Events
Department of Mathematics and Statistics
Texas Tech University
The present work aims to propose a computational technique for reconstructing the spatially complex-valued potential in the time-dependent Schrödinger equation from two time measurements and Cauchy boundary data. This reconstruction process also aids in determining the intensity function. Our inverse solver is derived based on some special transformations and the so-called Fourier--Klibanov basis, leading to a block system of coupled elliptic PDEs with Cauchy data. The block-coupled elliptic system is approximated by minimizing a weighted Tikhonov-like cost functional in a partially discrete setting. In this scenario, we rely on a general 1D Carleman estimate to prove the Fréchet differentiability, strong convexity of the functional, which further leads to the $L^2$-type error estimate of the gradient descent method. Some numerical results are provided to verify the performance of the proposed inversion. This is a joint work with Dr. Vo Anh Khoa, Texas Tech University.
This Analysis seminar may be attended Monday the 27th at 4:00 PM CDT (UTC-5) via this Zoom link.
Meeting ID: 947 7432 8481
Passcode: 745829
 | Wednesday
Oct. 29
|
| Algebra and Number Theory
No Seminar
Please attend the departmental colloquium
|
When a sheet of paper is crumpled, it spontaneously develops a network of creases. Despite the apparent disorder of this process, statistical properties of crumpled sheets exhibit striking reproducibility. Recent experiments have shown that when a sheet is repeatedly crumpled, the total crease length grows logarithmically [1]. This talk will offer insight into this surprising result by developing a correspondence between crumpling and fragmentation processes. We show how crumpling can be viewed as fragmenting the sheet into flat facets that are outlined by the creases, and we use this model to reproduce the characteristic logarithmic scaling of total crease length, thereby supplying a missing physical basis for the observed phenomenon [2].
This study was made possible by large-scale data analysis of crease networks from crumpling experiments. We will describe recent work to use the same data with machine learning methods to probe the physical rules governing crumpling. We will look at how augmenting experimental data with synthetically generated data can improve predictive power and provide physical insight [3,4,5].
[1] O. Gottesman et al., Commun. Phys. 1, 70 (2018).
[2] J. Andrejevic et al., Nat. Commun. 12, 1470 (2021).
[3] J. Hoffmann et al., Sci. Advances 5, eaau6792 (2019).
[4] J. Andrejevic and C. H. Rycroft, J. Comput. Phys. 471, 111607 (2022).
[5] M. Leembruggen et al., Phys. Rev. E 108, 015003 (2023).
When: 4:00 pm (Lubbock's local time is GMT -5)
Where: room Math 011 (Math Basement)
ZOOM details:
- Choice #1: use this
Direct Link that embeds meeting with ID and passcode.
- Choice #2: join meeting using this link or your standard Zoom login method, then you must input the ID and Passcode by hand:
* Meeting ID: 915 2866 2672
* Passcode: applied
First, a mathematical model for the spatiotemporal distribution of a migratory bird species is derived and analyzed. The birds have specific sites for breeding and winter feeding, and usually several stopover sites along the migration route, and therefore a patch model is the natural choice. However, we also model the journeys of the birds along the flyways, and this is achieved using a continuous space model of reaction-advection type. In this way proper account is taken of flight times and in-flight mortalities which may vary from sector to sector, and this information is featured in the ordinary differential equations for the populations on the patches through the values of the time delays and the model coefficients. The seasonality of the phenomenon is accommodated by having periodic migration and birth rates.
Second, another approach to modeling bird migration is proposed, in which there is a region where birds do not move but spend time breeding. Birds leave this breeding region and enter a migration flyway which is effectively a one-way corridor starting and ending at the breeding location. Mathematically, the flyway is a curve parametrized by arc-length. Flight speed depends on position along the flyway, to take account of factors such as wind and the pausing of birds at various locations for wintering or stopovers. Per-capita mortality along the flyway is both position and age-dependent, allowing for increased risks at stopover locations due to predation, and increased risks to immature birds. We also model indirect transmission, via contact with viruses, of avian influenza in migratory and nonmigratory birds, taking into account age structure. Sufficient conditions are obtained for the local stability of the disease-free equilibrium (for a species without migration) and for the disease-free periodic solution (for a migratory species). The birds have specific sites for breeding and winter feeding, and usually several stopover sites along the migration route, and therefore a patch model is the natural choice. However, we also model the journeys of the birds along the flyways, and this is achieved using a continuous space model of reaction-advection type. In this way proper account is taken of flight times and in-flight mortalities which may vary from sector to sector, and this information is featured in the ordinary differential equations for the populations on the patches through the values of the time delays and the model coefficients. The seasonality of the phenomenon is accommodated by having periodic migration and birth rates.
The Biomath seminar may be attended virtually Friday at 11:00 AM CDT (UTC-5) via this Zoom link.
Meeting ID: 938 8653 3169
Passcode: 883472
abstract noon CDT (UTC-5)
Zoom link available from Dr. Brent Lindquist upon request.
