Boundary Value Problems for Delay Differential Equations
Prof. Ravi P. Agarwal, Texas A&M University
 Kingsville, Texas, USA
We develop an upper and
lower solution method for second order boundary value
problems for nonlinear delay differential equations on an
infinite interval. Sufficient conditions are imposed on the
nonlinear term which guarantee the existence of a solution
between a pair of lower and upper solutions, and triple
solutions between two pairs of upper and lower solutions. An
extra feature of our existence theory is that the obtained
solutions may be unbounded. Two examples which show how
easily our existence theory can be applied in practice are
also illustrated.
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Dynamics of the
"Matryoshka" cavity generated
due to impact of highspeed train of microdrops on a liquid
pool
Prof.
Gautam Biswas, Indian Institute of Technology
Kanpur, India
Abstract:
When a drop of a liquid impacts on the liquidair interface of a liquid pool,
depending on the size and velocity of the drop, it may coalesce partially or
completely [1]. Based on the shape of the crater and its expansion and
contraction time, the final outcome can be coalescence, jet formation with or
without bubble entrapment and splashing. Speirs et al. [2] demonstrated
formation of long slender cavities due to multiple drop impact on a deep liquid
pool. Bouwhuis et al. [3] studied the same event for
microdroplets impacting with frequencies in the range of 1030 kHz.
Tongue shaped cavities are seen during the hydrophobic sphere impact, jet
impact, and impact of a train of microdrops on a deep liquid pool [4]. For the
impact of multiple microdrops, the mechanisms, which lead to deep cavity
formation and later bubble entrapment inside the liquid pool, are presented in
this work. A train of highspeed microdrops impacting on a liquid pool can
create a very deep and narrow cavity, leading to depths more than several
hundred times the size of the individual drops. Seemingly the deep cavity is
agglomeration of "matryoshka" cavities, named after the Russian nesting dolls.
We analyzed these nested cavities (matryoshka cavities) created by multidroplet
impacts. The investigations were performed in an airwater system at large
values of Froude numbers, thus having a negligible effect of gravity. Depending
on the train length, the capillary wave generating from each drop impact affects
the necking. The temporal variation of the neck radius reveals a power law
behavior. Pinchoff is observed when the penetration depth of the cavity is more
than three times the diameter of the cavity
References:
[1]. B. Ray, G. Biswas and A. Sharma,
"Regimes during liquid drop impact on a
liquid pool," Journal of Fluid Mechanics, Vol. 768, pp. 492523, (2015).
[2]. N. B. Speirs, Z. Pan, J. Belden
and T. T. Truscott, "The water entry of
multidroplet streams and jets", Journal of Fluid Mechanics, Vol. 844,
pp. 1084_1111, (2018).
[3]. W. Bouwhuis, X. Huang, C. U. Chan, P.E. Frommhold, C.D. Ohl, D. Lohse, D.,
J.H. Snoeuer, and D. van der Meer, 2016 "Impact of a highspeed train of
microdrops on a liquid pool", Journal of Fluid Mechanics, 792, 850868,
(2016).
[4]. H. Deka, B. Ray, G. Biswas and A. Dalal,
"Dynamics of tongue shaped cavity
generated during the impact of highspeed microdrops," Physics of Fluids,
Vol. 30, pp. 0421031 04210314, (2018).
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Runge  Kutta
methods and Bseries
Prof. John Butcher, The University of Auckland, Auckland, New Zealand
Abstract: In 1895 an important discovery was made [1].
It became possible to obtain second order Runge{Kutta
methods. A few years later Heun [2] and Kutta [3] raised the
order to 3 and 4 and ventually to 5 [4] and 6 [5]. A
Runge{Kutta method for a scalar initial value problem ...
(Read More)
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Invariant domain preserving approximation of
nonlinear conservation equations
Prof. JeanLuc Guermond, Texas A&M University, College Station, USA
Abstract: The objective of
this talk is to present a fullydiscrete approximation
technique for the compressible NavierStokes equations. The
method is implicit explicit, secondorder accurate in time
and space,and guaranteed to be invariant domain preserving.
The restriction on the timestep size is the standard
hyperbolic CFL condition. To the best of our knowledge, this
method is the first one that is guaranteed to be invariant
domain preserving under the standard hyperbolic CFL
condition and be secondorder accurate in time and space.
Of course there are countless papers in the literature
describing techniques to approximate the timedependent
compressible NavierStokes equations, but there are very few
papers establishing invariant domain properties. Among the
latest results in this direction we refer the reader to
Grapsas, Herbin, Kheriji, Latche (2016) where a firstorder
method using upwinding and staggered grid is developed (see
Eq.~(3.1) therein). The authors prove positivity of the
density and the internal energy (Lem.~4.4 therein).
Unconditional stability is obtained by solving a nonlinear
system involving the mass conservation equation and the
internal energy equation. One important aspect of this
method is that it is robust in the low Mach regime. A
similar technique is developed in Gallouet, Gastaldo, Herbin,
Latche (2008) for the compressible barotropic NavierStokes
equations (see \S3.6 therein). We also refer to Zhang (2017)
where a fully explicit dG scheme is proposed with positivity
on the internal energy enforced by limiting. The invariant
domain properties are proved there under the parabolic time
step restriction.
The key idea of the present talk is to build on Guermond,
Nazarov, Popov, Tomas (2019), Guermond, Popov, Tomas (2019)
and use an operator splitting technique to treat separately
the hyperbolic part and the parabolic part of the problem.
The hyperbolic substep is treated explicitly and the
parabolic substep is treated implicitly. This idea is not
new and we refer for instance to Demkowicz et al. (1990) for
an early attempt in this direction. The novelty of our
approach is that each substep is guaranteed to be invariant
domain preserving. In addition, the scheme is conservative
and fullycomputable (e.g. the method is fullydiscrete and
there are no openended questions regarding the solvability
of the subproblems). One key ingredient of our method is
that the parabolic substep is reformulated in terms of the
velocity and the internal energy in a way that makes the
method conservative, invariant domain preserving, and
secondorder accurate.
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Mathematical models and analytical methods
for the hydroelastic responses of a very large floating structure
Prof. D.Q. Lu,
Shanghai
University, Shanghai, China
Abstract: A vast natural ice
cover in the polar region and a manmade very large floating
structure (VLFS) in the offshore region are usually the
idealized as thin elastic plates floating on an inviscid
incompressible fluid. To consider the effects of density
stratification in the ocean, a simple but useful model,
namely a multiplelayer fluid, is often employed. For the
mathematical formulation, the Laplace equation is taken for
the governing equation, representing the continuity of the
mass. The dynamic condition on the fluid–plate interface
indicates the balance among the hydrodynamic pressure of the
fluid, the elastic and inertial forces of the plate, and
external moving loads, which forms a hydroelastic problem.
Under the assumptions of smallamplitude wave motion and
small deflection of plate, the fluid–plate model is
established within the linear potential theory. Dynamic
responses of the plate (namely the hydroelastic waves or
flexural–gravity waves), which are the key concerns of the
present study, occur as the structure is subjected to
incident ocean waves or an external downward load. For the
wave–plate interaction problems, the velocity potentials are
expressed by the eigenfunction expansions in the frequency
domain. We introduce some new inner products for the
multiplelayer fluid to obtain the expansion coefficients.
Thus the wave scattering and plate deflection are studied.
An object moving on or beneath the surface of VLFS can be
modeled as a concentrated load singularity, which
mathematically involves the Dirac delta function. Farfield
hydroelastic responses of the plate due to
translating/instantaneous singularities are analytically
investigated with the aid of integral transforms and the
asymptotic analysis.
To consider the nonlinear effects on the hydroelastic waves,
the convective term in the momentum equation for the fluid
motion and the PlotnikovToland model for the elastic
structure are employed. Semianalytical approximation for
the propagating characteristics of nonlinear hydroelastic
waves is obtained in terms of homotopy analysis method. For
the headon collision process of two hydroelastic solitary
waves, we utilize a singular perturbation method, namely the
Poincare–Lighthill–Kuo (PLK) method of strained coordinates,
to obtain the asymptotic solutions analytically. We mainly
examine the effects of important physical parameters,
including the density, the thickness and Young’s modulus of
the plate, the wave amplitude, larger density ratio or depth
ratio of the twolayer fluid, on hydroelastic dynamic
characteristics of flexible structures.
This research was sponsored by the
National Natural Science Foundation of China under Grant No.
11872239.
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A wellbalanced positivitypreserving
quasiLagrange moving mesh DG method for the shallow water
equations
Prof.
Weizhang Huang, Department of Mathematics, University of
Kansas, Lawrence, Kansas 66045, USA. Email:
whuang@ku.edu
JianxianQiu, School of Mathematical Sciences and Fujian
Provincial Key Laboratory of Mathematical Modeling and
HighPerformance Scientific Computing, Xiamen University,
Xiamen, Fujian 361005, China. Email:
jxqiu@xmu.edu.cn
Min Zhang, School of Mathematical Sciences, Xiamen
University, Xiamen, Fujian 361005, China.
Email:
minzhang2015@stu.xmu.edu.cn
Abstract:
In this talk we will present a highorder, wellbalanced,
positivitypreserving quasiLagrange moving mesh DG method
for the numerical solution of the shallow water equations
with nonflat bottom topography. The wellbalance property
is crucial to the ability of a scheme to simulate
perturbation waves over the lakeatrest steady state such
as waves on a lake or tsunami waves in the deep ocean. The
method combines a quasiLagrange moving mesh DG method, a
hydrostatic reconstruction technique, and a change of
unknown variables. We will discuss the strategies to use
slope limiting, positivitypreservation limiting, and change
of variables to ensure the wellbalance and positivitypreserving
properties. Compared to rezoningtype methods, the current
method treats mesh movement continuously in time and has the
advantages that it does not need to interpolate flow
variables from the old mesh to the new one and places no
constraint for the choice of a update scheme for the bottom
topography on the new mesh. A selection of one and
twodimensional examples are presented to demonstrate the
wellbalance property, positivity preservation, and
highorder accuracy of the method and its ability to adapt
the mesh according to features in the flow and bottom
topography.
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Nondegenerate solitons and their collisions
in the two component Manakov nonlinear partial differential
equations
Prof.
M. Lakshmanan, Department of Nonlinear Dynamics, School of
Physics, Bharathidasan University, Tiruchirappalli  620
024, India
Abstract:
Nonlinear Schrödinger equation is a wellknown soliton
possessing nonlinearpartial differential equation occurring
in many physical contexts. An importantvector generalization
of it is the Manakov system for two complex valued functions
and integrable by the inverse scattering transform method.
Recently, wehave shown that the Manakov equation can admit a
more general class of nondegenerate vector solitons, which
can undergo collisions without any intensity redistribution
in general among the modes, associated with distinct wave
numbers,besides the already known energy exchanging solitons
corresponding to identicalwave numbers. In my lecture, I
will discuss in detail the various special features ofthe
reported nondegenerate vector solitons. To bring out these
details, we derivethe exact forms of such vector one, two
and threesoliton solutions through Hirota bilinear method
and they are rewritten in more compact forms using
Gramdeterminants. The presence of distinct wave numbers
allows the nondegeneratefundamental soliton to admit various
profiles such as doublehump, tabletop andsinglehump
structures. We explain the formation of doublehump
structure inthe fundamental soliton when the relative
velocity of the two modes tends to zero.More critical
analysis shows that the nondegenerate fundamental solitons
can undergo shape preserving as well as shape altering
collisions under appropriate conditions. The shape changing
collision occurs between the modes of nondegeneratesolitons
when the parameters are fixed suitably. Then we observe the
coexistenceof degenerate and nondegenerate solitons when the
wave numbers are restrictedappropriately in the obtained
twosoliton solution.In such a situation we find
thedegenerate soliton induces shape changing behavior of
nondegenerate soliton during the collision process. By
performing suitable asymptotic analysis we analyse the
consequences that occur in each of the collision scenarios.
Finally we pointout that the previously known class of
energy exchanging vector bright solitons,with identical wave
numbers, turns out to be a special case of the newly
derivednondegenerate solitons.
References:
1. M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics:
Integrability, Chaos, andPatterns, (SpringerVerlag Berlin
Heidelberg) (2003).
2. S. V. Manakov, Sov. Phys. JETP38, 248 (1974).
3. R. Radhakrishnan, M. Lakshmanan and J. Hietarinta, Phys.
Rev. E56, 2213(1997).
4. T. Kanna and M. Lakshmanan, Phys. Rev. Lett.86, 5043
(2001).
5. S. Stalin, R. Ramakrishnan, M. Senthilvelan and M.
Lakshmanan, Phys. Rev. Lett.122043901 (2019).
6. S. Stalin, R. Ramakrishnan and M. Lakshmanan, Phys. Lett.
A384126201 (2019).
7. R. Ramakrishnan, S. Stalin and M. Lakshmanan, Submitted
for Publication inPhys. Rev. E.
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Development and analysis of an unconditional stable method
for Acoustic Wave Equations
Prof. Wenyuan
Liao, University of Calgary, Calgary, Canada
Abstract: In the field of
numerical simulation of seismic wave propagation, the
explicit finite difference scheme is a popular choice due to
its high efficiency and simple implementation. However,
because of the timestep constraint posed by CourantFridrichsLewy
(CFL) number, the explicit finite difference methods become
less efficient for timedomain acoustic wave equation,
mainly due to the stability limit, which requires very small
time step step. The situation is more severe when the wave
speed is larger. In this work we focused on the development
and analysis of highly accurate and unconditionally stable
numerical schemes for solving acoustic wave equations.
Firstly, we derived an unconditionally stable backward
difference formula (BDF) for solving secondorder ordinary
differential equation. The BDF is then applied to solve the
semidiscrete secondorder ordinary differential system,
which is the result of applying spatial discretization on
the acoustic wave equation. In addition to the conventional
secondorder finite difference
scheme, we also considered the higher order accuracy in
space, which is obtained by the utilization of Pad\'{e}
approximation of the convectional second order central
difference.
We tested the new unconditionally BDF on various models
through extensive numerical examples on the accuracy and
stability of the new method, such as secondorder ordinary
differential equation, 1D and 2D acoustic wave equations
with constant and variable wave speeds. The new method is
compact and fourthorder accurate in space, while the order
of convergence in time can be improved to fourthorder as
well. A rigorous stability analysis has been conducted to
show that the new scheme is unconditionally stable.
Moreover, the new scheme is very efficient, thus, can find
wide applications in various Geophysical inversion areas,
such as the full waveform inversion problems
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Computational Data Modelling: Methods and Applications
Prof. Chee Peng Lim,
Deakin University, Melbourne, Victoria, Australia
Abstract: Computational
intelligence is a broad discipline that encompasses a
variety of methodologies inspired by human and/or animal
intelligence. In this talk, the use of computational
intelligencebased methods for data modelling will be
described. The underlying algorithms comprising individual
and hybrid intelligent databased models, which include
artificial neural networks, fuzzy systems, and evolutionary
algorithms, will be explained. In addition, applications of
such intelligent databased models to different realworld
problems will be demonstrated.
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Epidemiological shortterm Forecasting with Model Reduction
of Parametric Compartmental Models.
(Application to the first pandemic wave of COVID19 in
France.)
Prof. Yvon
Maday, Universitè Pierre et Marie Curie, Paris,
France, Athmane Bakhta, Thomas Boiveau, Olga Mula
Abstract :
In this talk, I will present a
forecasting method for predicting epidemiological health
series on a twoweek horizon at the regional and
interregional level. The approach is based on model order
reduction of parametric compartmental models, and is
designed to accommodate small amount of sanitary data.
The efficiency of the method is examined in the case of the
prediction of the number of hospitalized infected and
removed people during the first pandemic wave of COVID19 in
France, which has taken place approximately between February
and May 2020. Numerical results illustrate the promising
potential of the approach.
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LaxPhillips Scattering Theory for Simple Wave Scattering
Prof. Mike Meylan, The University of New Castle, Callaghan, New South Wales,
Australia
Abstract:
LaxPhilips scattering theory is a method to solve for
scattering as an expansion over the singularities of the
analytic extension of the scattering problem to complex
frequencies. I will show how a complete theory can be
developed in the case of simple scattering problems. Even
for the simplest case, it requires a nontrivial generalised
eigenfunction transformation to project into the space of
analytic functions on the real line. The scattering operator
in this space is simply the complex exponential. I will
illustrate how this theory can be used to find a numerical
solution, and I will demonstrate the method by applying it
to the vibration of ice shelves.
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Modeling of fluidporoelastic structure interaction
Prof. Ivan Yotov, Department of
Mathematics, University of Pittsburgh, USA
Abstract: We study mathematical models and their finite element approximations for solving
the coupled problem arising in the interaction between a free fluid and a fluid
in a poroelastic material. Applications of interest include flows in fractured
poroelastic media and arterial flows. The free fluid flow is governed by the
NavierStokes or Stokes/Brinkman equations, while the poroelastic material is
modeled using the Biot system of poroelasticity. We present several approaches
to impose the continuity of normal flux, including an interior penalty method
and a Lagrange multiplier method. A dimensionally reduced fracture model based
on averaging the equations over the crosssections will also be presented.
Stability, accuracy, and robustness of the methods will be discussed.
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PDEs and Optimal Control Problems in Domains with Highly
Oscillating boundaries: Asymptotic Analysis
Prof. A.K. Nandakumaran, Indian Institute of Science, Bangalore, India
Abstract: In this talk,
we discuss the asymptotic analysis (homogenization) of
various optimal control problems defined in domains whose
boundary is rapidly (highly) oscillating. Such complex
domains appears in many real life applications like heat
radiators, flows in channels with rough boundaries,
propagation of electromagnetic waves in regions having
rough interface, absorption and diffusion in biological
structures, acoustic vibrations in medium with narrow
channels etc. We present the work which we are carrying out
in my group for the last 10 years. We introduce the so
called unfolding operators which we have developed for the
problems under study through which we characterize the
optimal controls. Finally, we do a homogenization process
and obtain the limit control problem.
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Semantic Technologies in a Decision Support System
Prof. Marcin Paprzycki,
Systems Research Institute of the
Polish Academy of Sciences, Warshaw, Poland
The aim of our work was to design a decision support system based
on
ontological representation of domain(s) and semantic technologies.
Specifically, we considered the case when Grid / Cloud users describe
their requirements regarding a "resource" as a semantic expression
(based on domain capturing ontology), while the instances
of (the same) ontology represent available resources.
The aim of the presentation is to discuss in what way semantic
technologies can and in what way they cannot be used in a decision
support system.
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Derivation of Ray Equations of a Polytropic Gas from
Fermat's Principle
Prof. Phoolan
Prasad,
Indian Institute of Science, Bangalore, India
Abstract: According to
Fermat's principle, a ray going from one point P_{0}
to another point P_{t} in space chooses a path such
that the time of transit is stationary. Given initial
position of a wavefront ...(Read
more)
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Recent advances in Numerical methods for singular PDEs
Prof. Tim Sheng, Center for Astrophysics, Space Physics and
Engineering Research (CASPER), Baylor University. Waco,
Texas, USA
Abstract: In this talk, we
will start with some interesting singular reactiondiffusion
problems in multiple scientific applications. An outline of
a theoretical background of exponential splitting approaches
will then be introduced. We will continue on typical
quenchingcombustion equations via decomposed finite
difference approaches. Straightforward numerical analysis on
the monotonicity, convergence and linear stability will be
discussed. The latest exponential evolving grid development
inspired by moving grid strategies will be proposed. Several
experimental results will be given. The general idea of
adaptative splitting can be extended for solving other
multiphysics equations in particular those in biophysics,
oil pipeline decay preventions and lasermaterials
interactions. Potentials of research collaborations will be
explored.
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Stability of time discretizations for
semidiscrete high order schemes for timedependent PDEs
Prof.
ChiWang Shu, Division of Applied Mathematics, Brown
University, Providence, RI 02912, USA
Abstract:
In scientific and engineering computing, we encounter
timedependent partial differential equations (PDEs)
frequently. When designing high order schemes for solving
these timedependent PDEs, we often first develop
semidiscrete schemes paying attention only to spatial
discretizations and leaving time $t$ continuous. It is then
important to have a high order time discretization to main
the stability properties of the semidiscrete schemes. In
this talk we discuss several classes of high order time
discretization, including the strong stability preserving (SSP)
time discretization, which preserves strong stability from a
stable spatial discretization with Euler forward, the
implicitexplicit (IMEX) RungeKutta or multistep time
marching, which treats the more stiff term (e.g. diffusion
term in a convectiondiffusion equation) implicitly and the
less stiff term (e.g. the convection term in such an
equation) explicitly, for which strong stability can be
proved under the condition that the time step is
upperbounded by a constant under suitable conditions, and
the explicit RungeKutta methods, for which strong stability
can be proved in many cases for seminegative linear
semidiscrete schemes. Numerical examples will be given to
demonstrate the performance of these schemes.
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The numerical solution of timefractional initialboundary
value problems
Prof. Martin Stynes, Beijing Computational Science Research
Center, Beijing, China
Abstract:
An introduction to fractional derivatives and some of their
properties will be presented. The regularity of solutions to
Caputo fractional initialvalue problems is then discussed;
it is shown that typical solutions have a weak singularity
at the initial time t=0. This singularity has to be taken
into account when designing and analysing numerical methods
for the solution of such problems. To address this
difficulty we use graded meshes, which cluster mesh points
near t=0, and answer the question: how exactly should the
mesh grading be chosen? Finally, initialboundary value
problems are considered, where the time derivative is a
Caputo fractional derivative. (This is a
fractionalderivative generalisation of the classical
parabolic heat equation.) Once again a weak singularity
appears at t=0, and the mesh in the time coordinate should
be graded to compute satisfactory numerical solutions.
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Stability of NatureInspired Algorithms Using
Dynamical System Theory
Prof.
XinShe Yang, Middlesex University, London, United Kingdom.
Abstract:
Natureinspired algorithms such as the particle swarm
optimization, bat algorithm and firefly algorithm have been
used to solve optimization problems quite efficiently.
However, it lacks some indepth mathematical analysis of
these algorithms. This talk summarizes the latest
developments, and provide some analysis of stability of
these algorithms using dynamical system theory. Some
challenges and open problems will also be highlighted.
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Godunov type solvers for Hyperbolic systems admitting
δ−shocks
Prof. G.D. Veerappa Gowda, Tata Institute of Fundamental Research  CAM,
Bangalore, India
Abstract: Discontinuous flux based
numerical schemes for the class of hyperbolic systems
admitting nonclassical δ−shocks are proposed, by extending
the theory of discontinuous flux for nonlinear conservation
laws. It is shown that the numerical scheme converges to the
solution which preserves the physical properties such as
positive density and bounded velocity. The numerical results
are compared with the existing literature and the schemes
are shown to capture the solution efficiently. This is a
joint work with Aekta Aggarwal and Ganesh Vaidya.
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Turnpike control and deep learning
Prof.
Enrique Zuazua, FriedrichAlexanderUniversität
ErlangenNürnberg, Germany, Deusto Foundation, Bilbao,
Spain, Universidad Autönoma de Madrid, Spain
Abstract:
The tunrpike principle asserts that in long time horizons
optimal control strategies are nearly of a steady state
nature.In this lecture we shall survey on some recent
results on this topic and present some its consequences
on deep supervised learning.
This lecture will be based in particular in recent joint
work with C: Esteve, B. Geshkovski and D. Pighin.
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Higher Order PDE Based Image Processing: Theory, Computation & Application
Prof. B.V.Rathish Kumar,
Department of Mathematics & Statistics, IIT Kanpur, India
Image processing is one of the interesting topics of research in mathematics and
engineering. In last few decades partial differential equation (PDE) based image
processing has attracted the researchers because of the sound theoretical and
numerical background of PDEs. The PDE models give the insight into the physical
phenomena and help to come up with new models and effective numerical methods to
solve it. Most of the PDE models of initial days are of lower order but they
have some drawbacks such as blocky effect in denoising, failure with large gap
in inpainting. Higher order PDE models have shown promise to overcome these
defects. So the idea is to look for appropriate higher order PDE models to deal
with the problems that occurred in the field of image processing. In this talk,
we will focus on three different types of image processing problems namely image
denoising, inpainting and segmentation via higher order PDE models and will
share with you the developments which we have made on theoretical and
computational fronts towards better PDE based image analysis
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