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Event(s) on April 2021
- Wednesday, 21st April, 2021
Title: Multiscale Analysis: Understanding And Improving Ensemble Methods For Inversion Speaker: Prof Andrew Stuart, Caltech, USA Time/Place: 10:00 - 11:00
Zoom, (Meeting ID: 945 6745 3135)
Abstract: I will describe the use of ensemble based particle methods to solve inverse problems, including ensemble and unscented Kalman methods. I will show how multiscale analysis can be used to understand the performance of these methods; and I will show how multiscale methods can be used to improve upon these methods.
- Wednesday, 21st April, 2021
Title: Learning Better Discretizations for Singular Variational Problems Speaker: Prof Antonin Chambolle, CMAP, Ecole Polytechnique, France Time/Place: 16:00 - 17:00
Zoom, (Meeting ID: 942 1567 1183)
Abstract: The total variation has been promoted for a long time as a regularizer for inverse problems in imaging, thanks to its ability to recover sharp edges and smooth regions. Yet, there are millions of different ways to represent it in a discrete form, yielding to quite different behaviours (sometimes desirable or not, depending on the application). In this talk, we will describe a few variants, their advantages and drawbacks, and how one can numerically "learn" improved discretizations with better precision, sharper results and good isotropy properties. This is based on joint works with Thomas Pock, TU. Graz.
- Thursday, 22nd April, 2021
Title: Mathematical Study on Several Inverse Problems and Invisibility Cloaking with Applications Speaker: Ms TSUI Wing Yan, Department of Mathematics, Hong Kong Baptist University, Hong Kong Time/Place: 15:00 - 17:00
Zoom, Meeting ID: 942 8042 5739 Password: 820242
Abstract: The study on inverse problems has played a pivotal role to various disciplines of science, technology, engineering and mathematics, including x-ray, ultrasound, magnetoencephalography, geophysical exploration, radar and criminal investigations. In the view of their novel promising applications, we investigate the potentials for several inverse scattering approaches and applications. In our first topic, we are concerned a new approach for generating two-dimensional or three-dimensional geometric shapes by inputting characteristic parameters of specific geometric shapes. Our study is combined with the machine learning approach and the inverse scattering techniques on the theory of wave propagation associated with the Helmholtz equation. We first introduce the important notations of the shape space and then the shape generators via inverse source scattering associated with Helmholtz equation for the generation of the geometric body shapes. Then, we develop a machine learning scheme for the generation of geometric body shape by using the setup of the shape generators and the shape space. That is, the input-output pairs of the training data set are formulated by the characteristic set and the shape generators. The predicted output, the new shape generator is computed by the training dataset and learning model. We finally reconstruct the new shape generator to geometric body shape by a stable multiple-frequency Fourier method and numerically simulate some examples. In our second topic, we are concerned with the three-dimensional elastic scattering coefficients (ESC) and the enhancement of the elastic near cloaking. We establish the ESC of arbitrary three-dimensional objects and some of their properties using the elements of the elastic layer potential theory and multipolar expansions. We then construct the enhanced near elastic cloaking at a fixed frequency by using the ESC-vanishing-structures and transformation-elastodyamics. We also study some numerical examples on three-dimensional ESC. In our third topic, we are concerned with the inverse problem of identifying magnetic anomalies with varying parameters beneath the Earth using geomagnetic monitoring. We study the information about the anomalies as well as their variations by the observations of the change in Earth's magnetic field, so called the secular variation. We rigorously establish the unique recovery results for this magnetic anomaly detection problem. We show that one can uniquely recover the locations, the variation parameters including the growth or decaying rates as well as their material parameters of the anomalies.
- Wednesday, 28th April, 2021
Title: Top Signs to Consider Nonlocal Modeling Speaker: Prof Qiang Du, Columbia University, USA Time/Place: 10:00 - 11:00
Zoom, (Meeting ID: 935 1964 4563)
Abstract: In recent years, nonlocality has been given increasing attention in the modeling of various complex systems, especially in the presence of anomalies and singularities. The effective modeling and simulation of nonlocal interactions have led to new challenges. In particular, it is interesting to ask what are the signs for one to consider nonlocal modeling as a potentially helpful approach in application. We will present examples to address this question. We will further discuss some related mathematical and computational issues, as well as recent works.