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
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).
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.
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.
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.
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.
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.
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.
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
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.
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.
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.
