueness theorem for an inverse boundary value problem" with J. Sylvester in 1987, "The Calderón problem with partial data" with C. Kenig and J. Sjöstrand in 2007 and a groundbreaking work on cloaking "Full-wave invisibility of active devices at all frequencies" with A. Greenleaf, Y. Kurylev and M. Lassis in 2007.

• Wednesday, February 1, 2012 at 4:00PM in Fretwell 379 (Math Conference Room)
  Prof. Xiuqing Chen, Beijing University of Posts & Telecommunications; Duke University
  
  Title: Global weak solution for kinetic models of active swimming and passive suspensions
  
  Abstract: We investigate two kinetic models for active suspensions of rod-like and ellipsoidal particles, and passive suspensions of dumbbell beads dimmers, which couple a Fokker-Planck equation to the incompressible Navier-Stokes or Stokes equation. By applying cut-off techniques in the approximate problems and using compactness argument, we prove the existence of the global weak solutions with finite (relative) entropy for the two and three dimensional models. For the second model, we establish a new compact embedding theorem of weighted spaces which is the key in the compactness argument. (Joint work with Jian-Guo Liu)

• Wednesday, January 25, 2012 at 4:00PM in Fretwell 379 (Math Conference Room)
  Prof. Guowei Wei, Michigan State University (Mathematics)
  
  Title: Variational multiscale models for ion channel transport
  
  Abstract: A major feature of biological science in the 21st Century will be its transition from a phenomenological and descriptive discipline to a quantitative and predictive one. Revolutionary opportunities have emerged for mathematically driven advances in biological research. However, the emergence of complexity in self-organizing biological systems poses fabulous challenges to their quantitative description because of the excessively high dimensionality. A crucial question is how to reduce the number of degrees of freedom, while returning the fundamental physics in complex biological systems. This talk focuses on a new variational multiscale paradigm for biomolecular systems. Under the physiological condition, most biological processes, such as protein folding, ion channel transport and signal transduction, occur in water, which consists of 65-90 percent of human cell mass. Therefore, it is desirable to describe membrane protein by discrete atomic and/or quantum mechanical variables; while treating the aqueous environment as a dielectric or hydrodynamic continuum. I will discuss the use of differential geometry theory of surfaces for coupling microscopic and macroscopic scales on an equal footing. Based on the variational principle, we derive the coupled Poisson-Boltzmann, Nernst-Planck (or Kohn-Sham), Laplace-Beltrami and Navier-Stokes equations for the structure, dynamics and transport of ion-channel systems. As a consistency check, our models reproduce appropriate solvation models at equilibrium. Moreover, our model predictions are intensively validated by experimental measurements. Mathematical challenges include the well-posedness and numerical analysis of coupled partial differential equations (PDEs) under physical and biological constraints, lack of maximum-minimum principle, effectiveness of the multiscale approximation, and the modeling of more complex biomolecular phenomena.
  
  References:
  • Guo-Wei Wei, Differential geometry based multiscale models, Bulletin of Mathematical Biology, 72, 1562-1622, (2010).
  
  • Zhan Chen, Nathan Baker and Guo-Wei Wei, Differential geometry based solvation model I: Eulerian formulation, Journal of Computational Physics, 229, 8231-8258 (2010).
  
  • Qiong Zheng and Guo-Wei Wei, Poisson-Boltzmann-Nernst-Planck model, Journal of Chemical Physics, 134 (19), 194101, (2011).

• Friday, November 4, 2011 at 2:00PM in Fretwell 379 (Math Conference Room)
  Prof. R. Bruce Kellogg, University of South Carolina
  
  Title: Boundary layers, corner singularities, and mesh refinements
  
  Abstract: We review recent and ongoing work on boundary value problems for a singularly perturbed convection diffusion equation in a domain with corners. The small singular perturbation parameter makes it necessary to use a stretched mesh along the boundary, refined in the direction normal to the boundary. (This is the "Shishkin" mesh.) The corner singularity makes it necessary to use a geometric refinement near the boundary. In each case, the error analysis of a finite element calculation requires good information on derivative bounds for the solution. This talk concerns the case when both a small singular parameter and corners are present in the same problem. Some recent work will be reviewed, our ongoing work will be discussed, and some open problems will be presented. Some ingredients in the error analysis of finite elements for these problems will be given.

• Friday, October 28, 2011 at 2:00PM in Fretwell 379 (Math Conference Room)
  Dr. Kui Ren, University of Texas at Austin
  
  Title: Reconstruction strategies in quantitative photoacoustic tomography in diffusive regime
  
  Abstract: The objective of quantitative photoacoustic tomography (qPAT) is to reconstruct physical parameters of biological tissues from data of absorbed radiation distribution inside the tissues. Mathematically, qPAT can be regarded as an inverse problem to the diffusion equation where physical parameters enter as coefficients. We present in this talk some efficient reconstruction strategies for the inverse problem in qPAT with mono-frequency and multi-frequency data.

• Monday, July 11, 2011 at 2:30PM in Fretwell 379 (Math Conference Room)
  Dr. Jianxian Qiu, Xiamen University, China
  
  Title: Runge-Kutta discontinuous Galerkin methods for simulations of multi-medium flow
  
  Abstract: In the presentation, we will describe our recent work on using Runge-Kutta discontinuous Galerkin (RKDG) finite element methods for multi-medium flow simulations, the treatment of moving material interfaces by both a conservative method and a non-conservative method based on ghost fluid method (GFM). In generally, the method for solving multi-medium compressible flow comprises two parts: one is the technique for solving in the single-medium and another is treatment of material interfaces. A relatively dominant difficulty for simulation of compressible multi-medium flow is the treatment of moving material interfaces and their vicinity. In the literature there are some methods to overcome this difficulty. The ghost fluid method (GFM) developed by Fedkiw et al. and modified GFM by Liu et al. have provided a flexible way to treat the two-medium flow. Although the GFM is simple and effective, it is still not a conservative method. In our proposed conservative method, we use both the exact Riemann solvers as fluxes for multi-medium flow at the interfaces and adaptive interface cell; that is every cell contains only one medium and the size of neighboring cell next to the interfaces is modified according to the movement of interface. For solving of single- medium flow, we adopt the RKDG method. RKDG method is a very popular method in solving single-medium flow; it is a high order finite element method suitable for hyperbolic conservation laws while encompassing the useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, TVD Runge-Kutta time discretizations, and limiters. Numerical results for both gas-gas and gas-water flow are provided to illustrate the validity of the RKDG with conservative treatment and GFM treatment of the interfaces. Joint work with Tiegang Liu, IHPC, Singapore and B. C. Khoo, NUS