Tiitle: What the second Strang lemma and the Aubin-Nitsche trick should be
Abstract: The second Strang lemma gives an error estimate for linear
problems written in variational formulation, such as elliptic equations.
It covers both conforming and non-conforming methods, it is widely spread
in the finite element community, and usually considered as the starting
point of any convergence analysis.
For all its potency, it has a number of limitations which prevents its
direct application to other popular methods, such as dG methods, Virtual
Element Methods, Hybrid High Order schemes, Mimetic Methods, etc. Ad-hoc
adaptations can be found for some of these methods, but no general `second
Strang lemma' has been developed so far in a framework that covers all
these schemes, and others, at once.
In this talk, I will present a `third Strang lemma' that is applicable to
any discretisation of linear variational problems. The main idea to
develop a framework that goes beyond FEM and covers schemes written in a
fully discrete form is to estimate, in a discrete energy norm, the
difference between the solution to the scheme and some interpolant of the
continuous solution. I will show that this third Strang lemma is much
simpler to prove, and use, than the second Strang lemma. It also enables
us to define a clear notion of consistency, including for schemes for
which such a notion was not clearly defined so far, and for which the Lax
principle `stability + consistency implies convergences' holds.
I will also extend the analysis to the Aubin-Nitsche trick, presenting a
generalisation of this trick that covers fully discrete schemes and
provides improved error estimates in a weaker norm than the discrete
energy norm. We will see that the terms to estimate when applying this
Aubin-Nitsche trick are extremely similar to those appearing when applying
the third Strang lemma; work done in the latter case can therefore be
re-invested when looking for improved estimates in a weaker norm.
I will conclude by briefly presenting applications of the third Strang
lemma and the abstract Aubin-Nitsche trick to discontinuous Galerkin and
Finite Volume methods.
Time:
4:00pm-5:00pm
Location:
Ramanujan Hall, Department of Mathematics
Description:
Statistics Seminar.
Speaker: Prof. Palash Ghosh
Centre for Quantitative Medicine
DUKE-NUS Medical School
National University of Singapore
Date: Monday, 15/10/18.
Time: 4:00--5:00 pm.
Venue: Ramanujan Hall
Title: Dynamic Generalized Odds-Ratio (dGOR): A novel approach to assess
Dynamic Treatment Regimes (DTR) with An Ordinal Outcome.
Abstract: See Attachment.
Time:
4:00pm
Location:
Ramanujan Hall, Department of Mathematics
Description:
PDE & Numerical Analysis Seminar
Date and Time: 16th October 2018, at 4PM
Venue: Ramanujan Hall, Mathematics Department.
Speaker: Prof. Ravi Prakash, University of Concepcion, Chile.
Title: "Homogenization: Oscillating Boundary Domains"
Abstract of the seminar:
Homogenization is a branch of science where we try to understand microscopic structures via a macroscopic medium. Hence, it has applications in various branches of science and engineering. This study is basically developed from material science in the creation of composite materials though the contemporary applications are much far and wide. It is a process of understanding the microscopic behavior of an in-homogeneous medium via a homogenized medium. Mathematically, it is a kind of asymptotic
analysis. We are interested in the asymptotic behavior of elliptic boundary value problems posed in domains with highly oscillating boundary. In fact, we will consider different types of unfolding operators to study many types of oscillating boundary domains with various model problems posed in them. We will also see some interesting optimal boundary control problem posed in such domains. In total, the presentation will start from the asymptotic behavior of Laplacian in a simple rectangular oscillating boundary domains to the future possibilities of the shapes of oscillations in the boundary keeping in mind the mathematical issues arise in topology optimization.
Time:
5:00pm
Location:
Ramanujan Hall, Department of Mathematics
Description:
CACAAG Seminar
Speaker: Neeraj Kumar.
Time: 5pm, Tuesday, 16 October 2018.
Venue: Ramanujan Hall.
Title: Wilf's conjecture on numerical semigroups
Abstract: The aim of the talk is to give a brief survey on the Wilf's
conjecture, and to present a commutative algebra formulation of it. We
will verify Wilf's conjecture in some cases.
A numerical semigroup $S$ is a subset of the nonnegative integers $N$ that
is closed under addition, contains 0, and has finite complement in $N$.
The Frobenius number $F$ of numerical semigroup $S$ is the largest integer
not in $S$. Let $d$ be the minimal number of generators of $S$ and $n$ be
the number of representable integers in the interval $[0, F]$. Wilf's
conjecture states that $F +1 \leq n d$.
Time:
2:30pm-3:30pm
Location:
Ramanujan Hall, Department of Mathematics
Description:
Speaker: Somnath Jha
Title: A duality for Selmer groups
Time: 14:30 - 15:30, Wednesday, October 17, 2018.
Venue: Ramanujan Hall, Department of Mathematics
Abstract: Selmer group is an important object of study in number theory.
We will discuss a twisting result in the setting of so called
"non-commutative" Iwasawa theory. We will further use this to deduce a
duality result for certain Selmer groups. (This talk is based on joint
works with T. Ochiai, G. Zabradi and S. Shekhar.
Time:
4:00pm
Location:
Ramanujan Hall, Department of Mathematics
Description:
Department Colloquium.
Speaker: Harsha Hutridurga.
Time: 4pm, Wednesday, October 17, 2018.
Venue: Ramanujan Hall.
Title: An overview of the theory of Hypocoercivity.
Abstract: In this talk, we attempt to give a brief introduction to the
theory of Hypocoercivity which has become an indispensable tool in the
study of relaxation to equilibrium states for mathematical models arising
in statistical physics. The essential ideas behind this theory will be
motivated via simple examples. The role of certain functional inequalities
while deriving explicit rates of convergence will be made precise during
this talk. This talk concludes by addressing a certain degenerate kinetic
Fokker-Planck equation. Incidentally, the study of the trend to
equilibrium for this degenerate model finds link to the acclaimed
Geometric Condition from the theory of control for wave propagation.
Time:
3:30pm-4:30pm
Location:
Ramanujan Hall, Department of Mathematics
Description:
Speaker: Parvez Sarwar
Title: Algebraic K-theory and homology stability
Time: 15:30 - 16:30, Thursday, October 18
Venue: Ramanujan Hall
Abstract: We shall begin with the homotopy invariance property of
K-theory. After reviewing monoids and monoid algebras, we present some
results which are monoid version of the homotopy invariance property in
K-theory. This answers a question of Gubeladze. Next, we will discuss the
monoid version of Weibel's vanishing conjecture and some results in this
direction. Finally, we will talk about the homology stability for groups.
Here we present a result which improves homology stability for symplectic
groups. If the time permits, some application of the homology stability
will be given to the hermitian K-theory.
Time:
4:00pm
Location:
Room No. 215, Department of Mathematics
Description:
PDE seminar on Control and homogenization:
Title: Control of wave equation.
Speaker: Debanjana Mitra,
Time: Thursday, 18-10-18, 4p.m.-5p.m.
Venue: Room 215, Department of Mathematics, IIT Bombay.
Abstract: In this talk, we will mainly discuss on the control of wave
equation. At the first part of the talk, we will give an overview of the
control of wave equation and mention some important results in this
direction. Then in the second part, the control of wave equation using
Hilbert uniqueness method will be discussed.