MS02: Recent advances in application-oriented numerical computation and optimization
Organizers:
- Xiang-Sheng Wang, University of Louisiana at Lafayette
- Qin Sheng, Baylor University
- Bruce Wade, University of Louisiana at Lafayette
Session A: Oct.1, 10:40am-12:00pm, Classroom Building 108
Session B: Oct.1, 3:10pm-4:30pm, Classroom Building 108
Session C: Oct.1, 5:00pm-6:20pm, Classroom Building 108
Session D: Oct.2, 10:30am-11:50am, Classroom Building 108
MS02-A-1 |
Bruce Wade, |
Splitting with and Exponential Time Differencing Scheme for Advection-Diffusion-Reaction Systems |
We describe a second order Exponential Time Differencing (ETD) scheme for advection-diffusion-reaction systems and analyze their convergence properties under different operator splitting scenarios when applied to systems with nonsmooth or mismatched data. |
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MS02-A-2
11:00am-11:20am (Oct 1) |
Emmanuel Asante-Asamani,
Clarkson University |
Exponential time differencing with real distinct poles for simulating chemotaxis problems. |
Collective migration of cells in response to a chemical gradient (chemotaxis) is important for many biological processes such as tumor angiogenesis and aggregation of unicellular organisms. Mathematical models of these processes fall under the class of advection-diffusion-reaction (ADR) equations posed in 2D or 3D, having positive solutions, variable speed advection, stiff linear diffusion and possibly stiff nonlinear reaction kinetics. In this work, we apply a second order exponential time differencing scheme (ETD-RDP) to solve chemotaxis problems. I will discuss the positivity and stability of ETD-RDP for ADR equations and illustrate its advantage over IMEX methods when applied to chemotaxis problems. |
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MS02-A-3
11:20am-11:40am (Oct 1) |
Olaniyi Samuel Iyiola, Clarkson University |
Subgradient extragradient method with double inertial and self-adaptive step size: applications in Dynamical systems. |
Dynamical systems are fundamental tools in modeling physical phenomena with applications in several fields of study. In recent decades, constructing fixed-point iterative schemes for solving variational inequality problems (VIP) has found applications in developing robust numerical schemes for dynamical systems and has become active research area for many applied Mathematicians. This talk will focus on the overview of the relationship that exists between dynamical system and several fixed point iterative schemes. In addition, I will discuss our recent results on the subgradient extragradient method with double inertial extrapolation terms and self-adaptive step sizes for solving VIP. Rate of convergence is enhanced in this version which is more relaxed with easy to implement conditions on the inertial-factor and relaxation parameter. Numerical examples are provided to demonstrate the accelerating behaviors of our method over other related methods in the literature. |
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MS02-A-4 11:40am-12:00pm (Oct 1) CLB 108 |
Sean R. Breckling, Department of Energy |
Scattered results on the recovery of 3D volumetric density information from a single projection view |
High-speed, high-energy flash X-ray radiography is a cornerstone diagnostic measurement for physics experiments studying hypersonic hydrodynamics. Given that precise timing and coordination of the X-ray light exposure is required for multi-projection view tomography, many premier experimental testbeds operate utilizing as few as one to two projection images. Experimentalists have coped with these limitations by designing studies in settings where recovery of 3D volumetric information remains mathematically possible. A common modality is one that exploits spherical or cylindrical symmetry. In idealized settings, performing tomographic reconstruction is equivalent to solving Abel’s integral equation. While this formulation is a fully-determined problem, the Abel transform is unbounded. Discretizations of the reconstruction procedure are frequently beleaguered with noise amplification and instability near the singularity at the axis of rotational symmetry. To address this, several regularization techniques (e.g. TVmin, L_{p}, L_{1}-L_{2}, L_{1}/L_{2}, etc) have been tested; many of which have shown promise. Additionally, quite recently, reframing the data discretization paradigm from the classic “rectangle of pixelated data” setting to a regression problem using high-order Radial Basis Functions (RBFs) or the Finite Element Method (FEM) have been considered. |
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MS02-B-1 3:10pm-3:30pm (Oct 1) CLB 108 |
Qin Sheng, Baylor University |
Notes on Global Error Analysis for Splitting Methods |
Splitting methods have been used for solving a broad spectrum of problems in scientific applications. They are designed for the numerical solutions to not only differential equations, but also optimization and machine learning procedures. A splitting method decomposes an original problem to several subproblems, computes separately the solution of each of them, and then combines all sub-solutions to form an approximation of the solution to the original problem. Motivations of different splitting methods are inspired by problems with multiple operators in natural ways. In all cases, the computational advantage is that it is faster to compute the solution of the split components separately, than to compute the solution directly when they are treated together. However, this comes at the cost of an error introduced by the splitting, so strategies must be devised for controlling the error. This talk studies splitting mechanisms via operator formulations. A survey will be conducted in global error estimates of popular exponential splitting strategies. Adaptive splitting, a highly effective decomposition collaborating with mesh adaptations, will also be briefly elaborated. |
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MS02-B-2 3:30pm-3:50pm (Oct 1) CLB 108 |
Eduardo Servin Torres, Baylor University |
A study of the numerical stability of a Crank-Nicolson method for solving singular Kawarada equations |
This presentation focuses on the proof of the numerical stability and convergence of a second-order semi-adaptive method for the numerical solution of nonlinear Kawarada equations. The nonlinear source terms will not be frozen. The convergence and the preservation features of the numerical method will be investigated. Order of convergence will be validated through improved Milne devices. Simulation experiments will be carried out to illustrate our theoretical results. The partial differential equations model a broad spectrum of important phenomena that occur in numerous industrial applications, including the combustion in thermal engines. |
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MS02-B-3 3:50pm-4:10pm (Oct 1) CLB 108 |
Lisa Kuhn, Southeastern Louisiana University |
Biquintic B-Spline Solutions of Smart Material Structures and Applications to Control Theory |
In distributed parameter control theory researchers employ Galerkin’s method in order to guarantee convergent solutions to the corresponding optimal control problem. Recent advances in smart material structures, such as weak internal damping and material continuities, have substantially increased the complexity of models and the computation time required for simulation. In this talk numerical computations are presented for clamped beam and plate smart material structures using quintic and biquintic B-spline basis functions. Convergence and computation time are compared with other known methods. |
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MS02-B-4 4:10pm-4:30pm (Oct 1) CLB 108 |
Zilong Song, Utah State University |
Solving A Friction Stir Welding Problem with Reduced Order Models and Neural Networks |
The friction stir welding process can be modelled using a system of heat transfer and Navier-Stokes equations with a shear-dependent viscosity. Finding numerical solutions to this system of nonlinear partial differential equations over a set of parameter space, however, is extremely time-consuming. Therefore, it is desirable to find a computationally efficient method that can be used to obtain an approximation of the solution with acceptable accuracy. In this talk, we present a reduced basis method for solving the parametrized coupled system of heat and Navier-Stokes equations using a proper orthogonal decomposition (POD). In addition, we apply a machine learning algorithm based on an artificial neural network (ANN) to learn (approximately) the relationship between relevant parameters and the POD coefficients. Our computational experiments demonstrate that substantial speed-up can be achieved while maintaining reasonable accuracy. |
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MS02-C-1 5:00pm-5:20pm (Oct 1) CLB 108 |
Tilsa Aryeni, University of Wyoming |
Stable Generalized Finite Element Method for Richards Equation in Heterogenous Soil in 1D |
This study concentrates on the numerical approximation of the one-dimensional Richards equation for unsaturated flow in heterogeneous soil layers. The nature of each different soil layer enforces the elliptic coefficient to be discontinuous with respect to the spatial variable. It is known that the standard finite element technique fails to maintain the convergence optimality for this type of problem. In particular, it happens when a node of spatial mesh configuration does not match the interface location. The Generalized FEMs (GFEMs) have been utilized to recover this issue. It is based on augmenting the standard finite element basis with some enrichment that can capture the behavior of the solution near the interfaces. Stable GFEM is one version of GFEMs that not only maintains the optimal rate of convergence but also possess the same order of the scaled condition number of the associated stiffness matrix as the standard FEM. The numerical examples are presented and tested with the available analytical solutions to illustrate the performance of the method. |
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MS02-C-2 5:20pm-5:40pm (Oct 1) CLB 108 |
Jaeun Ku, Oklahoma State University |
Convergence of adaptive least-squares finite element methods |
In this talk, we consider adaptive procedures for least-squares finite element methods. We establish that the sequence of the approximation solutions generated by adaptive procedures is a Cauchy sequence in a Banach space. This leads to the conclusion that the sequence converges. In order to force the sequence converging to the true solutions, we propose a refinement strategy using a weighted least-squares functional as an a posteriori error indicator to identify the local regions to refine the current underlying mesh. |
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MS02-C-3 5:40pm-6:00pm (Oct 1) CLB 108 |
Yunhui He, The University of British Columbia |
Block preconditioners for the MAC discretization of the Stokes-Darcy equations |
In this talk, we discuss preconditioning methods for solving the Stokes-Darcy equations, discretized by the Marker and Cell (MAC) finite difference method. A central challenge in the solution of the Stokes-Darcy equations is that the equations governing each domain are fundamentally different. This difficulty is particularly highlighted when the parameters involved, specifically the viscosity coefficient and permeability constant, differ from each other by a few orders of magnitude. We propose three block preconditioners and analyze the eigenvalue distribution of the preconditioned operators. Our proposed preconditioned operators have strongly clustered eigenvalues that are independent of physical parameters, and consequently, a minimum residual iterative method such as GMRES rapidly converges. Numerical results validate our theoretical observations. |
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MS02-C-4 6:00pm-6:20pm (Oct 1) CLB 108 |
Srijana Ghimire, Tulsa Community College |
Bivariate Lagrange interpolation at the checkerboard nodes |
An explicit formula for the bivariate Lagrange basis polynomials of a general set of checkerboard nodes is derived. This formula generalizes existing results of bivariate Lagrange basis polynomials at the Padua nodes, Chebyshev nodes, Morrow-Patterson nodes, and Geronimus nodes. In addition, a subspace that is spanned by linearly independent bivariate vanishing polynomials that vanish at the checkerboard nodes is also constructed. As a result, the uniqueness of the set of bivariate Lagrange basis polynomials is proved in the quotient space defined as the space of bivariate polynomials with a certain degree over the subspace of bivariate vanishing polynomials. |
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MS02-D-1 |
Erik Van Vleck, University of Kansas |
Dimension reduction for data assimilation: Particle filters with reduced order models and data |
Particle filters are a class of data assimilation techniques that can estimate the state and uncertainty of dynamical models by combining nonlinear evolution models with non-Gaussian uncertainty distributions. However, estimating high dimensional states, such as those associated with spatially-discretized PDE models, requires an exponentially-large number of ensemble members or particles in order to avoid the so-called filter collapse. This dramatically decreases the accuracy and efficiency in obtaining the estimation. By combining particle filters with projection-based data-driven model reduction techniques, such as Proper Orthogonal Decomposition and Dynamic Mode Decomposition, we demonstrate that it is possible to reduce the effective dimension of the models and reduce the occurrences of filter collapse for a class of dynamical models relevant to forecasting of geophysical fluid flows. This technique can be adapted to account for models with transient change in parameters, by developing time dependent projection matrices using window snapshots. We demonstrate several variants of the technique on Lorenz’96-type models and on a simulation of shallow-water equations. |
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MS02-D-2 10:50am-11:10am (Oct 2) CLB 108 |
Sidney Shields, Sandia National Laboratories |
Modeling a cylindrical end irradiated cavity with EMPIRE: investigating the effects of approximating geometric features |
When comparing experimental data with simulation results, great care must be taken in choosing which physics and geometric features to model. The task of simplifying a model for the sake of simulation with minimal loss of fidelity can be quite challenging for even the simplest of problems. This talk will focus on using Sandia's plasma code EMPIRE to model a cylindrical end irradiated cavity fielded at experimental facilities, such as Z and NIF. Photoelectric diodes generate an intense beam of electrons by the interaction of soft X-rays with an emitter surface driving an electromagnetic mode in the cavity. These electromagnetic modes can be quite sensitive to various geometric features one might want to approximate away in the model. Using various configurations of this cavity, we can quantify some of the sensitivities in the electromagnetic modes due to the geometric description of the problem. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525. |
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MS02-D-3 11:10am-11:30am (Oct 2) CLB 108 |
Xiang-Sheng Wang, University of Louisiana at Lafayette |
The condition number of a Vandemonde-like matrix arising from a direct parallel-in-time algorithm |
We study a direct parallel-in-time (PinT) algorithm for first- and second-order time-dependent differential equations. We use a second-order boundary value method as the time integrator. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix B, which yields a direct parallel implementation across all time steps. A crucial issue of this methodology is how the condition number (denoted by Cond(V)) of the eigenvector matrix V of B behaves as n grows, where n is the number of time steps. A large condition number leads to a large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present an explicit formula for V as a Vandemonde-like matrix and prove that Cond(V) = O(n^{2}). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows, and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. |