Spring 2023

March 30, 2023 (Frederic and Julia Wan Lecture): Qing Nie, UC Irvine

Location: Smith Hall 205

Title: Multicellular organization: from biophysical models to single-cell genomics

Abstract: Fred Wan’s career has been defined by ‘mathematics in the service of humanity’, from neurobiology and developmental biology to solid mechanics and economics, from the classroom and the University to the National Science Foundation.” as best described by Simon Levin from Princeton University. I was fortunate to be a colleague and a close collaborator of Fred for nearly 20 years, witnessing Fred’s passion and dedication to applied mathematics on research, education, and service. As a small testimony to Fred’s impact on the community, the joint work with Fred on morphogenesis for multicellular organization has shaped my research career. It is in the spirit of Fred that applied mathematics adapts to the ever-changing science, this lecture is focused on new mathematical approaches in analyzing multicellular organization in the era of big data.

Cells make fate decisions in response to their dynamic environments, and multicellular structures emerge from close interplays among cells and genes in space and time. Mechanistic models, based on a small number of selected biochemical and physical regulators, have provided many insights into how cells organize their spatiotemporal patterns for relatively simple biological systems. The recent single-cell genomics technologies provide an unprecedented opportunity to explore complex spatial tissues systematically and comprehensively from in vivo animal models and human diseases. However, since its start emergence five years ago, spatiotemporal analysis of multi-modal single-cell genomics datasets is still at its early stage and many major challenges remain. In this talk, by motivating via mechanistic models of “small” data of genes and cells for multicellular systems, I will present new methods to reconstruct spatiotemporal tissue properties from large single-cell genomics datasets. Specifically, we derive dynamic transitions of cell fate from static measurements, infer cell-cell communication from nonspatial data, and uncover spatiotemporal cellular interaction and organization from spatial data. Through several applications to development, regeneration, and disease, we show the discovery power of these methods as well the need of new sophisticated mechanistic models and inference tools that enable better understanding of principles governing the complex multicellular organization as new data grow exponentially in many forms.

April 6, 2023: Andrew Christlieb, Michigan State University (Note special time)

Location and time: Smith Hall 205, 2:30-3:30pm

Title: Taming The Curse of Dimensionality to Enable ‘First Principle’ Optimal Design in Fusion Energy

Abstract: Recent advances at the National Ignition Facility and their achievement of thermal nuclear burn on December 5th of 2022 represents a great achievement and an exciting advancement in the state of fusion energy systems. A close synergy between simulation, theory, and experiment, (including data assimilation) led to this advance. However, if we were to seek to design a new device of this kind in a new operating point outside parameters covered by the experiments, we lack ‘first principle’ predictive capabilities that would enable design of such systems. This is because inertial confined fusion systems, as well as magnetically confined fusion systems, are far from the equilibrium state and can span many plasma regimes within the device during the evolution of the plasma.

Behind all of these challenges in building effective ‘first principle’ models is the curse of dimensionality. To address this fundamental challenge, the Center for Hierarchical and Robust Modeling of Non-Equilibrium Transport (CHaRMNET) was created. CHaRMNET is one of only 4 DoE funded Mathematical Multifaceted Integrated Capability Centers (MMICC). CHaRMNET seeks to develop the mathematical tools that will enable the inclusion of ‘first principle’ effects within the optimal design loop of fusion energy systems. CHaRMNET seeks to build a first-of-its-kind holistic approach that will exploit structure within models to mitigate the curse of dimensionality and to bridge a wide range of length and time scales in plasma science. The curse of dimensionality is a critical challenge that is pervasive throughout computational science and refers to the observation that the resources needed to solve a problem on a computer scale exponentially with the dimension of the problem. Fundamental plasma models are seven-dimensional and are presently computationally intractable (with existing mathematical methods and computational resources) to drive optimization and uncertainty quantification at the engineering scale of plasma systems.

To achieve our goals, CHaRMNET has four synergistic thrusts: ‘Beyond Forward Simulation’, development of new theory for UQ and optimization with ensembles of models; ‘Multi-Scale Modeling’, development of structure preserving surrogates; ‘Simulation Acceleration’, development of structure preserving sparce representations and blended computing; and ‘Self-Consistency’, development of structure preserving and asymptotic preserving discretization’s. In this talk, I will give an overview of these four key thrusts, and how they interact. Next I will focus on one area, the teams work on Multi-Scale Modeling. This will include an overview of our recent work on structure preserving ML surrogates for closure of moment expansions of kinetic systems and the outlook for this approach. I will conclude with pointing to next steps and challenges that are on the horizon for the team.

April 6, 2023 (Boeing Distinguished Colloquium): Simon Tavare, Columbia University

Location and time: Smith Hall 205, 4:00 pm

Title: Modeling and simulation of cancer evolution in single cells

Abstract: Understanding the role of copy number variation in tumor evolution has long been a challenge, in part because identifying subclones from bulk DNA sequencing data is hard. Recent advances in single-cell whole genome sequencing, however, enable proﬁling of copy number aberrations at high resolution in thousands of cells. Single-cell genomics data from these technologies has enabled quantitative measurements of tumor dynamics, and measurements of the rate of chromosomal aneuploidy, whole-genome duplications and replication errors in tumors.

To better understand clonal evolution, we have developed a detailed model for studying single-cell dynamics in a population of cells, incorporating somatic copy number changes, clonal selection of driver mutations and accumulation of neutral passenger mutations. Simulation of the model follows population dynamics as input by the user, generates the clonal evolution forward in time, where clones are deﬁned by their copy number and driver mutation proﬁles. The phylogeny of a sample is then constructed backward in time. The algorithm is designed to be eﬃcient for large cell populations while maintaining statistical accuracy.

We present two examples exploiting the simulation. The ﬁrst follows the neutral evolution of copy number events in the population of epithelial cells in the fallopian tube, the second investigates the evolution of high-grade serous ovarian cancer (HGSOC) driven by genomic instability. I will outline how Approximate Bayesian Computation may be used to estimate model parameters. By simulating real sequencing reads from the ideal data provided by the model, the simulator may also be used to calibrate clonal reconstruction algorithms used on single-cell DNA sequencing data, with some useful consequences.

April 13, 2023: Ka-Kit Tung, UW

Location: Smith Hall 205

Title: Catastrophe Collapse of the Atlantic Overturning Circulation: Is it Eminent?

Abstract: Theory of catastrophe was a popular area of study for applied mathematicians a few decades ago, and then it fell out of fashion. Now it is brought back into the limelight by climate scientists, who are concerned with “tipping points” in climate under global warming. A major tipping point is thought to exist in the Atlantic Meridional Overturning Circulation (AMOC), which includes the Gulf Stream. It transports massive amount of heat from the tropics to high northern latitudes. Global warming may inhibit sinking in its subpolar branch as warmer and fresher water is less dense, thus slowing the overturning. It is thought that its collapse would bring back the ice age, as dramatized in the movie “The Day After Tomorrow”. Are there Early Warning Signs to such a pending catastrophe? Some say yes, and that AMOC is already collapsing. We shall examine the theory and observational evidence in this talk.

April 20, 2023: Samuel Van Fleet, UW

Location: Smith Hall 205

Title: Genuinely multi-dimensional, maximum Taylor discontinuous Galerkin schemes for solving linear hyperbolic systems of conservation laws

Abstract: In this work we develop the maximum Taylor discontinuous Galerkin (MTDG) method for solving linear systems of hyperbolic partial differential equations (PDEs). The proposed method is a variant of the Lax-Wendroff discontinuous Galerkin (LxW-DG) method from the literature. The process by which the Lax-Wendroff DG method is obtained can be summarized as follows: 1. Compute a truncated Taylor series in time that relates the solution that is being sought to the known solution at the previous time-step. 2. Replace all the temporal derivatives in this Taylor expansion by spatial derivatives by repeatedly invoking the underlying PDE. 3. Multiply this expansion by appropriate test functions, integrate over a finite element, and perform a single integration-by-parts that places a derivative on the test functions as well as introducing boundary terms. 4. Replace the boundary terms by appropriate numerical fluxes. The key innovation in the newly proposed method is that we replace the single integration-by-parts step by an approach that moves all spatial derivatives onto the test functions; this process introduces many new terms that are not present in the Lax-Wendroff DG approach. The regions of stability various MTDG methods are compared to the LxW-DG stability regions. It is shown that compared to the Lax-Wendroff DG method, the maximum Taylor DG method has a larger region of stability and has improved accuracy. These properties are demonstrated by applying MTDG to several numerical test cases.

April 26, 2023: Hau-Tieng Wu, Duke University (special seminar)

Location: SMI 205

Title: Turning nonstationary biomedical signals into useful clinical information by denoising manifolds

Abstract: In the clinical arena we are moving beyond snapshot health data. Physicians are now provided and confronted by multimodal physiological data collected over long stretches of time, and some are of low quality or contaminated by undesired information. The nonstationarity and heterogeneity nature of these datasets can impose a serious challenge for health care providers and medical researchers, particularly when they need clinically useful and actionable information at the bedside. I will discuss recent progress in signal processing dealing with some of these challenges. The main tool is circling around the mission called manifold denoising, and our solution depends on random matrix theory and spectral geometry. Clinical examples and current progress will be structured toward providing practical solutions.

May 4, 2023: Kevin Lu, UW

Location: Smith Hall 205

Title: Parameter Estimation and Pairs Trading for Some Lévy-driven Ornstein-Uhlenbeck Processes

Abstract: We discuss parameter estimation using maximum likelihood and Fourier inversion for Lévy-driven Ornstein-Uhlenbeck processes, where the stationary distribution or background driving Lévy process is a weak variance alpha-gamma distribution, a multivariate generalization of the variance gamma distribution. These processes allow for the modeling of possibly infinite activity mean reverting price processes with jumps, and we then study how to perform pairs trading in this framework in the univariate case. Specifically, we use simulation methods to demonstrate how to find the optimal level of the process to enter and exit trades, with control variate as a variance reduction technique.

May 11, 2023: Jeffrey Oval, Portland State University

Location: Smith Hall 205

Title: Computational Tools for Exploring the Spatial Localization of Eigenvectors (and Waves) in Complex Media

Abstract: It is well-known that a function u = u(t, x) describing the behavior of acoustic or electromagnetic waves in time and space can often be decomposed as an infinite sum

u(t, x) = \sum_{n=1}^\infty c_n(t) \psi_n(x),

where each term in the sum is a product of a function c_n varying only in time and a function \psi_n varying only in space. The standing waves \psi_n are eigenvectors of a spatial differential operator associated with the medium through which the waves are propagating. It is not as well-known that properties of the medium can cause some eigenvectors to be strongly spatially localized. A practical consequence of eigenvector localization is that waves at certain frequencies can be “trapped” at some location or “channelled” along some favorable path. Such features are of interest in the design of structures having desired acoustic or electromagnetic properties: sound-mitigating outdoor barriers and next generation organic LEDs and solar cells are examples of this design principle in action. There remain many open problems related to understanding and exploiting this kind of localization, and we will discuss a computational approach that we hope will provide useful insight. More specifically, we focus on the issue of eigenvector localization, outlining our computational approach and providing theoretical, heuristic, and empirical support for it through several examples (with many pictures).

May 18, 2023 (Boeing Distinguished Colloquium): Beatrice Pelloni, Heriot-Watt University

Location: Smith Hall 205

Title: Non-smooth behaviour of linearised water waves: the Talbot effect revisited

Abstract: I will describe some surprising phenomena that appear when periodic linearised wave models such as the linear KdV, BO, or ILW equations are considered with a discontinuous initial condition. These phenomena are analogous to the “Talbot effect” observed in the 1850’s in linear optics.

I will show the result of experiments as well as the mathematical analysis of the models, and discuss the dependence of this effect on the linearity and the boundary conditions, to conclude that it is remarkably robust, leaving its echo in the solution of a wide variety of problems.

May 19, 2023: Ricardo Baptista, Caltech (Note special time and date)

Location: Smith Hall 205, 3-4pm

Title: Toward consistent nonlinear filtering and smoothing via measure transport

Abstract: Solving filtering and smoothing problems for geophysical applications involve estimating the hidden states of complex systems and accurately characterizing their uncertainty. Popular algorithms for tackling these problems include ensemble Kalman methods such as the EnKF, EnKS and RTS smoother. While these algorithms yield robust state estimates for high-dimensional models with non-Gaussian statistics, ensemble Kalman methods are limited by linear transformations and are generally inconsistent with the true Bayesian solution. In this presentation, I will discuss how measure transport can be used to consistently transform a prior ensemble into samples from a filtering or smoothing distribution. This approach provides a natural generalization of Kalman methods to nonlinear transformations, thereby reducing the intrinsic bias of classic algorithms with a marginal increase in computational cost. In small-sample settings, I will show how to estimate transport maps for high-dimensional inference problems by exploiting low-dimensional structure in the target distribution. Finally, I will demonstrate the benefit of this framework for filtering and smoothing on chaotic dynamical systems and aerodynamic flows.

May 25, 2023: Konstantinos Mamis, UW

Location: Smith Hall 205

Title: Stochastic model for cell population dynamics quantifies homeostasis in colonic crypts and its disruption in early tumorigenesis

Abstract: The questions of how healthy colonic crypts maintain their size, and how homeostasis is disrupted by driver mutations, are central to understanding colorectal tumorigenesis. We propose a three-type stochastic branching process, which accounts for stem, transit- amplifying (TA) and fully differentiated (FD) cells, to model the dynamics of cell populations residing in colonic crypts. Our model is simple in its formulation, allowing us to easily estimate all but one of the model parameters from the literature. Fitting the single remaining parameter, we find that model results agree well with data from healthy human colonic crypts, capturing the considerable variance in population sizes observed experimentally. Importantly, our model predicts a steady state population in healthy colonic crypts for relevant parameter values. We show that APC and KRAS mutations, the most significant early alterations leading to colorectal cancer, result in increased steady-state populations in mutated crypts, in agreement with experimental results. Finally, our model predicts a simple condition for unbounded growth of cells in a crypt, corresponding to colorectal malignancy. This is predicted to occur when the division rate of TA cells exceeds their differentiation rate, with implications for therapeutic cancer prevention strategies.

June 1, 2023: Lee Ricketson, Lawrence Livermore National Laboratory

Location: Smith Hall 205

Title: An implicit, asymptotic-preserving time integration scheme for charged particle motion in arbitrary electromagnetic fields

Abstract: In magnetic confinement fusion reactors, the strong background magnetic field used for confinement also induces a fast oscillation time-scale that can be on the order of nanoseconds. Meanwhile, global reactor codes must simulate scales on the order of seconds. While asymptotic limits that average over the fast oscillatory motion are known, these approximations break down in some critical physical regimes. Thus, one is motivated to seek asymptotic preserving schemes that can take large time-steps when physically permissible, but still recover accurate particle trajectories when the asymptotics break down. I will present such a scheme, along with a sketch of its derivation. Particular attention will be paid to energy conservation, as it will be shown that this has enormous consequences on the long-term accuracy of particle trajectories. I will also report on recent progress on adaptive time-stepping, efficient solution of the nonlinear system of equations the scheme requires, and capturing of so-called finite Larmor radius effects when electric fields feature small length scales.

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