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UID:15@cds.iisc.ac.in
DTSTART;TZID=Asia/Kolkata:20231102T140000
DTEND;TZID=Asia/Kolkata:20231102T150000
DTSTAMP:20231205T090948Z
URL:https://cds.iisc.ac.in/events/m-tech-research-thesis-defense-cds-02-no
 vember-2023%e2%80%b3semi-analytical-solution-for-eigenvalue-problems-of-la
 ttice-models-with-boundary-conditions/
SUMMARY:M.Tech Research: Thesis Defense: CDS: 02\, November 2023″Semi-ana
 lytical solution for eigenvalue problems of lattice models with boundary c
 onditions”.
DESCRIPTION:02 Nov @ 2:00 PM -- 3:00 PM\n\nDEPARTMENT OF COMPUTATIONAL AND 
 DATA SCIENCES\nM.Tech Research Thesis Defense\n\n\n\nSpeaker : Ms. Athira 
 Gopal\n\nS.R. Number : 06-18-00-10-12-19-2-17782\n\nTitle : “Semi-analyt
 ical solution for eigenvalue problems of lattice models with boundary cond
 itions”\n\nResearch Supervisor: Prof. Murugesan Venkatapathi\n\nDate &am
 p\; Time : November 02\, 2023 (Thursday)\, 02:00 PM\n\nVenue : Room No. 10
 2 (CDS Seminar Hall)\n\n\n\nAbstract\nClosed-form relations for limiting e
 igenvalues of an infinite k-periodic spatial lattice in any number of dime
 nsions ‘d’\, and its semi-analytical extensions for any given size ‘
 n’ of the lattice with free-free boundary conditions\, are known. These 
 are based on the eigenvalues of tridiagonal k-Toeplitz matrices (represent
 ing chains and d=1)\, and their tensor products or sums. These semi-analyt
 ical methods for eigenvalues incur drastically lower computing costs than 
 the direct numerical methods i.e. O(n) vs. O(n^2) for the latter\, and fur
 ther they are more accurate for sufficiently large lattices approaching th
 e limiting case (n &gt\; 100). This advantage in computing cost\, accuracy
 \, and numerical stability results as the original eigenvalue problem of n
 k in size is reduced to n eigenvalue problems each k in size\, further mak
 ing this approach very amenable to parallel computation when required. In 
 this work\, their errors in eigenvalues are compared with the errors of th
 e direct numerical methods using special examples with high condition numb
 ers. Secondly\, in the absence of such analytical methods\, one also resor
 ts to periodic boundary conditions to limit the size of the numerical mode
 l representing a very large system. The convergence of numerical models wi
 th periodic boundary conditions to the limiting eigenvalues is highlighted
 \, to emphasize the utility of the closed-form solution for the limiting e
 igenvalues. Thirdly\, the fixed-fixed boundary conditions on a finite chai
 n and their counterpart for periodic spatial lattices in higher dimensions
  (d&gt\;1) are addressed using perturbations to tridiagonal k-Toeplitz mat
 rices on their main diagonal. Extensions of the semi-analytical methods fo
 r these cases by applying numerical methods to update only the few perturb
 ed eigenvalues is proposed. An efficient extension for evaluating the eige
 nvectors in the case of real eigenvalues as required in most physical syst
 ems\, is also presented.\n\n\n\nALL ARE WELCOME
CATEGORIES:Events,Thesis Defense
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