SPUDS Seminar

We will use this time to set up the calendar for the semester. If you are interesting in speaking in SPUDS, but unable to attend this meeting, please send an email to mperlmutter@boisestate.edu.

Abstract: Graph-structured data can be found in a variety of domains such as social and information networks, transportation and infrastructure, finance, and biology. In this talk, we will explore machine learning for graph-structured data using various techniques, most of which are developed from spectral graph theory. For the purpose of image classification, we construct similarity graphs using Wasserstein distance between images, a concept from optimal transport. We demonstrate that graph neural networks trained on these graphs outperform dense neural networks in the semi-supervised setting. Lastly, we explore signed directed graphs, and how they are encoded in matrices via the Magnetic Signed Laplacian. 

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Abstract: I continue the informal discussion on integral equations from last week.

Abstract: The complexity of solving the Laplace or Poisson equations can be greatly reduced if certain properties of the boundary conditions and, in the case of the Poisson equation, the source term are known. In the case of open boundary conditions and a source term that is localized around a finite number of centers, the solution can be formulated in terms of a multipole expansion, where the angular part is expanded in a series of analytic expressions: the spherical harmonics. Convergence to good accuracy can be achieved at finite expansion orders. A numeric integration involving the Green's function is necessary only in the radial direction. In the case of boundary conditions on a convex boundary and no source term, a similar series expansion is possible. Here, the full basis functions, the solid harmonics, are known analytically, including the radial parts. The challenge in this case lies in finding the expansion coefficients. This can be achieved by casting the boundary conditions into a system of linear equations. Finally, I will show how both approaches can be combined to solve a generalized Poisson problem with a piecewise constant dielectric function.

Abstract:  The integration of observational data with dynamical models through data assimilation has become an essential tool across a wide range of scientific disciplines.  Data assimilation is typically achieved through two main methods: sequential and variational methods. The Kalman filter is sequential and essentially iterates on three-dimensional (3DVar).  Variational methods, 3DVar and four-dimensional (4DVar) variational data assimilation, involve optimizing a cost function over the entire domain or time period.  All methods require prior specification of the model error variance, which is difficult to do.  We have recently received feedback on a manuscript where we apply Tikhonov regularization techniques to estimate model error variance. In this manuscript we focus on 4DVar using the representer method.  The reviewers expressed that while this is a natural approach to specifying error variance, they think the approach can be more generally applied to 3DVar, 4DVar and the Kalman filter.  In this talk I will work through applying regularization parameters selection techniques to 3DVar, Kalman filter and 4DVar with the goal of giving a clear explanation as to why the approach only works with 4DVar with representers. 

Abstract: The terrestrial water cycle plays a pivotal role in Earth's climate system, with soil moisture acting as a critical variable influencing land-atmosphere interactions, ecosystem dynamics, and hydrological processes. However, accurate representation of soil moisture in Land Surface Models (LSMs), such as the Community Land Model Version 5 (CLM5), remains a challenge due to uncertainties in soil hydraulic parameterization. These uncertainties propagate errors in hydrological variables, including total water storage (TWS), evapotranspiration, and leaf area index (LAI), limiting predictive capabilities, particularly in ungauged basins. This dissertation systematically evaluates and mitigates parameterization uncertainties in CLM5 by employing empirical orthogonal function (EOF) analysis, self-organizing maps (SOM), and evidential deep neural networks (EDNNs) to enhance hydrological simulations. First, EOF analysis is used to assess the influence of soil hydraulic parameters on soil moisture variability across the Contiguous United States (CONUS), identifying dominant spatial and temporal patterns while distinguishing parameter-driven effects from climatic influences. Next, an integrated EOF-SOM approach clusters soil moisture patterns based on nonlinear relationships, refining parameter calibration strategies for diverse climatic and geographic regions. To address uncertainty quantification, EDNNs are applied to model LAI, capturing both epistemic and aleatoric uncertainties in CLM5 simulations, thereby improving the robustness of vegetation and water cycle predictions. Finally, the EDNN framework is extended to predict hydrological variables in ungauged basins, leveraging perturbed parameter ensembles to enhance TWS forecasts in data-scarce regions. Collectively, these methodological advancements contribute to improving soil moisture representation in LSMs, providing a scalable framework for hydrological predictions, climate adaptation strategies, and water resource management. By integrating statistical techniques and machine learning, this research enhances the accuracy of CLM5 simulations, bridging gaps between model uncertainty and real-world hydrological processes.

Abstract: In this talk, I will explore a straightforward but powerful approach to predicting March Madness game outcomes, leveraging power rankings alongside various statistical techniques. By combining historical performance data, team strength metrics, and advanced analytics, we will dive into how these tools can offer insights into tournament predictions and improve our understanding of the dynamics that drive upsets and victories.

Abstract: Computing a function of matrix times a vector has applications in many areas of scientific computing and data science, for example, in exponential integrators for stiff differential equations, simulating non-local diffusion, and signal processing on graphs.  The last of these motivates this talk, in which we focus on the problem of computing $A^{1/2} b$, where $A$ is a large, sparse, symmetric positive definite matrix. In such settings, the standard approach of computing $A^{1/2} b$ from the spectral decomposition of $A$ is computationally impractical. Instead, we discuss an alternative approach using rational approximation of the square root function that only involves matrix vector products and solving shifted linear systems involving $A$. We compare this a more direct approach of using polynomial approximation of the square root function and illustrate the improved convergence for the rational function case. 

Abstract: I will discuss deep learning methods for solving graph combinatorial optimization problems such as the finding the largest fully connected sub-graph of a large graph. 

Abstract: We discuss the famous PageRank Algorithm, which revolutionized how society interfaces with the internet, and how it may be adapted to rank universities. 

Abstract: Finite element simulations are increasingly used in orthopaedic and biomechanics applications to investigate injury, disease, and degeneration of our joints. They are an ideal complement to experimental simulations where we can use the experimental data to validate a subset of our modeling conditions, and then apply these validated models to investigate many more conditions (including design-phase evaluation of implants or anatomic variation in the patient population) than is feasible experimentally. However, finite element analyses require expertise to develop, specialized software to run, and computational and post-processing time to analyse. In this talk, I will discuss a coupled approach to combine finite element simulations with surrogate mathematical models to achieve the best of both worlds by integrating the complexity of finite element simulations with instantaneous solutions that surrogate models can provide.

Abstract: The accurate prediction of solvation free energies is pivotal in computational chemistry, impacting drug discovery, materials science, and the study of chemical equilibria. Continuum solvation models are computationally efficient for these predictions but often fail to capture critical solute-solvent interactions, such as hydrogen bonding and solvent structuring, leading to significant errors in predictions for acids, bases, and ions. To address these limitations, we propose a hybrid approach combining classical molecular dynamics via thermodynamic integration with continuum solvation models and machine learning. Using a large dataset of solvation free energies, we define environment-based atomic descriptors (symmetry functions) and employ an atom-based neural network to systematically correct errors in continuum models. Our framework is expected to enable accurate solvation free energy predictions for molecules of diverse sizes, overcoming the limitations of existing models constrained to ~30 heavy atoms. This approach significantly enhances computational efficiency while offering deeper insights into solute-solvent interactions, paving the way for broader applications in molecular property prediction.

Abstract: We give a broad overview of finite difference methods for numerically solving partial differential equations.  The goal is to introduce the basic ideas with the hope making future topics in the seminar more approachable for non-experts.

Abstract: The Generalized Poisson Equation (GPe) is a fundamental partial differential equation (PDE) that arises in diverse fields such as electrostatics, fluid dynamics, and biological modeling. When discontinuities or irregular interfaces are present within the computational domain, conventional numerical methods such as finite difference or finite element methods often fail to deliver accurate solutions without significant computational cost. The Immersed Interface Method (IIM) provides an efficient framework for addressing these challenges by incorporating interface conditions directly into the numerical scheme.

In this talk, we present an implementation of the IIM to solve the GPe in complex geometries with embedded interfaces. The method modifies the standard finite difference scheme by introducing jump conditions at the interface, ensuring second-order accuracy even in the presence of discontinuous coefficients and irregular domains. We shall detail the mathematical formulation of the interface conditions, derive the discretization for irregular points.

Abstract: Finite volume methods are a method of choice for many engineering and science applications where conservation of mass/momentum/energy and other conserved quantities is important.  The key goal of FV methods is to preserve this conservation property at the discrete level. This is important for computational models tracking tracer particles such as smoke or pollution, but also important for resolving shock speeds accurately.  I’ll discuss what we mean by “numerical conservation”, a discretization principle based on a “difference of fluxes” approach,  and then show some discretizations for 1d elliptic, parabolic and hyperbolic problems. 

Abstract: I will discuss the fundamental topics in machine learning, starting with classical methods for regression and classification such as linear / logistic regression. We will then move to deep learning, showing that it may be interpretted as combining classical methods with the kernel trick.

As time allows, I will deep learning methods designed for computer vision and graph-related tasks, discussing the relation between these models and group symmetries. Lastly, I will talk about the connections between graph neural networks and PDEs.

Abstract: On January 15th, 2022 around 04:05 UTC the undersea volcano Hunga Tonga-Hunga Ha’apai located near the South Pacific island of Tonga violently erupted, and a large amount of energy was released into the atmosphere. The atmospheric disturbances generated by this event were detected by equipment all around the world. Analysis of this data revealed that some of these atmospheric disturbances were Lamb waves generated by the volcanic eruption. Previous works have successfully modeled these types of waves by using a shallow water approximation. In this talk I will present results for the homogeneous shallow water equations at earth scale using high resolution finite volume methods and adaptive mesh refinement. I will also present a model for the source terms needed for the shallow water approximation of the Lamb waves generated by the Hunga Tonga-Hunga Ha’apai eruption.

Abstract: I will continue my introduction to machine learning talk. After a brief recap of my talk from 10/1, I will talk about methods in deep learning which aim to utilize the intrinsic structure of the data by producing data representations with desirable symmetries. Lastly, I will talk about the relationship between neural networks for graph structured data and the heat equation. 

Abstract: This dissertation focuses on improving the understanding of soil moisture, total water storage (TWS), and leaf area index (LAI) within the Community Land Model Version 5 (CLM5) simulations over the contiguous United States (CONUS). The research investigates the influence of soil hydraulic parameter set-ups on soil moisture variability through Empirical Orthogonal Function (EOF) analysis, identifying dominant spatial and temporal patterns. Integrating Self-Organizing Maps (SOM), the project clusters these EOF modes, enhancing insights into soil moisture dynamics across different soil textures and climatic regions. Additionally, the dissertation explores the application of Evidential Deep Neural Networks (EDNNs) to improve uncertainty quantification in CLM5 simulations. The EDNNs capture both aleatoric and epistemic uncertainties, providing a more robust framework for handling the complexities of perturbed parameter ensemble (PPE) simulations. Besides that, the research also assesses the transferability of EDNNs for hydrological modeling in unseen regions, using the CAMELS basins as a testbed. In addition, this research uses advanced statistical methods, machine learning techniques, and climate model simulations to enhance the accuracy and reliability of climate and hydrological predictions, informing better water resource management and climate adaptation strategies. Lastly, the results from this dissertation will significantly contribute to parameter optimization in CLM5 and improve predictive models for soil moisture, TWS, LAI, and the rest of the hydrological variables under varying environmental conditions.

Abstract: Genetics, the study of how traits are passed down through generations, plays an important role in health and disease. While many types of genetic variants are widely analyzed, other types, such as copy number variants (CNVs) and rare single nucleotide variants (SNVs), remain understudied due to their added complexity within analysis. Here, I focus on two questions surrounding these understudied genetic variants.

I first discuss the relationship between CNVs and normal facial variation in a cohort of Bantu African children. Similarity in facial characteristics between relatives suggests a strong genetic component, but little is known about the role of CNVs in facial variation. I present a genome wide association study (GWAS) and gene set analysis of the relationship between normal facial variation and CNVs in our Bantu sample. We find that CNVs play a role in normal facial variation. Ultimately, our findings suggest that the study of CNVs, including the re-evaluation of existing GWAS data, is likely beneficial for the study of complex traits, such as facial variation.

Secondly, I present RAREsim, a flexible and scalable genetic simulation method designed for accurate simulation of rare variants. I demonstrate RAREsim’s ability to simulate the expected distribution and total number of variants while maintaining correlation structure and ability to annotate variants. I show the mathematical models behind these simulations. RAREsim is easily implemented within the accompanying R package, enabling previously unavailable accurate simulation of large samples of rare variant data.

Abstract: This seminar will cover an overview of simulation methods connected to atomistic simulations of matter. I will review the main principles of electronic structure and statistical mechanics simulations and present the foundations of continuum embedding models for solvent environments. 

Abstract: The accurate prediction of solvation free energies is fundamental in computational chemistry, with substantial applications in drug discovery, materials science, and the study of chemical equilibria in solutions. Continuum models have proven valuable in capturing solvation effects and are extensively explored in the literature. However, their predictive accuracy is often limited by common approximations that may oversimplify or overlook key interactions, such as hydrogen bonding or the structural organization of solvent molecules around the solute. As a result, continuum models frequently yield less accurate solvation energy predictions for acids, bases, and ions. In contrast, classical molecular dynamics (MD) simulations, combined with thermodynamic integration (TI), can enable solvation free energy calculations for a wide range of compounds, given that accurate intermolecular and intramolecular force fields are available.

This seminar will introduce classical MD simulations and free energy methodologies as applied to organic compounds. I will discuss TI and free energy perturbation theory, demonstrating their applications in investigating solvation and electrolyte effects to advance our understanding of solvation dynamics and interactions.

Abstract: This seminar will cover an overview of simulation methods connected to atomistic simulations of matter. I will review the main principles of electronic structure and statistical mechanics simulations and present the foundations of continuum embedding models for solvent environments. - This talk will be a continuation of my talk from December 12. 

Abstract: I will lead an informal discussion on integral equations.