Invited Speakers

Invited Speakers (Confirmed)

Title and Abstract

Time (In Indian Standard Time: GMT+5.5 hours)

1. Prof. Ravi P. Agarwal, Texas A&M University - Kingsville, Texas, USA

Boundary Value Problems for Delay Differential Equations

8:30-9:15 AM, October 15

2. Prof. Gautam Biswas, Indian Institute of Technology Kanpur, India

Dynamics of the "Matryoshka" cavity generated due to impact of high-speed train of microdrops on a liquid pool

2:50-3:35 PM, October 15

3. Prof. John Butcher, The University of Auckland, Auckland, New Zealand

Runge-Kutta methods and B-series

10:10-10:55 AM, October 12

4. Prof. Jean-Luc Guermond, Texas A&M University, College Station, USA

Invariant domain preserving approximation of nonlinear conservation equations 9:20-10:05 AM, October 15

5. Prof. Weizhang Huang, University of Kansas, Kansas, USA

A well-balanced positivity-preserving quasi-Lagrange moving mesh DG method for the shallow water equations

9:20-10:05 AM, October 12

6. Prof. M. Lakshmanan, Bharathidasan University, Tiruchirappalli, India

Nondegenerate solitons and their collisions in the two component Manakov nonlinear partial differential equations

2:50-3:35 PM, October 13

7. Prof. Wenyuan Liao, University of Calgary, Calgary, Canada

Development and analysis of an unconditional stable method for Acoustic Wave Equations 9:20-10:05 AM, October 13

8. Prof. Chee Peng Lim, Deakin University, Melbourne, Victoria, Australia

Computational Data Modelling: Methods and Applications 10:10-10:55 AM, October 15

9. Prof. D.Q. Lu, Shanghai University, Shanghai, China

Mathematical models and analytical methods for the hydroelastic responses of a very large floating structure 2:00-2:45 PM, October 12

10. Prof. Yvon Maday, Universitè Pierre et Marie Curie, Paris, France

Epidemiological short-term Forecasting with Model Reduction of Parametric Compartmental Models. (Application to the first pandemic wave of COVID-19 in France.) 3:40-4:25 PM, October 14

11. Prof. Mike Meylan, The University of New Castle, Callaghan, New South Wales, Australia 

Lax-Phillips Scattering Theory for Simple Wave Scattering 10:10-10:55 AM, October 14

12. Prof. Ivan Yotov, Department of Mathematics, University of Pittsburgh, USA

Modeling of fluid-poroelastic structure interaction 8:30-9:15 AM, October 14

13. Prof. A.K. Nandakumaran, Indian Institute of Science, Bangalore, India

PDEs and Optimal Control Problems in Domains with Highly Oscillating boundaries: Asymptotic Analysis 3:40-4:25 PM, October 13

14. Prof. Marcin Paprzycki, Systems Research Institute of the Polish Academy of Sciences, Warshaw, Poland

Semantic Technologies in a Decision Support System 2:00-2:45 PM, October 15

15. Prof. Phoolan Prasad, Indian Institute of Science, Bangalore, India

Derivation of Ray Equations of a Polytropic Gas from Fermat's Principle 10:00-10:45 AM, October 13

16. Prof. B.V Rathish Kumar, Indian Institute of Technology Kanpur, India

Higher Order PDE Based Image Processing: Theory, Computation & Application  2:00-2:45 PM, October 14

17. Prof. Qin (Tim) Sheng, Baylor University, Waco, Texas, USA

Recent advances in Numerical methods for singular PDEs 9:20-10:05 AM, October 14

18. Prof. Chi - Wang Shu, Brown University, Providence, Rhode Island, USA

Stability of time discretizations for semi-discrete high order schemes for time-dependent PDEs 8:30-9:15 AM, October 12

19. Prof. Martin Stynes, Beijing Computational Science Research Center, Beijing, China

The numerical solution of time-fractional initial-boundary value problems 2:00-2:45 PM, October 13

20. Prof. Roger Temam, Indiana University, Bloomington, USA

  8:30-9:15 AM, October 13

21. Prof. Xin-She Yang, Middlesex University, London, United Kingdom.

Stability of Nature-Inspired Algorithms Using Dynamical System Theory

2:50-3:35 PM, October 14

22. Prof. G.D. Veerappa Gowda, Tata Institute of Fundamental Research - CAM, Bangalore, India

Godunov type solvers for Hyperbolic systems admitting δ−shocks 2:50-3:35 PM, October 12
23. Prof. Enrique Zuazua, Universidad Autònoma de Madrid, Madrid, Spain Turnpike control and deep learning 3:40-4:25 PM, October 12

 

 

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 high-speed 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 liquid-air 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 10-30 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 high-speed 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 multi-droplet impacts. The investigations were performed in an air-water 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. Pinch-off 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. 492-523, (2015).

[2]. N. B. Speirs, Z. Pan, J. Belden and T. T. Truscott, "The water entry of multi-droplet 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 high-speed train of microdrops on a liquid pool", Journal of Fluid Mechanics, 792, 850-868, (2016).

[4]. H. Deka, B. Ray, G. Biswas and A. Dalal, "Dynamics of tongue shaped cavity generated during the impact of high-speed microdrops," Physics of Fluids, Vol. 30, pp. 042103-1- 042103-14, (2018).

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Runge - Kutta methods and B-series

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. Jean-Luc Guermond, Texas A&M University, College Station, USA

Abstract: The objective of this talk is to present a fully-discrete approximation technique for the compressible Navier-Stokes equations. The method is implicit explicit, second-order accurate in time and space,and guaranteed to be invariant domain preserving. The restriction on the time-step 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 second-order accurate in time and space.

Of course there are countless papers in the literature describing techniques to approximate the time-dependent compressible Navier-Stokes 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 first-order 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 Navier-Stokes 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 sub-step is treated explicitly and the parabolic sub-step 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 sub-step is guaranteed to be invariant domain preserving. In addition, the scheme is conservative and fully-computable (e.g. the method is fully-discrete and there are no open-ended questions regarding the solvability of the sub-problems). One key ingredient of our method is that the parabolic sub-step is reformulated in terms of the velocity and the internal energy in a way that makes the method conservative, invariant domain preserving, and second-order 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 man-made 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 multiple-layer 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 small-amplitude 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 multiple-layer 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. Far-field 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 Plotnikov-Toland model for the elastic structure are employed. Semi-analytical approximation for the propagating characteristics of nonlinear hydroelastic waves is obtained in terms of homotopy analysis method. For the head-on 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 two-layer 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 well-balanced positivity-preserving quasi-Lagrange moving mesh DG method for the shallow water equations 

Prof. Weizhang Huang, Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, USA. E-mail: whuang@ku.edu 

JianxianQiu, School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian 361005, China. E-mail: jxqiu@xmu.edu.cn 

Min Zhang, School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China.

E-mail: minzhang2015@stu.xmu.edu.cn 

Abstract: In this talk we will present a high-order, well-balanced, positivity-preserving quasi-Lagrange moving mesh DG method for the numerical solution of the shallow water equations with non-flat bottom topography. The well-balance property is crucial to the ability of a scheme to simulate perturbation waves over the lake-at-rest steady state such as waves on a lake or tsunami waves in the deep ocean. The method combines a quasi-Lagrange moving mesh DG method, a hydrostatic reconstruction technique, and a change of unknown variables. We will discuss the strategies to use slope limiting, positivity-preservation limiting, and change of variables to ensure the well-balance and positivity-preserving properties. Compared to rezoning-type 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 two-dimensional examples are presented to demonstrate the well-balance property, positivity preservation, and high-order 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 well-known 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 non-degenerate vector solitons, which can undergo collisions without any intensity re-distribution 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 three-soliton solutions through Hi-rota 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 double-hump, table-top andsingle-hump structures. We explain the formation of double-hump 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 two-soliton 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, (Springer-Verlag 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 time-step constraint posed by Courant-Fridrichs-Lewy (CFL) number, the explicit finite difference methods become less efficient for time-domain 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 second-order ordinary differential equation. The BDF is then applied to solve the semi-discrete second-order ordinary differential system, which is the result of applying spatial discretization on the acoustic wave equation. In addition to the conventional second-order 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 second-order ordinary differential equation, 1D and 2D acoustic wave equations with constant and variable wave speeds. The new method is compact and fourth-order accurate in space, while the order of convergence in time can be improved to fourth-order 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 intelligence-based methods for data modelling will be described. The underlying algorithms comprising individual and hybrid intelligent data-based models, which include artificial neural networks, fuzzy systems, and evolutionary algorithms, will be explained. In addition, applications of such intelligent data-based models to different real-world problems will be demonstrated.

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Epidemiological short-term Forecasting with Model Reduction of Parametric Compartmental Models.
(Application to the first pandemic wave of COVID-19 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 two-week 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 COVID-19 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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Lax-Phillips Scattering Theory for Simple Wave Scattering

Prof. Mike Meylan, The University of New Castle, Callaghan, New South Wales, Australia

Abstract: Lax-Philips 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 non-trivial 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 fluid-poroelastic 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 Navier-Stokes 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 cross-sections 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 electro-magnetic 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 P0 to another point Pt 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 reaction-diffusion problems in multiple scientific applications. An outline of a theoretical background of exponential splitting approaches will then be introduced. We will continue on typical quenching-combustion 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 laser-materials interactions. Potentials of research collaborations will be explored.

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Stability of time discretizations for semi-discrete high order schemes for time-dependent PDEs

Prof. Chi-Wang Shu, Division of Applied Mathematics, Brown University, Providence, RI 02912, USA 

Abstract: In scientific and engineering computing, we encounter time-dependent partial differential equations (PDEs) frequently.  When designing high order schemes for solving these time-dependent PDEs, we often first develop semi-discrete 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 semi-discrete 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 implicit-explicit (IMEX) Runge-Kutta or multi-step time marching, which treats the more stiff term (e.g. diffusion term in a convection-diffusion 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 upper-bounded by a constant under suitable conditions, and the explicit Runge-Kutta methods, for which strong stability can be proved in many cases for semi-negative linear semi-discrete schemes.  Numerical examples will be given to demonstrate the performance of these schemes.

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The numerical solution of time-fractional initial-boundary 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 initial-value 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, initial-boundary value problems are considered, where the time derivative is a Caputo fractional derivative. (This is a fractional-derivative 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 Nature-Inspired Algorithms Using Dynamical System Theory

Prof. Xin-She Yang, Middlesex University, London, United Kingdom. 

Abstract: Nature-inspired 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 in-depth 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 non-classical δ−shocks are proposed, by extending the theory of discontinuous flux for non-linear 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, Friedrich-Alexander-Universität Erlangen-Nürnberg, 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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