Applied Math Seminar

Prediction intervals for random forests

04/16/2018 - 4:15pm
04/16/2018 - 5:15pm
Speaker: 
Jo Hardin (Pomona College)
Abstract: 

Although random forests are commonly used for regression, our understanding
of the prediction error associated with random forest predictions of individual re-
sponses is relatively limited. We introduce a novel measure of this error and evaluate
its properties, comparing it with the out-of-bag mean of squared residuals estimator
that, to our knowledge, is the only measure of random forest prediction error that
has been introduced in the literature thus far. We show that our proposed estimator
provides an individualized estimate of the error associated with a particular random
forest prediction, while the out-of-bag mean of squared residuals estimator provides
a more general estimate of the random forest's prediction error as a whole. Through
simulations on benchmark and simulated datasets, we also demonstrate that both
estimators of prediction error may form the bases for valid random forest predic-
tion intervals. Empirically, these prediction intervals performed as well as quantile
regression forest prediction intervals.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Elastoplasticity Simulation with the Material Point Method

04/23/2018 - 4:15pm
04/23/2018 - 5:15pm
Speaker: 
Joseph M. Teran (UCLA)
Abstract: 

Hyperelastic constitutive models describe a wide range of materials. Examples include biomechanical soft tissues like muscle, tendon, skin etc. Elastoplastic materials consisting of a hyperelastic constitutive model combined with a notion of stress constraint (or feasible stress region) describe an even wider range of materials. A very interesting class of these models arise from frictional contact considerations. I will discuss some recent results and examples in computer graphics and virtual surgery applications. Examples include simulation of granular materials like snow in Walt Disney's ``Frozen" as well as frictional contact between thin elastic membranes and shells for virtual clothing simulation. I will also discuss practical simulation of these materials with some recent algorithmic modifications to the Particle-In-Cell (PIC) technique, the Material Point Method (MPM).

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Solitons for a nonlinear Schr\"odinger equation with a slowly varying potential

04/09/2018 - 4:15pm
04/09/2018 - 5:15pm
Speaker: 
Ivan B Ventura (Cal Poly Pomona)
Abstract: 

We study the dynamics of solitary waves for a nonlinear Schr\"odinger equation, with an $L^2$-subcritical power nonlinearity. We show that with an initial condition $\eps \le \sqrt h$ away in $H^1$ from a soliton that, up to time $\sim |\log h|/h$, the solution evolves according to the equations from the effective Hamiltonian up to errors of size $\eps+h^2$. We achieve this result using the methods of Holmer-Zworski and the spectral results of Weinstein.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

A Central Scheme for Two-layer Shallow-water Flows along Channels with Arbitrary Geometry

05/07/2018 - 4:15pm
05/07/2018 - 5:15pm
Speaker: 
Jorge Balbas (California State University, Northridge)
Abstract: 

We present a new high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional two-layer shallow-water flows along channels with irregular cross sections and bottom topography. The scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and it enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, {\it i.e.}, the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. Along with a detailed description of the scheme and proofs of these two properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Time-delayed feedback at the edge of stability: Expert stick balancing

04/02/2018 - 4:15pm
04/02/2018 - 5:15pm
Speaker: 
John Milton (Keck Science Department)
Abstract: 

Stabilizing unstable states is a challenging task for control engineers and, at the same time, it is an exciting problem for computational neuroscientists. The mathematics involve investigating the properties of delay, or functional, differential equations. The benchmark paradigm is the stabilization of the upright position of an inverted pendulum. This talk reviews 14 years of student research at the Claremont Colleges aimed at determining the nature of the control mechanisms for human expert stick balancing. For seated expert stick balancers the time delay is 0.23s, the shortest stick that can be balanced for 240s is 0.32m and there is a sensory dead zone of 1-3 degrees for the estimation of the vertical displacement angle in the sagittal plane. These observations motivate a switching-type, pendulum-cart model for balance control which utilizes an internal model to compensate for the time delay by predicting the sensory consequences of the stick's movements. Numerical simulations using the semi-discretization method suggest that the feedback gains are tuned near the edge of stability. For these choices of the feedback gains the cost function which takes into account the position of the fingertip and the corrective forces is minimized. Surprisingly this model also explains why for the most expert, the stick always falls!

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Mathematical models of targeted cancer treatments

04/30/2018 - 4:15pm
04/30/2018 - 5:15pm
Speaker: 
Dominik Wodarz (UC, Irvine)
Abstract: 

The talk will discuss the use of mathematical models for understanding targeted cancer therapies. One area of focus is the treatment of chronic lymphocytic leukemia with tyrosine kinase inhibitors. I will explore how mathematical approaches have helped elucidate the mechanism of action of the targeted drug ibrutinib, and will discuss how evolutionary models, based on patient-specific parameters, can make individualized predictions about treatment outcomes. Another focus of the talk is the use of oncolytic viruses to kill cancer cells and drive cancers into remission. These are viruses that specifically infect cancer cells and spread throughout tumors. I will discuss mathematical models applied to experimental data that analyze virus spread in a spatially structured setting, concentrating on the interactions of the virus with innate immune mechanisms that determine the outcome of virus spread.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Modeling and Analysis of Thin Viscous Liquid Films in Spherical Geometry

03/26/2018 - 4:15pm
03/26/2018 - 5:15pm
Speaker: 
Di Kang (CGU)
Abstract: 

This talk studies the dynamics of a thin viscous liquid film coating the inner or outer surface of a sphere in the presence of gravity, surface tension and Marangoni effects. We also allow the sphere to rotate around its vertical axis. The surface tension coefficient can be considered as a constant, or a function of temperature or surfactant concentration. An asymptotic model describing the evolution of the film thickness is derived based on the lubrication approximation.

When the surface tension coefficient is a constant, the model includes the centrifugal and gravity forces and the stabilizing effect of surface tension. This thesis shows that the steady states are of three different types: uniformly positive film thickness, or the states with one or two dry zones on the sphere, depending on the relative strength of the centrifugal force to that of gravity. The transient dynamics in approaching those states are also described. This thesis also provides a constructive proof for the existence of non-negative weak solutions in a weighted Sobolev space.

When the surface tension coefficient is a non-constant function, an additional term representing the Marangoni effect is added to the equation. This thesis studies the cases when the surface tension gradient is due to an externally imposed temperature field or the presence of surfactant molecules. In the former case, we consider two different heating regimes with axial or radial thermal gradients and discuss the resulting dynamics. In the latter case, this thesis derives and studies a model for the coating flow inside the alveolar compartment of the lungs, taking into account the effect of pulmonary surfactant and its production and degradation. We derive a degenerate system of two coupled parabolic partial differential equations that describe the time evolution of the thickness of the coating film together with that of the surfactant concentration at the liquid-air interface. This thesis presents numerical simulations of the dynamics using parameter values consistent with experimental measurements.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Two Parameter Eigenvalue Problems

03/05/2018 - 4:15pm
03/05/2018 - 5:15pm
Speaker: 
Stephen B. Robinson (Wake Forest University)
Abstract: 

Consider a small variation of a standard first semester linear algebra problem. Find the {\em eigenpairs} $(\lambda,\mu)$ such that the problem

\begin{equation} \label{e.matrixproblem}

Ax \; = \; \lambda\, Bx \, + \, \mu\, x

\end{equation}

has a nontrivial vector solution $x\in\mathbb{R}^N$, where $A$ is a symmetric positive definite matrix and $B$ a symmetric matrix. The eigenpairs are on {\em eigencurves} in the plane.

When $A, B$ are, for example, the $3\times 3$ matrices given by

\begin{equation}\label{e.matrixvalues}

A\; = \; \begin{bmatrix}

2 & -1 & 0 \\

-1 & 2 & -1 \\

0 & -1 & 2

\end{bmatrix}

\qquad \text{and}\qquad

B\; = \; \begin{bmatrix}

2 & -1 & 0 \\

-1 & 2 & -1 \\

0 & -1 & 0

\end{bmatrix}

\end{equation}

the eigencurves are easily determined and are plotted in Figure \ref{matrix}.

 

\begin{figure}[h] %%% GENERALIZED MATRIX FIGURE

\begin{center}

\includegraphics[width=110mm, height=75mm]{matrix}

\caption{Eigencurves for the matrix eigenproblem \eqref{e.matrixproblem}-\eqref{e.matrixvalues}.}

\label{matrix}

\end{center}

\end{figure}

 

The ideas that arise in this finite dimensional example provide good motivation for a more complicated version of the eigenpair problem.

\[

\begin{aligned}

-\Delta u(x) \; = & \; \mu\, m_0(x)\, u(x) \quad\text{for }x\in \Omega \\

\frac{\partial u}{\partial\nu}(x) \; + \; c(x)\,u(x)\; = & \; \lambda\, b_0(x)\, u(x)\quad \text{for }x\in \partial\Omega,

\end{aligned}

\]

where $c, b_0, m_0$ are given functions in appropriate $L^p$-spaces on a smooth bounded region $\Omega$ in $\mathbb{R}^N$, and $\lambda, \mu$ are real eigenparameters.

Here, $m_0$ is assumed to be strictly positive, $b_0$ may be sign-changing, and $\nu$ denotes the outward normal vector.

The weak formulation of this problem leads to an analysis of abstract eigencurve problems associated with triples $(a, b, m)$ of continuous symmetric bilinear forms on a real Hilbert space $V$.

 

In this talk I will decribe how the eigenpairs form {\em variational eigencurves} with nice properties. In particular, the curves satisfy some convexity properties that are easy to describe. For example, the first eigencurve is convex, i.e. any straight line intersects the curve at most twice, and the second eigencurve is a little less convex, i.e. any straight line interesects the curve at most four times. In general, any line intersects the $n$th eigencurve at most $2n$ times.


 

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Organizational meeting

02/19/2018 - 4:15pm
02/19/2018 - 5:15pm
Speaker: 
TBA
Abstract: 

 TBA

Where: 
Emmy Noether Rm Millikan 1021 Pomona College

Cardiac arrhythnmias. Using mathematics to help prevent stroke and sudden cardiac death.

02/12/2018 - 4:15pm
02/12/2018 - 5:15pm
Speaker: 
Leon Glass (McGill University)
Abstract: 

In the normal human heart, a specialized region of the heart
called the sinus node sets the rhythm of the entire heart. However, in
some circumstances the normal sinus rhythm is disrupted and abnormal
cardiac arrhythmias arise. This talk will give a quick introduction to
some rhythms that are particularly important in medicine and
interesting in mathematics. One rhythm, called atrial fibrillation, is
associated with an irregular rhythm. This rhythm is generally not
fatal, but leads to an increased risk for stroke. Other rhythms, such
as ventricular tachycardia and ventricular fibrillation are life
threatening or fatal. In this talk, directed towards a general
audience, I will give a brief introduction to reading
electrocardiograms and then describe some of the mathematical
approaches that are being used to diagnose, predict and control these
abnormal rhythms.

Where: 
Emmy Noether Rm Millikan 1021 Pomona College
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