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UID:6ba51518921c076783460437868a0ac6
CATEGORIES:Applied and Computational Math Seminar
CREATED:20230125T004659
SUMMARY:Rigorous topological dynamics for Gaussian processes and Brownian motion
LOCATION:Hill 705
DESCRIPTION:We consider the problem of understanding a dynamical system via finite samp
 les of that system.  The most straightforward approach to addressing this p
 roblem is to use the data to generate a model and then analyze the dynamics
  of that model using standard techniques.  However, because long term dynam
 ics may qualitatively change under arbitrarily small perturbations of a sys
 tem it is unclear how reliable the conclusions that we arrive at in this ma
 nner will be; that is, it is difficult to quantify the probability that the
  predictions we make are correct.  In recent work, Batko et al. address thi
 s problem by combining Gaussian processes with multivalued dynamics.  This 
 talk will discuss this general framework as well as considering the special
  case of a Weiner process that is conditioned to pass through a finite set 
 of points and the dynamics generated by iterating a sample path from this p
 rocess.  In both the general and special cases, topological techniques (Con
 ley theory) are used to characterize the global dynamics and deduce the exi
 stence, structure and approximate location of invariant sets.  Most importa
 ntly, these techniques determine the probability (or confidence) that this 
 characterization is correct.  
X-ALT-DESC;FMTTYPE=text/html:<div style="border: 0px; font-style: normal; font-weight: 400; font-size: 1
 6px; line-height: inherit; font-family: Calibri, Arial, Helvetica, sans-ser
 if; margin: 0px; padding: 0px; vertical-align: baseline; color: #000000; le
 tter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text
 -transform: none; white-space: normal; widows: 2; word-spacing: 0px;"><span
  style="border: 0px; font-style: inherit; font-variant: inherit; font-weigh
 t: inherit; font-size: 12pt; line-height: inherit; font-family: Calibri, Ar
 ial, Helvetica, sans-serif; margin: 0px; padding: 0px; vertical-align: base
 line; color: black; background-color: white;">We consider the problem of un
 derstanding a dynamical system via finite samples of that system.&nbsp; The
  most straightforward approach to addressing this problem is to use the dat
 a to generate a model and then analyze the dynamics of that model using sta
 ndard techniques.&nbsp; However, because long term dynamics may qualitative
 ly change under arbitrarily small perturbations of a system it is unclear h
 ow reliable the conclusions that we arrive at in this manner will be; that 
 is, it is difficult to quantify the probability that the predictions we mak
 e are correct.&nbsp;&nbsp;<br aria-hidden="true"></span></div><div style="b
 order: 0px; font-style: normal; font-weight: 400; font-size: 16px; line-hei
 ght: inherit; font-family: Calibri, Arial, Helvetica, sans-serif; margin: 0
 px; padding: 0px; vertical-align: baseline; color: #000000; letter-spacing:
  normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: n
 one; white-space: normal; widows: 2; word-spacing: 0px;"><span style="borde
 r: 0px; font-style: inherit; font-variant: inherit; font-weight: inherit; f
 ont-size: 12pt; line-height: inherit; font-family: Calibri, Arial, Helvetic
 a, sans-serif; margin: 0px; padding: 0px; vertical-align: baseline; color: 
 black; background-color: white;"></span></div><div style="border: 0px; font
 -style: normal; font-weight: 400; font-size: 16px; line-height: inherit; fo
 nt-family: Calibri, Arial, Helvetica, sans-serif; margin: 0px; padding: 0px
 ; vertical-align: baseline; color: #000000; letter-spacing: normal; orphans
 : 2; text-align: start; text-indent: 0px; text-transform: none; white-space
 : normal; widows: 2; word-spacing: 0px;"><span style="border: 0px; font-sty
 le: inherit; font-variant: inherit; font-weight: inherit; font-size: 12pt; 
 line-height: inherit; font-family: Calibri, Arial, Helvetica, sans-serif; m
 argin: 0px; padding: 0px; vertical-align: baseline; color: black; backgroun
 d-color: white;">In recent work, Batko et al. address this problem by combi
 ning Gaussian processes with multivalued dynamics.&nbsp; This talk will dis
 cuss this general framework as well as considering the special case of a We
 iner process that is conditioned to pass through a finite set of points and
  the dynamics generated by iterating a sample path from this process.&nbsp;
  In both the general and special cases, topological techniques (Conley theo
 ry) are used to characterize the global dynamics and deduce the existence, 
 structure and approximate location of invariant sets. &nbsp;Most importantl
 y, these techniques determine the probability (or confidence) that this cha
 racterization is correct. &nbsp;</span></div>
CONTACT:Cameron Thieme
DTSTAMP:20260828T185345
DTSTART;TZID=America/New_York:20230131T110000
DTEND;TZID=America/New_York:20230131T120000
SEQUENCE:0
TRANSP:OPAQUE
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