Mathematical Sciences Colloquium at WPI
In 2018/2019 I co-ordinated the Colloquium in the Department of Mathematical Sciences at WPI. This is a record of the talks - in the unlikely event that it becomes interesting to someone.
The Mathematical Sciences Colloquium at WPI is held at 11am in Room 203 of Stratton Hall. In the 2018/9 academic year, Prof Andrea Arnold and I are organising the lecture series. Titles and Abstracts will be filled in below as they are received.
Spring 2019
10 January 2019 (Thursday)
Long Feng, Yale University
Title: Sorted Concave Penalized Regression
Abstract: The Lasso is biased. Concave penalized lease squares estimation (PLSE) takes advantage of signal strength to reduce this bias, leading to sharper error bounds in prediction, coefficient estimation and variable selection. For prediction and estimation, the bias of the Lasso can be also reduced by taking a smaller penalty level than what selection consistency requires, but such smaller penalty level depends on the sparsity of the true coefficient vector. The sorted L1 penalized estimation (Slope) was proposed for adaptation to such smaller penalty levels. However, the advantages of concave PLSE and Slope do not subsume each other. We propose sorted concave penalized estimation to combine the advantages of concave and sorted penalizations. We prove that sorted concave penalties adaptively choose the smaller penalty level and at the same time benefits from signal strength, especially when a significant proportion of signals are stronger than the corresponding adaptively selected penalty levels. A local convex approximation, which extends the local linear and quadratic approximations to sorted concave penalties, is developed to facilitate the computation of sorted concave PLSE and proven to possess desired prediction and estimation error bounds. We carry out a unified treatment of penalty functions in a general optimization setting, including the penalty levels and concavity of the above mentioned sorted penalties and mixed penalties motivated by Bayesian considerations. Our analysis of prediction and estimation errors requires the restricted eigenvalue condition on the design, not beyond, and provides selection consistency under a required minimum signal strength conditionin addition. Thus, our results also sharpens existing results on concave PLSE by removing the upper sparse eigenvalue component of the sparse Riesz condition.
18 January 2019
Chenlu Ke, University of Kentucky
Title: A New Class of Independence Measures and Its Application in Ultrahigh Dimensional Data Analysis
Abstract: In this talk, we first introduce a novel class of measures for testing independence between two random variables of arbitrary dimensions based on the discrepancy between the conditional and the marginal characteristic function. Theoretical properties and numerical studies are included to demonstrate the advantages of our method. Based on the proposed measure, we then develop a two-stage sufficient variable selection (SVS) procedure for the large p, small n problem and a sufficient dimension reduction (SDR) approach, which can have wide applications in Machine Learning, Genomics, image processing, pattern recognition, and medicine. Simulations and a real data example illustrate the efficacy of our approaches.
22 January 2019 (Tuesday)
Fangfang Wang, University of Wisconsin
Statistical Modelling of Multivariate Time Series of Counts
Abstract: In this presentation, I will talk about a new parameter-driven model for non-stationary multivariate time series of counts. The mean process is formulated as the product of modulating factors and unobserved stationary processes. The former characterizes the long-run movement in the data, while the latter is responsible for rapid fluctuations and other unknown or unavailable covariates. The unobserved stationary vector process is expressed as a linear combination of possibly low-dimensional factors that govern the contemporaneous and serial correlation within and across the count series. Regression coefficients in the modulating factors are estimated via pseudo maximum likelihood estimation, and identification of common factor(s) is carried out through eigen-analysis on a positive definite matrix that aggregates the autocovariances of the count series at nonzero lags. The two-step procedure is fast to compute and easy to implement. Appropriateness of the estimation procedure is theoretically justified, and simulation results corroborate the theoretical findings in finite samples. The model is applied to time series data consisting of the numbers of National Science Foundation funding awarded to seven research universities from January 2001 to December 2012. The estimated parsimonious and easy-to-interpret factor model provides a useful framework for analyzing the interdependencies across the seven institutions.
25 January 2019 - No colloquium
1 February 2019
Yevgeniy Ptukhin, WPI
A derivation of the percentile based Turkey distributions and a comparison of monotonic vs nonmonotonic and rank transformations
Abstract: The Method of Moments (MOM) has been extensively used in statistics for obtaining conventional moment-based estimators of various parameters. However, the disadvantage of this method is that the estimates “can be substantially biased, have high variance, or can be influenced by outliers” (Headrick & Pant, 2012). The Method of Percentiles (MOP) provides a useful alternative to the MOM when the distributions are non-normal, specifically being more computationally efficient in terms of estimating population parameters. Examples include the generalized lambda distribution (Karian & Dudewicz, 1999), third order power method (Koran, Headrick & Kuo, 2015) and fifth order power method (Kuo & Headrick, 2017). Further, the HH, HR and HQ distributions, as extensions of the Tukey g-h (GH) family, are of interest for investigation using the MOP in this study. More specifically, closed form solutions are obtained for left-right tail-weight ratio (a skew function) and tail-weight factor (a kurtosis function).
A Monte Carlo simulation study which includes the comparison of monotonic and nonmonotonic transformation scenarios is also performed. The effect on Type 1 error and power rates under severely nonmonotonic scenarios are of special interest in the study. Dissimilarities of not strictly monotonic scenarios are discussed. The empirical confirmation that Rank Transform (RT) is appropriate for 2x2 designs is obtained.
8 February 2019, no colloquium
15 February 2019
Hussein Nasralah, WPI
Portfolio optimization for small time horizons
Abstract: The problem of portfolio optimization is widely studied in mathematical finance and concerns finding an optimal trading strategy for a given investor on a specified time horizon. We study this problem in an incomplete financial market in which the stock price has stochastic volatility. Under mild assumptions on the investor's utility function, we exhibit a closed-form formula for a close-to-optimal portfolio, with the approximate optimality valid for small time horizons. A heuristic scheme extending these small-time results to any finite horizon will then be discussed.
22 February 2019
Qing Nie, UC Irvine
Data-driven multiscale modeling of cell fate dynamics
Cells make fate decisions in response to different and dynamic environmental and pathological stimuli. Recent technological breakthroughs have enabled biologists to gather data in previously unthinkable quantities at single cell level. However, synthesizing, analyzing, and understanding such data require new mathematical and computational tools, and in particular, dissecting cellular dynamics emerging from molecular and genomic scale details demands novel multiscale models. In this talk, I will present our recent works on analyzing single-cell molecular data, and their connections with cellular and spatial tissue dynamics. Our mathematical approaches bring together optimization, statistical physics, ODEs/PDEs, and stochastic simulations along with machine learning techniques. By utilizing our newly developed computational tools along with their close integrations with new datasets collected from our experimental collaborators, we are able to investigate several complex systems during development and regeneration to uncover new mechanisms, such as novel beneficial roles of noise and intermediate cellular states, in cell fate determination.
1 March 2019
Jean King, WPI
Brain and Behavior: Mathematical Modeling and Computation
Dean King will provide an overview of the role of mathematical modeling and computation in the field of neuroscience, as well as describe her own neuroscience research. In addition, she will provide an update on the ongoing neuroscience initiative at WPI, including the role of the mathematical sciences.
neuroscience initiative at WPI, including the role of the mathematical sciences
8 March 2019 - Spring Break, no colloquium
15 March 2019
Reginald McGee, College of the Holy Cross
Title: Uncovering linear and nonlinear relationships in leukemia
Complex protein interaction networks complicate the understanding of what most promotes the rate of cancer progression. High dimensional data provides new insights into possible mechanisms for the proliferative nature of aggressive cancers, but these datasets often require fresh techniques and ideas for exploration and analysis. In this talk, we consider expression levels of tens of proteins that were recorded in individual cells from acute myeloid leukemia (AML) patients via mass cytometry. After identifying immune cell subpopulations in this data using an established clustering method, we use topological data analysis to search for subpopulations that are most actively proliferating. To conduct the search within these subpopulations, we build on the differential geometric perspective that led to our recent statistic for testing aggregate differences in protein correlations between patients with different subtypes of AML.
22 March 2019
Rosemary Bailey, University of St Andrews
Finding good designs for experiments
Suppose that there are N experimental units available for an experiment to compare v treatments. The experimental units may be all alike, or they make be partitioned into blocks, or there may be rows and columns. The design is the function allocating treatments to units. It is said to be optimal if it minimizes the average value of the variance of the estimator of the difference between two treatments.
How should we find an optimal design for any given situation, with specified values of N and v? There are some theorems that cover a few cases. These lead on to some general folklore that is not always correct. One combinatorial approach is to make use of pretty patterns, to find designs with high symmetry or regularity. Another is to make a computer search. Sometimes a good design is found by a lucky accident.
29 March 2019
Peter Cameron, University of St Andrews
The Random graph
A large random finite graph, with high probability, has no non-trivial symmetry. However, Paul Erdos and Alfred Renyi discovered in 1963 that a random countable graph has an infinite group of automorphisms. The reason for this is even more surprising: there is only one countable random graph (that is, there is a graph which occurs with probability 1 up to isomorphism). This graph, and its automorphism group, have a rich structure, and make occurrences in several areas including set theory, number theory and topology. However, the graph is not an isolated phenomenon. In the late1940s, Roland Fraısse gave a necessary and sufficient condition for the existence of a countable homogeneous structure with prescribed finite substructures; the random graph is an example of his theory. But Fraısse was not the first to take this road. A posthumous paper of Pavel Urysohn (who was drowned in the Bay of Biscay in 1924 at the age of 26) constructed a homogeneous Polish space(complete separable metric space) using methods similar to those later developed by Fraısse.
) using methods similar to those later developed
by Fra ̈ıss ́e.
5 April 2019
Bruce Sagan, Michigan State University
The protean chromatic polynomial
Let t be a positive integer and let G be a combinatorial graph with vertices V and edges E. A proper coloring of G from a set with t colors is a function c : V → {1, 2, …, t} such that if uv ϵ E then c(u) ≠ c(v), that is, the endpoints of an edge must be colored differently. These are the colorings considered in the famous Four Color Theorem. The chromatic polynomial of G, P(G; t), is the number of proper colorings of G from a set with t colors. It turns out that this is a polynomial in t with many amazing properties. One can characterize the degree and coefficients of P(G; t). There are also connections with acyclic orientations, increasing spanning forests, hyperplane arrangements, symmetric functions, and Chern classes in algebraic geometry. This talk will survey some of these results.
12 April 2019
Nicola Garofalo, University of Padova.
Title: Nonlocal Sobolev and isoperimetric inequalities for a class of non-symmetric and non-doubling semigroups
Abstract: In his seminal 1934 paper on Brownian motion and the theory of gases Kolmogorov introduced a second order evolution equation which displays many challenging features. Despite the large amount of work done by many people over the past thirty years, some basic questions presently remain unsettled such as Hardy-Littlewood-Sobolev and Isoperimetric inequalities, a Calder\’on-Zygmund theory, and the study of local and nonlocal minimal surfaces. In this lecture I will present a fractional calculus adapted to a class of equations modelled on Kolmogorov’s, and using such calculus I will discuss some interesting developments in the above program.
19 April 2019 - Project Presentation day, no colloquium
26 April 2019
Yousef Marzouk, MIT
Nonlinear filtering and smoothing with transport maps
Bayesian inference for non-Gaussian state-space models is a ubiquitous problem, arising in applications from geophysical data assimilation to mathematical finance. We will present a broad introduction to these problems and then focus on high dimensional models with nonlinear (potentially chaotic) dynamics and sparse observations in space and time. While the ensemble Kalman filter (EnKF) yields robust ensemble approximations of the filtering distribution in this setting, it is limited by linear forecast-to-analysis transformations. To generalize the EnKF, we propose a methodology that transforms the non-Gaussian forecast ensemble at each assimilation step into samples from the current filtering distribution via a sequence of local nonlinear couplings. These couplings are based on transport maps that can be computed quickly using convex optimization, and that can be enriched in complexity to reduce the intrinsic bias of the EnKF. We discuss the low-dimensional structure inherited by the transport maps from the filtering problem, including decay of correlations, conditional independence, and local likelihoods. We then exploit this structure to regularize the estimation of the maps in high dimensions and with a limited ensemble size.
We also present variational methods---again based on transport maps---for smoothing and sequential parameter estimation in non-Gaussian state-space models. These methods rely on results linking the Markov properties of a target measure to the existence of low-dimensional couplings, induced by transport maps that are decomposable. The resulting algorithms can be understood as a generalization, to the non-Gaussian case, of the square-root Rauch--Tung--Striebel Gaussian smoother.
This is joint work with Ricardo Baptista, Daniele Bigoni, and Alessio Spantini.
3 May 2019
Alexander Mamonov, University of Houston
Inversion and Imaging with Waves Via Model Order Reduction
Joint work with Liliana Borcea, Vladimir Druskin, Mikhail Zaslavsky and Jörn Zimmerling.
We consider the problem of determining the structure (the presence and strength of reflectors)
in a medium occupying a domain of interest from the surface measurements of the scattered waves
(acoustic, elastic, electromagnetic) induced by surface sources. The numerous applications include geophysical exploration,
medical diagnostics, non-destructive evaluation and testing, etc. Our proposed framework for inversion and
imaging with waves is based on model order reduction. The reduced order model (ROM) is a projection of the
wave equation propagator on the subspace of time domain wavefield snapshots. Even though neither the propagator
nor the wavefields are known in the bulk, the projection can be computed just from the time-domain surface waveform data.
Once the ROM is computed, its use is trifold.First, the projected propagator can be backprojected to obtain an image of
subsurface reflectors directly. ROM computation is a highly nonlinear procedure unlike the conventional linear migration (Kirchhoff, RTM) imaging methods.
This allows to untangle the interactions between the reflectors, i.e., the mulitple scattring events. Consecutively,
the resulting images are almost completely free from multiple scattring artifacts that often appear in conventional linear migration.
Second, the property of the ROM to untangle the multiple scattering interactions allows to generate the Born data,
i.e. the data that the measurements would produce if wave propagation in the domain of interest obeyed Born
approximation instead of the wave equation. Obviously, such data only contains primary reflections and the multiples
are removed. We refer to such procedure as the Data-to-Born (DtB) transform. Once the multiply scattered data is
transformed to Born data via DtB, existing linear imaging and/or inversion methods can be applied to obtain the
reconstructions of the subsurface structure.
Third, the ROM can be used to perform quantitative inversion, i.e., not only to determine the reflector locations,
but also their strength. The conventional quantitative inversion approaches (e.g., the full waveform inversion -
FWI, widely employed in exploration geophysics) are based on minimizing the misfit between the measured and
predicted data, a highly non-convex objective that causes the optimization to converge slowly or even to get stuck
in abundant local minima. In contrast, we propose to minimize the ROM-misfit in the least square sense. Such
procedure can be set up as a fixed-point iteration that does not require derivative computations and converges
quickly to a high quality solution.
Fall 2018
24 August 2018 POSTPONED TO SPRING SEMESTER
Jana Gevertz, The College of New Jersey
Title: Robust optimization of cancer immunotherapy
Abstract: Mathematical models of biological systems are often validated by fitting the model to the average of an often small experimental dataset. Here we ask the question of whether predictions made from a model fit to the average of a dataset are actually applicable in samples that deviate from the average. We will explore this in the context of a mouse model of melanoma treated with two forms of immunotherapy. We have hierarchically developed a system of ordinary differential equations to describe the average of this experimental data, and optimized treatment subject to clinical constraints. Using a virtual population method, we explore the robustness of treatment response to the predicted optimal protocol; that is, we quantify the extent to which the optimal treatment protocol elicits the same qualitative response in virtual populations that deviate from the average. We find that our predicted optimal is not robust and in fact is potentially a dangerous protocol for a fraction of the virtual populations. However, if we consider a different drug dose than used in the experiments, we are able to identify an optimal protocol that elicits a robust anti-tumor response across virtual populations. Time permitting, we will consider how personalized optimal treatment protocols compare in efficacy to robust optimal protocols. This is joint work with Eduardo Sontag (Northeastern University) and Joanna Wares (Richmond University).
31 August 2018
Labor Day weekend - no colloquium
7 September 2018
Qing Han, Notre Dame
Title: The isometric embedding of abstract surfaces in the 3-dim Euclidean space
Abstract: A surface in the 3-dim Euclidean space can be viewed as the image of a map from a planar domain to the 3-dim Euclidean space, at least locally. The standard metric in the Euclidean space induces a metric on the surface, which allows us to compute the lengths of curves on the surface and to compute the distance of any two points on the surface. For example, the distance of two points on a sphere is the length of the small arc on the great circle through these two points. The induced metric on the surface can be transformed to an abstract metric by the abovementioned map. Now, we consider the converse question. Given an abstract metric on a planar domain, can we find a surface in the 3-dim Euclidean space whose induced metric is the given abstract metric? This is the isometric embedding problem we will discuss. It started with a conjecture by Schlaefli in 1873 that this can always be achieved near any given point. This conjecture is widely open and there are only a few results under various conditions. The question can be reformulated in terms of partial differential equations. Despite the technical description, the underlying equation has a simple form. In this talk, I will give a historical account and explain why this equation is hard to solve. The talk is aimed at a general audience.
14 September 2018
No Colloquium
21 September 2018
Daniela Calvetti, Case Western Reserve University
Title: Inverse Problems, Bayesian inference and Sparse Solutions: a bit of magic in L2.
Abstract: Recasting a linear inverse problems within the Bayesian framework makes it possible to use partial or qualitative information about the solution to improve the computed solution in spite of the inherent ill-posedness of the problem and noise in the data. In this talk we will show how a suitably chosen probabilistic setting can lead to a very efficient algorithm for the recovery of sparse solutions that only requires the solution of a sequence of linear least squares problems. The fast converge rate of the algorithm and its low computational cost will be discussed and illustrated with computed examples.
28 September 2018
Jeffrey Case, Penn State
Title: Sharp Sobolev trace inequalities via conformal geometry
Abstract: Escobar proved a sharp Sobolev inequality for the embedding of $W^{1,2}(X^{n+1})$ into $L^{2n/(n-1)}(\partial X)$ by exploiting the conformal properties of the Laplacian in X and the normal derivative along the boundary. More recently, an alternative proof was given by using a Dirichlet-to-Neumann operator along the boundary and its close relationship to the 1/2-power of the Laplacian. In this talk, I describe a new relationship between the conformally covariant fractional powers of the Laplacian due to Graham--Zworski and higher-order Dirichlet-to-Neumann operators in the interior, and use it to prove sharp Sobolev inequalities for embeddings of $W^{k,2}$. Other consequences of this relationship, such as a surprising maximum principal for the conformal 3/2-power of the Laplacian, will also be discussed.
5 October 2018
Carolyn Mayer, WPI
Title: Partial Erasure Channels
Abstract: Partial erasure channels model situations in which some information may remain after an erasure event occurs. These types of erasure events can happen in applications such as flash memory storage. After reviewing recently introduced partial erasure channels, we present results in two directions. First, we show how multilevel coding and multistage decoding may be applied to break partial erasure channels into simpler subchannels. Second, we consider Luby Transform (LT) codes, a class of codes without fixed rates, over partial erasure channels. We present a modified decoding process for these codes and compare their efficiency with standard LT codes on the q-ary erasure channel.
12 October 2018
Fall break - no colloquium
19 October 2018
Fall break - no colloquium
26 October 2018
Juan Manfredi, University of Pittsburgh
Title: Dynamic Programming principles for nonlinear elliptic equations
2 November 2018
Levi Conant Lecture
(NB: 4pm, Higgins 218)
Henry Cohn, Microsoft Research
Title: A conceptual breakthrough in sphere packing
9 November 2018
TBA
16 November 2018
Jiashun Jin, Carnegie Mellon University
Talk title: Co-authorship and Citation Networks of statisticians
Abstract: We have collected a data set for the networks of statisticians, consisting of titles, authors, abstracts, MSC numbers, keywords, and citation counts of papers published in representative journals in statistics and related fields. In Phase I of our study, the data set covers all published papers from 2003 to 2012 in Annals of Statistics, Biometrika, JASA, and JRSS-B. In Phase II of our study, the data set covers all published papers in 36 journals in statistics and related fields, spanning 40 years. The data sets motivate an array of interesting problems in social networks, topic learning, and knowledge discovery. In the first part of the talk, I will discuss the problem of network membership estimation. We propose a new spectral approach called Mixed-SCORE, and reveal a surprising simplex structure underlying the networks. We explain why Mixed- SCORE is the right approach and use it to investigate two networks constructed from the Phase I data. In the second part of the talk, I will report some Exploratory Data Analysis
(EDA) results including productivity, journal ranking, topic learning, citation patterns. This part of result is based on Phase II data.
23 November 2018
Thanksgiving - no colloquium
30 November 2018
Anastasios Matzavinos, Brown University
Title: Mesoscopic modeling of polymer transport in an array of entropic barriers
Abstract: In this talk, we discuss dissipative particle dynamics (DPD) simulations of biological polymers (e.g., DNA molecules) dispersed by a pressure-driven fluid flow across a periodic array of entropic barriers. We compare our simulations with nanofluidic experiments, which show polymers transitioning between various types of behaviors as pressure increases, and discuss physical insights afforded by the ability of the DPD method to model flows at the nanoscale. We also consider anomalous diffusion phenomena that emerge in both experiment and simulation, and illustrate similarities between this system and Brownian motion in a tilted periodic potential. Finally, we formulate and analyze a continuous-time Markov process modeling the motion of the polymer across the entropic barriers. Our main result is a functional central limit theorem for the position of the polymer with an explicit formula for the effective diffusion coefficient in terms of the parameters of the model. A law of large numbers for the asymptotic velocity and large deviation estimates are also obtained.