TBA
Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.
On June 10, 2000, the Millennium Bridge in London opened to the public. As people crossed the bridge, it wobbled. The sway of the bridge was large enough that prompted many on the bridge to hold on to the rails. Three days later, the bridge closed. It reopened only after modifications to prevent the wobbling were made, eighteen months later. We develop and study a model motivated by this event
Please Note: This is a part of IEEE Signal Processing Society Lecture Series, organized by Dr. Alessio Medda (alessiomedda@ieee.org). PLEASE RSVP to https://events.vtools.ieee.org/m/222947
The set of diffeomorphisms of the unit interval, or “warping functions,” plays an important role in many in functional data analysis applications. Most prominently, the problem of registering, or aligning, pairs of functions depends on solving for an element of the diffeomorphism group that, when applied to one function, optimally aligns it to the other.
The registration problem is posed as the unconstrained minimization of a cost function over the infinite dimensional diffeomorphism function space. We make use of its well-known Riemannian geometry to implement an efficient, limited memory Riemannian BFGS optimization scheme. We compare performance and results to the benchmark algorithm, Dynamic Programming, on several functional datasets. Additionally, we apply our methodology to the problem of non-parametric density estimation and compare to the benchmark performance of MATLAB’s built-in kernel density estimator ‘ksdensity’.
The past few years have seen the advent of big data, which brings unprecedented convenience to our daily life. Meanwhile, from a computational point of view, a central question arises amid the exploding amount of data: how to tame big data in an economic and efficient way. In the context of matrix computations, the question consists in the ability to handle large dense matrices. In this talk, I will first introduce data-sparse hierarchical representations for dense matrices. Then I will present recent development of a new data-driven algorithm called SMASH to operate dense matrices efficiently in the most general setting. The new method not only outperforms existing algorithms but also works in high dimensions. Various experiments will be provided to justify the advantages of the new method.
The most popular model for Image Denoising is without any doubt the ROF (for Rudin-OsherFatemi) model. However, since the ROF approach has some drawbacks (the stair-case effect being one of them) practitioners have been looking for alternatives. One of them is the Elastica model, relying on the minimization in an appropriate functional space of the energy functional $J$ defined by
$$ J(v)=\varepsilon \int_{\Omega} \left[ a+b\left| \nabla\cdot \frac{\nabla v}{|\nabla v|}\right|^2 \right]|\nabla v| d\mathbf{x} + \frac{1}{2}\int_{\Omega} |f-v|^2d\mathbf{x} $$
where in $J(v)$: (i) $\Omega$ is typically a rectangular region of $R^2$ and $d\mathbf{x}=dx_1dx_2$. (ii) $\varepsilon, a$ and $b$ are positive parameters. (iii) function $f$ represents the image one intends to denoise.
Minimizing functional $J$ is a non-smooth, non-convex bi-harmonic problem from Calculus of Variations. Its numerical solution is a relatively complicated issue. However, one can achieve this task rather easily by combining operator-splitting and finite element approximations. The main goal of this lecture is to describe such a methodology and to present the results of numerical experiments which validate it.
It's know that when discretizing stochastic ordinary equation with non-globally Lipschitz coefficient, the traditional numerical method, like
Euler method, may be divergent and not converge in strong or weak sense. For stochastic partial different equation with non-globally Lipschitz
coefficient, there exists fewer result on the strong and weak convergence results of numerical methods. In this talk, we will discuss several numerical schemes approximating stochastic Schrodinger Equation. Under certain condition, we show that the exponential integrability preserving schemes are strongly and weakly convergent with positive orders.
Kinetic equations play an important role in multiscale modeling hierarchy. It serves as a basic building block that connects the microscopic particle models and macroscopic continuum models. Numerically approximating kinetic equations presents several difficulties: 1) high dimensionality (the equation is in phase space); 2) nonlinearity and stiffness of the collision/interaction terms; 3) positivity of the solution (the unknown is a probability density function); 4) consistency to the limiting fluid models; etc. I will start with a brief overview of the kinetic equations including the Boltzmann equation and the Fokker-Planck equation, and then discuss in particular our recent effort of constructing efficient and robust numerical methods for these equations, overcoming some of the aforementioned difficulties. This is joint work with Ruiwen Shu (University of Maryland).
Consider a stochastic process (such as a stochastic differential equation) arising from applications. In practice, we are interested in many things like the invariant probability measure, the sensitivity of the invariant probability measure, and the speed of convergence to the invariant probability measure. Existing rigorous estimates of these problems usually cannot provide enough details. In this talk I will introduce a few data-driven computational methods that solve these problems for a class of stochastic dynamical systems, including but not limited to stochastic differential equations. All these methods are driven by the simulation data, and are less affected by the curse-of-dimensionality than traditional grid-based methods. I will demonstrate a few high (up to 100) dimensional examples in my talk.
The talk is concerned with low multilinear rank approximations to antisymmetric tensors, that is, multivariate arrays for which the entries change sign when permuting pairs of indices. Such tensors play a major role in quantum chemistry. We show which ranks can be attained by an antisymmetric tensor and discuss the adaption of existing approximation algorithms to preserve antisymmetry, most notably a Jacobi-type algorithm. Particular attention is paid to the special case when choosing the rank equal to the order of the tensor. It is shown that this case can be addressed with an unstructured rank-1 approximation. This allows for the straightforward application of the higher-order power method, for which we discuss effective initialization strategies. This is a joint work with Daniel Kressner (EPFL).
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