## Applied Math Seminar at Colorado State University## Thursday 3:00-4:00PM, Weber 223 |

Feb 14 Back to top

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

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

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

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

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

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

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

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

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

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

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

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

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Oct 24 Back to top

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Oct 31 Back to top

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Dec 12 Back to top

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