BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Sabre//Sabre VObject 4.5.8//EN CALSCALE:GREGORIAN BEGIN:VTIMEZONE TZID:Europe/Zurich X-LIC-LOCATION:Europe/Zurich TZURL:http://tzurl.org/zoneinfo/Europe/Zurich BEGIN:DAYLIGHT TZOFFSETFROM:+0100 TZOFFSETTO:+0200 TZNAME:CEST DTSTART:19810329T020000 RRULE:FREQ=YEARLY;BYMONTH=3;BYDAY=-1SU END:DAYLIGHT BEGIN:STANDARD TZOFFSETFROM:+0200 TZOFFSETTO:+0100 TZNAME:CET DTSTART:19961027T030000 RRULE:FREQ=YEARLY;BYMONTH=10;BYDAY=-1SU END:STANDARD END:VTIMEZONE BEGIN:VEVENT UID:news1840@dmi.unibas.ch DTSTAMP;TZID=Europe/Zurich:20250313T150350 DTSTART;TZID=Europe/Zurich:20250320T121500 SUMMARY:Bernoullis Tafelrunde: Remo Von Rickenbach (Universität Basel) DESCRIPTION:Many problems from physics or engineering  result in partial d ifferential equations.   In both cases\, the unknown function \\(u\\) can \,  in most practically relevant applications\, only be approximated nume rically. Therefore\, it is essential to use efficient algorithms to approx imate \\(u\\) at as low costs as possible.\\r\\nTo build efficient algorit hms\, one first needs to understand how well a function of a given regular ity can be approximated. For example\, by using finite elements of grid si ze \\(h\\)  on the unit interval \\(I\\)\,  it is well established that the approximation error decays as \\begin{align*}     \\inf_{v_h \\in V _h} \\|u - v_h\\|_{L^2(I)}     \\leq C h^s \\|u\\|_{H^{s}(I)}\,      \\quad 0 \\leq s \\leq d\, \\end{align*} where \\(d \\in \\mathbb{N}\\) is the polynomial order  of the trial functions involved.\\r\\nHowever\, wh at can we say about the approximation order if  \\(u\\) is only piecewise regular\,  but admits a singularity and is therefore not in \\(H^{s}(I)\ \)? For this\,  adaptive schemes\, which rely on the concept of \\emph{no nlinear approximation}\, need to be studied.\\r\\nIn this talk\, some basi c concepts and examples of approximation theory will be introduced and we will characterise the approximation spaces with respect to some common bas is systems. In particular\, we will see that the approximation spaces with respect to nonlinear approximation by wavelet bases contain Besov spaces which are strictly larger than the corresponding\, classical finite elemen t approximation spaces\,  showing that adaptive schemes strictly outperfo rm classical schemes in the case of limited regularity.\\r\\nabstract [t3: //file?uid=3857] X-ALT-DESC:

Many problems from physics or engineering \;
result in partial differential equations.  \;
In both cases\, the unkn own function \\(u\\) can\, \;
in most practically relevant appli cations\,
only be approximated numerically.
Therefore\, it is essential to use efficient algorithms to approximate
\\(u\\) at as l ow costs as possible.

\n

To build efficient algorithms\, one first ne eds to understand how well a function
of a given regularity can be a pproximated.
For example\, by using finite elements of grid size \\( h\\) \;
on the unit interval \\(I\\)\, \;
it is well e stablished that the approximation error decays as
\\begin{align*}
 \; \;  \;\\inf_{v_h \\in V_h} \\|u - v_h\\|_{L^2(I)}
 \; \;  \;\\leq C h^s \\|u\\|_{H^{s}(I)}\,
 \;&nbs p\;  \;\\quad 0 \\leq s \\leq d\,
\\end{align*}
where \\(d \\in \\mathbb{N}\\) is the polynomial order \;
of the trial fun ctions involved.

\n

However\, what can we say about the approximation order if \;
\\(u\\) is only piecewise regular\, \;
bu t admits a singularity and is therefore not in \\(H^{s}(I)\\)?
For t his\, \;
adaptive schemes\, which rely on the concept of \\emph{ nonlinear approximation}\,
need to be studied.

\n

In this talk\ , some basic concepts and examples of approximation theory will be
i ntroduced and we will characterise the approximation spaces with respect t o
some common basis systems.
In particular\, we will see that the approximation spaces with respect to
nonlinear approximation by wavelet bases contain Besov spaces which are strictly
larger than th e corresponding\, classical finite element approximation spaces\, \; showing that adaptive schemes strictly outperform classical schemes i n the case
of limited regularity.

\n

abstract

DTEND;TZID=Europe/Zurich:20250320T130000 END:VEVENT END:VCALENDAR