applied math

Applied Math Seminar at Colorado State University

Thursday 3:00-4:00PM, Weber 223

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Spring 2019

Feb 14   Feb 28   Mar 07   Mar 14   Mar 28   Mar 29   Apr 04   Apr 11   Apr 18   Apr 22   Apr 25   May 02  

Fall 2019

Oct 10   Oct 24   Oct 31   Dec 12  


 
 
 
Feb 14   Back to top

Paul Constantine

Department of Computer Science, University of Colorado Boulder

Title  Tools and techniques for subspace-based parameter reduction in computational science models

Abstract  Scientists and engineers use computer simulations to study relationships between a physical model's input parameters and its output predictions. However, thorough parameter studies - e.g., constructing response surfaces, optimizing, or averaging - are challenging, if not impossible, when the simulation is expensive and the model has several inputs. To enable parameter studies in these instances, the engineer may attempt to reduce the dimension of the model's input parameter space using techniques such as sensitivity analysis or variable screening to identify unimportant variables that can be fixed for model analysis. Generalizing classical coordinate-based reduction, there are several emerging subspace-based parameter reduction tools, such as active subspace and sufficient dimension reduction, that identify important directions in the input parameter space with respect to a particular model output. I will motivate and provide an overview of subspace-based parameter reduction techniques and discuss strategies for exploiting such low-dimensional structures - including analysis and computation - to enable otherwise infeasible parameter studies. For more information, see activesubspaces.org  
 
 
Feb 28   Back to top

Zack Kilpatrick

Department of Applied Mathematics, University of Colorado Boulder

Title  Analyzing decision making in dynamic environments using Chapman Kolmogorov equations

Abstract  To make decisions organisms often accumulate information across multiple timescales. To understand decision-making under these conditions we derive and analyze stochastic differential equations that model ideal observers accumulating evidence to make binary choices in dynamic environments. We use principles of probabilistic inference to show how the observer incorporates old observations into their belief depending on how rapidly the environment changes. When the environment switches between trials, the observer carries information forward from one trial to the next by biasing the initial condition of the drift diffusion model determining their evidence accumulation process. Analyzing the corresponding first passage time problem for decisions, we show this decreases the average time, but not the accuracy of the next decision. When the environment switches within trials, we can describe the ensemble stochastic dynamics of the the observer's belief using differential Chapman Kolmogorov equations. This allow us to efficiently compute the response accuracy of the ideal observer. We also find there are multiple approximate models that perform nearly as well as the optimal model, some of which are simpler to implement. Our results provide clues as to the why humans exhibit common cognitive biases in decision making experiments, and suggest task parameters ranges to better identify strategies humans use to make decisions. This is joint work with Nick Barendregt (CU Boulder PhD Student), Kate Nguyen (U Houston PhD Student), and Kreso Josiic (U Houston, Math).  
 
 
Mar 07   Back to top

Hamza Mahmoud

Department of Mathematics, University of Tennessee, Knoxville

Title  Rejection off-lattice kinetic Monte Carlo method

Abstract  While most kinetic Monte Carlo (KMC) simulations are lattice based, many important technological applications involve multi-component systems in which lattice mismatch leads to elastic strain and crystal defects, neither of which can be accurately modeled with a lattice-based approach. Off-lattice kinetic Monte Carlo (OLKMC) aims to overcome these limitations. In OLKMC one needs to calculate the rates for all possible moves from the current minimum state by searching the energy landscape for index-1 saddle points surrounding the current basin of attraction. We introduce a rejection scheme for OLKMC where the true rates are replaced by rate estimates and the saddle point searches are done locally. Our numerical results show that our algorithm is stochastically equivalent to the original OLKMC. However, our scheme allows a performance boost that scales with the number of particles in the system. We test the method on a growing two-species nanocluster with an emerging core-shell structure bound by Lennard-Jones potential and find we can reduce computation time (compared to the fully off-lattice KMC) by 90% for clusters that contain around 55 particles, and 96% percent for clusters that contain around 65 particles.  
 
 
Mar 14   Back to top

Greg Fesshauer

Department of Applied Mathematics and Statistics, Colorado School of Mines

Title  Some Remarks on the Construction of Designer Kernels and Their Applications

Abstract  Positive definite reproducing kernels (or covariance kernels) play a central role in many applications in numerical analysis, spatial statistics, as well as statistical learning. They appear in methods known, e.g., as radial basis functions, kriging, Gaussian processes, or simply kernel-based methods. Some kernels, such as the Gaussian kernel, multiquadric kernel or the family of Matern kernels, are very popular and are often used in a "one-size-fits-all" general purpose strategy. In this talk I will emphasize a different approach; that of custom-built designer kernels that have certain desirable built-in properties such as, e.g., periodicity, satisfaction of boundary conditions, or non-stationarity. After introducing a few different types of designer kernels I will illustrate their use with some examples from data fitting, the numerical solution of PDEs, and electrical power demand forecasting.  
 
 
Mar 28   Back to top

Jinbo Xu

Toyota Technological Institute at Chicago, and University of Chicago

Title  Distance-based protein folding powered by deep learning

Abstract  Accurate description of protein structure and function is a fundamental step towards understanding biological life and highly relevant in the development of therapeutics. Although greatly improved, experimental protein structure determination is still low-throughput and costly, especially for membrane proteins. As such, computational structure prediction is often resorted. Predicting the structure of a protein without similar experimental structures is very challenging and usually needs a large amount of computing power. We have developed a deep learning method for protein contact and distance prediction that won the CASP (Critical Assessment of Structure Prediction) in both 2016 and 2018 in the category of contact prediction. In this talk we show that by using this powerful deep learning technique, even with only a personal computer we can predict the structure of a protein much more accurately than ever before. Due to it success in 2016, this deep learning technique has been adopted widely by the structure prediction community and thus, resulted in the largest progress in CASP13 (2018) in the history of CASP.  
 
 
Mar 29 (2-3PM)   Back to top

Thomas Conrad Clevenger

Department of Mathematics, Clemson University

Title  Adaptive, parallel, matrix-free geometric multigrid for Stokes equations with large viscosity contrast

Abstract  Problems arising in the earth's mantle convection involve finding the solution to Finite Element Stokes systems with large viscosity contrasts. These systems contain localized features which, even with adaptive mesh refinement, result in linear systems that can be on the order of 100+ million unknowns. One common approach while preconditioning the velocity space of these systems is to apply an Algebraic Multigrid (AMG) v-cycle (as is done in the ASPECT software, for example), however, with AMG, robustness can often be difficult with respect to problem size and number of parallel processes. Additionally, we have seen an increase in iteration counts during steps adaptive refinement when using AMG. In contrast, the Geometric Multigrid (GMG) method, by using information about the geometry of the problem, should offer a more robust option, i.e., should have convergence properties independent of the mesh size and be equivalent across processor counts. Here we present a matrix-free variant of the GMG v-cycle which will work on adaptively refined, distributed meshes, and we will compare it against the current AMG preconditioner used in the ASPECT software. We will demonstrate the robustness of GMG with respect to problem size and show scaling up to 1024 cores, as well as show other advantages of matrix-free computations.  
 
 
Apr 04   Back to top

Max Gunzburger

Department of Scientific Computing, Florida State University

Title  Integral equation modeling for anomalous diffusion and nonlocal mechanics

Abstract  We use the canonical examples of fractional Laplacian and peridynamics equations to discuss their use as models for nonlocal diffusion and mechanics, respectively, via integral equations with singular kernels. We then proceed to discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. In particular, we discuss the recently developed asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems.  
 
 
Apr 11   Back to top

Varis Carey

Department of Mathematics and Statistics Sciences, University of Colorado Denver

Title  Derivative-Free Optimization For Solving Stochastic Inverse Problems

Abstract  We discuss the approximate solution of data-consistent stochastic inverse probems(Butler et al, 2018) via derivative-free optimization techniques. In particular, we apply a modification of the STARS algorithm(Chen and Wild 2015) in the context of the active subspace(Constantine 2015) dimension-reduction technique. The talk is self-contained, giving a brief introduction to data-consistent inversion, derivative-free optimization, and active subspace analysis.  
 
 
Apr 18   Back to top

Varun Gupta

Pacific Northwest National Laboratory

Title  Recent advances in Generalized/Extended Finite Element Method (GFEM/XFEM) and its application to fracture mechanics

Abstract  The Generalized/eXtended Finite element method (GFEM/XFEM) has become extremely popular in last two decades for modeling problems involving moving interfaces, crack propagation or material discontinuities. It can be regarded as an extension of the finite element method (FEM), obtained by enriching the approximation space for solutions to differential equations with enrichment functions, typically selected based on the problem at hand. It offers several advantages over the standard FEM, including significant improvement in the numerical accuracy for a given computational cost. The efficacy of GFEM/XFEM relies, to a great extent, on the proper selection of enrichment functions. This talk will feature a multi-scale GFEM for the accurate and efficient computation of the numerical solution for the problems where only limited a-priori knowledge about the solution is available. This method, termed as the Generalized FEM with global-local enrichments (GFEM gl) is based on the solution of interdependent global and local scale problems, and can be applied to a broad class of multi-scale problems. In this approach, the enrichment functions are obtained from the numerical solution of a fine-scale boundary value problem defined around a localized region of interest. The local problems focus on the resolution of fine scale features of the solution while the global problem addresses the macro-scale structural behavior.

Albeit the popularity of GFEM/XFEM, it has been well known that these methods suffer from a major drawback of numerical ill-conditioning that limited the use of these methods for large-scale problems. The recently developed Stable GFEM (SGFEM) will also be presented in this talk which is an improvement to GFEM/XFEM and provides a robust, yet simple solution to this ill-conditioning. The SGFEM involves a simple local modification of the enrichments employed in the GFEM, which near-orthogonalizes the enrichment space to the finite element approximation space. Other bonus features of this method include improved accuracy and solution efficiency compared to the GFEM/XFEM  
 
 
Apr 22   Back to top

Xiu Ye

Department of Mathematics and Statistics, University of Arkansas at Little Rock

Title  Finite element methods with discontinuous approximations

Abstract  In this talk, different finite element methods with discontinuous approximations will be discussed including IPDG, HDG and specially WG finite element methods as well as the relations between them. In addition, a new conforming DG finite element method will be introduced which combines the features of both conforming finite element method and discontinuous Galerkin method.  
 
 
Apr 25   Back to top

Mark Hoefer

Department of Applied Mathematics, University of Colorado Boulder

Title  Dispersive hydrodynamics: the mathematics and physics of nonlinear waves in dispersive media

Abstract  Dispersive hydrodynamics - modeled by hyperbolic conservation laws with dispersive perturbation - has emerged as a unified mathematical framework for the description of multiscale nonlinear wave phenomena in dispersive media and accurately describes a plethora of physical systems. This talk will be a tour through some recent mathematical and physical results in this growing field of research. Parallels and analogies to classical hydrodynamics will be presented such as the generation of shock waves subject to appropriate regularization and their description in terms of characteristics. From the existence of expansion shocks to the generation of viscous shock waves in a conservative medium, dispersive regularization also leads to a number of counterintuitive, effects, which will also be described.  
 
 
May 02   Back to top

Jared Bronski

Department of Mathematics, University of Illinois Urbana-Champaign

Title  Coupled Oscillators, Random networks, and Association Schemes

Abstract  We will present some ideas around dynamics of oscillators on networks. In particular we will discuss some algebraic structure that arose somewhat unexpectedly in the analysis of a certain algorithm for random graph generation that has been considered in the context of mathematical biology.  
 
 
Oct 10   Back to top

Valeria Barra

Department of Computer Science, University of Colorado Boulder

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Oct 24   Back to top

Runchang Lin

Department of Mathematics and Physics, Texas A&M International University

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Oct 31   Back to top

Zixuan Cang

Department of Mathematics, University of California Irvine

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Dec 12   Back to top

James Liu

Department of Mathematics, Colorado State University

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