## 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

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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

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