BEGIN:VCALENDAR VERSION:2.0 PRODID:-//jEvents 2.0 for Joomla//EN CALSCALE:GREGORIAN METHOD:PUBLISH BEGIN:VTIMEZONE TZID:America/New_York BEGIN:STANDARD DTSTART:20241103T010000 RDATE:20250309T030000 TZOFFSETFROM:-0400 TZOFFSETTO:-0500 TZNAME:America/New_York EST END:STANDARD BEGIN:STANDARD DTSTART:20251102T010000 RDATE:20260308T030000 TZOFFSETFROM:-0400 TZOFFSETTO:-0500 TZNAME:America/New_York EST END:STANDARD BEGIN:STANDARD DTSTART:20261101T010000 RDATE:20270314T030000 TZOFFSETFROM:-0400 TZOFFSETTO:-0500 TZNAME:America/New_York EST END:STANDARD BEGIN:STANDARD DTSTART:20271107T010000 RDATE:20280312T030000 TZOFFSETFROM:-0400 TZOFFSETTO:-0500 TZNAME:America/New_York EST END:STANDARD BEGIN:STANDARD DTSTART:20281105T010000 RDATE:20290311T030000 TZOFFSETFROM:-0400 TZOFFSETTO:-0500 TZNAME:America/New_York EST END:STANDARD BEGIN:DAYLIGHT DTSTART:20241020T110000 RDATE:20241103T010000 TZOFFSETFROM:-0500 TZOFFSETTO:-0400 TZNAME:America/New_York EDT END:DAYLIGHT BEGIN:DAYLIGHT DTSTART:20250309T030000 RDATE:20251102T010000 TZOFFSETFROM:-0500 TZOFFSETTO:-0400 TZNAME:America/New_York EDT END:DAYLIGHT BEGIN:DAYLIGHT DTSTART:20260308T030000 RDATE:20261101T010000 TZOFFSETFROM:-0500 TZOFFSETTO:-0400 TZNAME:America/New_York EDT END:DAYLIGHT BEGIN:DAYLIGHT DTSTART:20270314T030000 RDATE:20271107T010000 TZOFFSETFROM:-0500 TZOFFSETTO:-0400 TZNAME:America/New_York EDT END:DAYLIGHT BEGIN:DAYLIGHT DTSTART:20280312T030000 RDATE:20281105T010000 TZOFFSETFROM:-0500 TZOFFSETTO:-0400 TZNAME:America/New_York EDT END:DAYLIGHT END:VTIMEZONE BEGIN:VEVENT UID:334dc2d278d5a16b0accaf31c50ff3b4 CATEGORIES:Applied and Computational Math Seminar CREATED:20251009T235806 SUMMARY:Randomized Multiscale Methods for Heterogeneous Nonlinear Partial Differential Equations LOCATION:Hill 525 DESCRIPTION:To construct localizable multiscale methods for nonlinear partial different ial equations we consider a transfer operator that maps arbitrary admissibl e boundary data on the boundary of an oversampling domain to the respective (local) solution on the target subdomain; here the boundary of the latter must have a distance greater than zero from the boundary of the oversamplin g domain. Then, we try to approximate the set of all local solutions on the target subdomain. Interpreting the boundary data as some input parameter, we can view this set of local solutions as a set of solutions depending on a parameter. This motivates using methods from model order reduction such a s the proper orthogonal decomposition (POD) or the Greedy algorithm to appr oximate this set. However, both the POD and the Greedy algorithm rely on a training set of finite cardinality that is chosen such that every point in the admissible parameter set is close to a point in the training set. There fore, both algorithms suffer from the curse of dimensionality. We thus empl oy randomization and consider the parameter (here: boundary data) as a rand om variable with values in a Hilbert space. By choosing a suitable distribu tion we can then exploit the concentration of measure phenomenon, which is also sometimes called the "blessing of dimensionality" to break the curse.  In detail, we will present a randomized greedy algorithm that provides with high probability a certification for the whole parameter set rather than o nly for the parameters in the training set. Moreover, we will present a ran domized POD and a corresponding error analysis that shows that for exponent ially decaying eigenvalues of the randomized POD which uses the exact corre lation operator (integral in the expectation) the approximation error betwe en any solution corresponding to a parameter in the admissible parameter se t and the approximation with the POD that uses a Monte-Carlo approximation converges exponentially as well. X-ALT-DESC;FMTTYPE=text/html:
To construct localizable multiscale methods for nonlinear partial diff erential equations we consider a transfer operator that maps arbitrary admi ssible boundary data on the boundary of an oversampling domain to the respe ctive (local) solution on the target subdomain; here the boundary of the la tter must have a distance greater than zero from the boundary of the oversa mpling domain. Then, we try to approximate the set of all local solutions o n the target subdomain. Interpreting the boundary data as some input parame ter, we can view this set of local solutions as a set of solutions dependin g on a parameter. This motivates using methods from model order reduction s uch as the proper orthogonal decomposition (POD) or the Greedy algorithm to approximate this set. However, both the POD and the Greedy algorithm rely on a training set of finite cardinality that is chosen such that every poin t in the admissible parameter set is close to a point in the training set. Therefore, both algorithms suffer from the curse of dimensionality. We thus employ randomization and consider the parameter (here: boundary data) as a random variable with values in a Hilbert space. By choosing a suitable dis tribution we can then exploit the concentration of measure phenomenon, whic h is also sometimes called the "blessing of dimensionality" to break the cu rse.
 
In detail, we will present a randomized gre edy algorithm that provides with high probability a certification for the w hole parameter set rather than only for the parameters in the training set. Moreover, we will present a randomized POD and a corresponding error analy sis that shows that for exponentially decaying eigenvalues of the randomize d POD which uses the exact correlation operator (integral in the expectatio n) the approximation error between any solution corresponding to a paramete r in the admissible parameter set and the approximation with the POD that u ses a Monte-Carlo approximation converges exponentially as well.
CONTACT:Kathrin Smetana (Stevens Institute of Technology) DTSTAMP:20260827T115847 DTSTART;TZID=America/New_York:20251021T110000 DTEND;TZID=America/New_York:20251021T120000 SEQUENCE:0 TRANSP:OPAQUE END:VEVENT END:VCALENDAR