Poster
POSTERS
FOR POSTER PRESENTERS:
The conference will begin with a poster session. All posters will be on display throughout the conference. Poster presenters are expected to be present, with their poster set up on the poster boards we will provide, during the first poster session at 9am. Afterward, during the coffee and lunch breaks, poster presenters may work on coffee and their presentation in serial or parallel. All poster boards have the size of 4' x 6'. Poster presentations should have dimensions less than 104 cm high x 145 cm wide (41" x 57") and be in landscape orientation.
If you would like to present a poster, please e-mail a title, abstract, and your affiliation to Bin Jiang (bjiang@pdx.edu) by September 28, 2014, unless seeking travel support (see below).
Travel support
We are especially soliciting posters from students, postdocs, and junior researchers. For such participants, we expect to be able to provide some funding to partially cover travel expenses.
For students and postdocs at US institutions, limited travel support can be provided thanks to a grant from the National Science Foundation.
For students and postdocs at Canadian institutions, limited travel support is available via a grant from the Pacific Institute for the Mathematical Sciences.
If you wish to apply for travel support, please indicate it in your poster submission e-mail and include a short CV.
The deadline for receiving poster submissions seeking travel support is Sunday, September 14, 2014. Priority will be given to earlier submitted posters.
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Titles and Abstracts for Poster Presentations
at the 27th Pacific Northwest Numerical Analysis Seminar
Portland, Oregon, October 18, 2014
Jason Albright, Unversity of Utah
Title: High-order Accurate Difference Potentials Methods for Parabolic Interface Problems
Abstract: The Difference Potentials Method (DPM) was originally designed as a computationally efficient framework for the numerical approximation of solutions to elliptic problems in irregular domains in 2D and 3D. Additionally, DPM can handle general boundary conditions with equal ease. Recently, the Difference Potentials approach was extended to the spatial discretization of parabolic problems. I will present the development of high-order accurate DPM for variable coefficient parabolic problems with interfaces in 1D. The performance and flexibility of the obtained schemes will be illustrated numerically, as well as compared to the Immersed Interface Method (IIM). Based on the success of the aforementioned high-order DPM approach in 1D and several preliminary results in 2D, this approach remains highly generalizable and current work is centered upon the development of DPM for variable coefficient parabolic problems in arbitrary domains in 2–D and 3–D. This is joint work with Y. Epshteyn.
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Liran Barokas and Ofer Levi, Ben Gurion University of the Negev, Israel
Title: SPARSE REPRESENTATION OF ECG SIGNALS FOR PAIN ASSESSMENT
Abstract: Physiological signals such as ECG consist of mixtures of patterns and phenomena occurring at different times. Traditional signal processing and analysis methods are optimized to handle signals that include a single class of patterns, such as Fourier Representation for pure harmonics or Wavelets Representation for piece-wise smooth functions. In simplified and unreal scenarios, simple operations such as thresholding or filtering in the appropriate space can be very effective for separation of signal and noise. However, using a single representation method usually yields mediocre results on real-life signals. Matching Pursuit and Basis Pursuit use the idea of merging several different representation methods to create a so-called over-complete dictionary. These methods can decompose a signal into relatively few meaningful components by searching for the sparsest possible representation, given the right choice of dictionary.
We show that such tools can provide much better insight into a heart-beat signal's basic components and their relation to different physiological states in general, and pain in particular, than traditional analysis methods based on a single dictionary. We have analyzed the inter-beat time series (known as R-R signals) measured during a controlled pain-related experiment. A simultaneous Wavelet and Fourier analysis was applied using both Orthogonal Matching Pursuit and Basis Pursuit to project all wavelet-like features into the wavelet domain and Fourier-like features into the frequency domain. The analysis clearly reveals pain-related events, and our proposed method outperforms traditional spectral analysis methods with respect to both sensitivity and time delay. Joint work with Tobias Moeller and Shai Tejman-Yarden (UCSD Medical center), Michael Saunders, Jiyan Yang (ICME, Stanford University).
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Vrushali Bokil, Oregon State University
Title: Fast Estimating Current Densities in Equilibrium Magnetohydrodynamic Generator Channels
Authors: V. A. Bokil, N. L. Gibson, D. A. McGregor, C. R. Woodside
Abstract: Direct power extraction via magnetohydrodynamic (MHD) principles offers a potential step improvement in thermal efficiencies over energy systems utilizing traditional turbomachinery. This is principally due to the lack of moving parts in a MHD generator, as the temperature limits of the moving parts tend to limit cycle temperatures in traditional combustion driven systems. It was established that a major weakness toward commercialization of MHD power generation was the durability of the current collectors on the walls of the generator (electrodes). The electrodes must withstand harsh conditions, and the most damaging and perhaps most difficult to predict phenomenon experienced in the generator was arcing. For the expected temperature range of these generators, the current passing through ceramic electrodes can be expected to transition from a diffuse state to a highly dense arc. From a design perspective, there is a desire to be able to measure and then prevent these damaging arcs. In the arc state we expect the current densities in the channel to be many orders of magnitude larger than in the diffuse state.
Given these large differences in current density, the induced magnetic fields are measurably different close to the arc. The idea of reconstructing current densities from external magnetic flux density measurements has been successfully applied to fuel cells and vacuum arc remelters. We model the induced fields in an MHD generator channel, terms usually neglected in low magnetic Reynold's number flows and present a Mimetic Finite Difference method for the solution of the forward problem.
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Thomas Cameron, Washington State University
Title: Fast Computation of the Eigenvalues and Eigenvectors for Tridiagonal Matrix Polynomials
Abstract: The Ehrlich-Aberth method is a third order method for finding all the roots of a polynomial, p(t). At the heart of each Ehrlich-Aberth iteration is the computation of the Newton correction term, N(t) = p(t)/p'(t). A generalization of Hyman's method is developed for upper Hessenberg matrix polynomials, which makes it possible to accurately evaluate N(t) in O(mn2) time, where m is the degree and n the size of the matrix polynomial. Then we apply this method to an algorithm based on the Ehrlich-Aberth method to find both the eigenvalues and eigenvectors of a tridiagonal matrix polynomial in O(m2n2) time.
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Jummy Funke David, University of British Columbia
Title: Mathematical Modeling of HIV-TB Co-infection
Abstract: The project deals with the analysis of a dynamical model for the spread of tuberculosis
and HIV Co-infection. We capture in the model the dynamics of HIV infected individuals and investigate their impacts in the progression of tuberculosis. We proved analytically that the model has six equilibrium points where one of the equilibrium points is a disease-free equilibrium point and the rest are endemic equilibrium points which represented various hypothetical scenarios that inuence the progression of TB infection. We found that when the basic reproduction ratio is less than unity, then the disease-free equilibrium is locally asymptotically stable and when the basic reproduction number is greater than unity, a unique endemic equilibrium exists and is locally asymptotically stable under certain conditions. The stability of equilibria is derived through the use of Routh hurwitz stability criterion and Van den Driessche method of generational matrix. Numerical simulations are provided to illustrate the theoretical results and these showed that individual infected with HIV is at a high risk of being infected with TB. Also, Co-infection is higher in developing countries than developed countries due to TB exogenous reinfection.
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Tilmann Glimm, Jianying Zhang, Western Washington University
Title: Stability of Turing-like patterns in a reaction-diffusion system with an external gradient
Abstract: We investigate a generic model of a reaction-diffusion system consisting of an activator and an inhibitor molecule in the presence of a linear morphogen gradient. This morphogen gradient is established independently of the reaction-diffusion system and acts by varying the production rate of the activator and the inhibitor proportional to the morphogen concentration. It is motivated by models in developmental
biology in which a Turing patterning mechanism has been proposed and various chemical gradients are known to be important for development, for instance in the process of pre-cartilage condensation in embryonic limb development. Mathematically, this leads to reaction-diffusion equations with explicit spatial dependence. We investigate analytically the dependence of the Turing-like bifurcation on two small parameters: the ratio of diffusion coefficients and the slope of the chemical gradient. A complete bifurcation diagram is provided and verified numerically.
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Jay Gopalakrishnan, Ignacio Muga, Nicole Olivares, Portland State University and Pontificia Universidad Cat´olica de Valpara´ıso
Title: Dispersive and Dissipative Errors in the DPG Method with Scaled Norms for Helmholtz Equation
Abstract: I will present a study of dispersive and dissipative errors in the lowest order Discontinuous Petrov-Galerkin (DPG) [3, 4] solution to the Helmholtz equation, using a modified graph norm for the test space. The modification scales one of the terms of the graph norm by an arbitrary positive parameter.
Through mathematical analysis we show that as the parameter approaches zero, the error in an ideal DPG method must improve. However a typical practical implementable DPG method differs from the ideal DPG method. Whether the actually observed numerical error of the practical DPG method improves is investigated through a dispersion analysis [1, 5]. Since the DPG method has multiple interacting stencils, to find the discrete wavespeed, we must solve a nonlinear system for every given wavespeed. The dispersion analysis indicates that the performance does improve in certain cases. It also shows that the discrete wavenumbers of the method are complex, which explains the numerically observed artificial dissipation in the computed wave approximations. The performance of the DPG method is compared with a standard least-squares method [2] and finite element method.
References
1. M. AINSWORTH. Discrete Dispersion Relation for hp-Version Finite Element Approximation at High Wave Number. SIAM J. Numer. Anal. 42-2 (2004) 553-575.
2. Z. CAI AND R. LAZAROV AND T. A. MANTEUFFEL AND S. F. MCCORMICK. First-Order System Least Squares for Second-Order Partial Differential Equations: Part I. SIAM J. Numer. Anal. 31-6 (1994) 1785-1799.
3. L. DEMKOWICZ AND J. GOPALAKRISHNAN. A Class of Discontinuous Petrov-Galerkin Methods. Part II: Optimal Test Functions. Numer. Methods Partial Differential Eq. 27 (2011) 70-105.
4. L. DEMKOWICZ AND J. GOPALAKRISHNAN AND I. MUGA AND J. ZITELLI. Wavenumber Explicit Analysis of a DPG Method for the Multidimensional Helmholtz Equation. Comput. Methods Appl. Mech. Engrg. 213-216 (2012) 126–138.
5. A. DERAEMAEKER AND I. BABUˇSKA AND P. BOUILLARD. Dispersion and Pollution of the FEM Solution for the Helmholtz Equation in One, Two and Three Dimensions. Int. J. Numer. Meth. Engng. 46 (1999) 471-499.
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Ammar Harb, Portland State University
Title: Convergence Rates of the DPG Method with Different Spaces.
Abstract: This poster presents the main results obtained in the paper titled "Convergence rates of the DPG method with reduced test space degree" by Boumaa, Gopalakrishnan, and Harb. Considering the specific example of the mild-weak (or primal) discontinuous Petrov Galerkin (DPG) method for the Laplace equation, two results are obtained. First, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree. This poster refers to a duality theorem of the Aubin-Nitsche type for DPG methods. This explains the numerically observed higher convergence rates in weaker norms.
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Natalia Iwanski, Simon Fraser University
Title: Integral Equation Methods for Point Vortex Motion on the Surface of the Sphere
Abstract: Our research uses integral equation methods to study point vortex motion on the surface of the sphere. Equations that govern uid ow on the sphere can model the large-scale behaviour of the earth's oceans. Point vortex motion in particular can be used to examine how vortices travel over large distances and how they behave in the presence of coastlines. Although vortex motion has already been studied with various techniques, integral equation methods provide fast, highly accurate solutions to the steady state behaviour of uid and are being developed to study the evolution of point vortices. This poster considers the solution of the Laplace-Beltrami equation in the presence of a simply-connected island. The equation is first reformulated into an integral equation using the double layer potential, and then it can be discretized using the boundary of the island alone and solved with the trapezoid rule. This gives super-algebraic convergence for any smooth, closed boundary that can be described parametrically. A test case is illustrated showing the steady state behavior of a point vortex as well as the process for evolving the vortex over time.
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Milagros Loreto, University of Washington at Bothell
Title: Study of spectral step on Projected Subgradient Methods
Abstract: The spectral step is applied on a classical projected subgradient approach to create the
Spectral Projected Subgradient (SPS) and a modication of it (MSPS) with variations on the
direction. These methods are proposed to solve the problem: {min f(x), x ∈ Ω} where Ω is a
convex set and f(x) is a piece-wise linear, non-differentiable at some points. The spectral step
has been widely studied and applied to gradient and projected gradient methods, it requires little
computational work and it doesn't dependent on the optimal value of the function. A momentum
term is also added to the subgradient direction in order to accelerate the convergence process
towards the global solution. Encouraging numerical results are obtained and discussed.
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Duncan McGregor, Oregon State University
Title: M-Adaptation for the Electric Vector Wave Equation
Abstract: We present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time-domain electric vector wave equation on square meshes based on the mimetic
finite difference (MFD) method. We compare this strategy, called M-adaptation, to the lowest order N ed elec edge element discretization. Both discrete methods use the same edge-based degrees of freedom, while the temporal discretization is performed using the standard explicit Leapfrog scheme. To obtain efficient and explicit time stepping methods, both schemes are further mass lumped. We perform a dispersion and stability analysis for the presented schemes and compare all three methods in terms
of their stability regions and phase error. Our results indicate that the N ed elec method is second order accurate for numerical dispersion. The result of M-adaptation is a discretization that is fourth order accurate for numerical dispersion as well as numerical anisotropy. Numerical simulations are
provided that illustrate the theoretical results.
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Daniel Shapero, University of Washington
Title: Optimizing sparse matrix operations with just-in-time compilation
Abstract: Sparse matrix-vector multiplication is one of the most common computational kernels and yet, for many implementations, it achieves less than 10% of peak serial efficiency. Many libraries, like OSKI and PETSc, use an optimized block matrix format in order to improve performance. However, the optimizations used typically only occur at compile-time, which necessitates generating hundreds of functions which do essentially the same thing. I have developed a new approach, which generates and compiles the desired function at program run-time. This affords greater flexibility, and can be adapted to other kernels like FFT for which the greater number of parameters to optimize over precludes generating all possible functions ahead of time.
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Kyle Steffen, University of Utah
Title: The Difference Potentials Method for the Stokes--Darcy problem
Abstract: The Difference Potentials Method is a framework for developing efficient, high-order accurate numerical methods for the solution of boundary value problems (BVPs) on complex geometry. It was originally introduced by Viktor S. Ryaben'kii in his Doctor of Science (Habilitation) thesis in 1969, and then extensively developed by him and his collaborators (Ryaben'kii (2002)). It reduces well-posed BVPs to well-posed pseudo-differential boundary equations, constructed by the solution of auxiliary difference problems, which permit the use of fast numerical methods such as FFT-based or multigrid solvers. It does not require knowledge of the fundamental solution (Green's function) of the underlying problem, nor does it require the numerical solution of boundary integral equations.Recently, the Difference Potentials Method was extended to elliptic interface problems by Ryaben'kii, Turchaninov, and Epshteyn (2003, 2006); high-order methods have been developed in 1-D by Epshteyn and Phippen (2014). Parabolic interface problems have also been considered (Epshteyn (2014), Albright, Epshteyn, and Steffen (2014)).
We now consider the Stokes--Darcy problem, which is a multiphysics model of fluid motion coupling free flow, modeled by the linear Stokes equations, with flow in porous media, modeled by Darcy's law. The Difference Potentials Method reduces the numerical solution of this coupled problem to the numerical solutions of two separate auxiliary difference problems (corresponding to the linear Stokes equation and to Darcy's law, in their respective subdomains) which are then coupled by the boundary equations at the interface.For my poster I will present the numerical methods for each component (the linear Stokes equations, and Darcy's law). These methods use uniform Cartesian grids which are not required to conform to boundaries or interfaces. I will also present numerical results to test the developed methods. This poster is based on joint work with Y. Epshteyn.
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Sheldon Wang, Midwestern State University
Title: Singular Value Decomposition and Its Application in the Multi-Scale Modelling of Soft Biological Materials
Abstract: A major challenge in the modeling of biological systems is to handle complex materials. The accurate description of physical, chemical, and biological phenomena over a wide range of spatial and temporal scales is extremely difficult. In practice, various hierarchical modeling techniques have been explored. However, the concurrent coupling which is more relevant and ubiquitous in biological systems has not yet been fully understood. It appears that various modeling techniques eventually depend on the singular value decomposition, a common method to identify rank, left null space, column space, right null space, and row space for any rectangular matrix. Using the singular value decomposition, it is possible to identify the hidden spatial and temporal correlations and patterns between variables and material properties. It is hopeful that based on the singular value decomposition, a computational protocol for complex dynamical systems can be established to handle complex biological materials. In an example published earlier in Refs. [1] [2], sickle cell anemia, one of the first diseases pinpointed to the genetic cause at the DNA level, was modeled. Hemoglobin in its quaternary molecular structure is very much like a bead. The red blood cell has many such beads within the cell cytoskeleton. The cause of the sickle cell disease is a simple switch of the DNA base pair from A to T, with this switch, the codon will be changed from GAG to GTG. The normal hemoglobin at this particular location is slightly hydrophilic, thus tends to form a protective layer with the surrounding water molecules and is separated from each other. As a consequence, the normal red blood cell membrane is flexible and fluidic. Due to the sickle cell mutation, the hydrophilic spot becomes slightly hydrophobic and during the deoxygenated state, it tends to lose the protective layer of water molecules and consequently forms a chain of hemoglobin beads. Moreover, such chains will continue to form bundles and eventually yield a very stiff and sticky material property for the sickle cell membrane. In the end, these sickle cells tend to block the capillary vessels and cause the sickle cell anemia. We start with a series of molecular dynamics simulations of hemoglobin-hemoglobin interactions coupled with surrounding water molecules. Different level of coarse graining models will employ, employed immersed boundary/continuum methods for the direct simulation of phase transitions of normal and sickle red blood cell membranes [3].
REFERENCES
[1] T. Wu, S. Wang, B. Cohen, and H. Ge, “Molecular Modeling of Normal and Sickle
Hemoglobins,” International Journal for Multiscale Computational Engineering, 8, pp 237-244, 2010.
[2] T. Wu, S. Wang, and B. Cohen “Modeling of Proteins and Their Interactions with Solvent,” in Advances in Cell Mechanics. Chapter 3, pp. 55-116, Springer, ISBN 978-3-642-17589-3, 2011.
[3] S. Wang, “Immersed Methods for Compressible Fluid-Solid Interactions” in Multiscale Simulation and Mechanics of Biological Materials. Chapter 12, Wiley, 2012.
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Bin Zheng, Pacific Northwest National Laboratory
Title: Two-Level Stochastic Collocation Method
Abstract: In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh with a low level stochastic collocation (corresponding to the polynomial space) and solve linearized equations on a fine mesh using high level stochastic collocation (corresponding to the polynomial space). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method. The two-level method is computationally more efficient, especially for nonlinear problems with high random dimensions. Numerical experiments are also provided to verify the theoretical results.