Fall 2016 - wiki.math.ntnu.no

Speaker: Karl Kristian Brustad (NTNU)
Title: Superposition Principles in the non-linear \(p\)-Laplace Equation.

Abstract: A main feature of linear PDE's is that solutions may be superimposed. That is, a sum of, say, supersolutions is still a supersolution.

The prime example is the translates of the fundamental solution to the Laplace Equation. They are harmonic away from the poles, but superharmonic when
considered as functions on the whole Euclidean space – a Dirac Delta is produced. This is why a convolution solves the Poisson Problem.

However, when the PDE is non-linear, there is no reason to believe that any kind of superposition principle for solutions should hold.
It was therefore a great surprise when Crandall & Zhang in 2003 discovered that every sum of translates of the fundamental solution to the \(p\)-Laplace Equation
is again a \(p\)-superharmonic function.

In this talk I shall present a new and simple proof of this fact. I shall categorize the superposition principles in the \(p\)-Laplace Equation,
and generalize the results of Crandall & Zhang. The expedient tool is a newly discovered auxiliary operator.

Time/place: Friday 25th of November at 13:15–14:00 in room 734
Speaker: Aingeru Fernández (University of Bordeaux) Title: The Hardy Uncertainty Principle and Schrödinger evolutions.
Abstract:In Mathematics, there are different properties that relate the behavior of a function to the behavior of its Fourier transform. In this talk, we will focus on Uncertainty Principles, which state that a function and its Fourier transform cannot decay simultaneously too fast at infinity. Among the different known uncertainty principles, the one which will be of interest was given by Hardy in 1933, with a proof based on complex analysis arguments. However, in the last years, Escauriaza, Kenig, Ponce and Vega developed a proof of this principle, in terms of solutions to Schrödinger equations, by using real analysis arguments. The aim of the talk is to review this theory and adapt it to other settings. On the one hand, we will discuss discrete versions of the principle, assuming that our space variable is not in Rn, but it is a point of the lattice Zn. On the other hand, if time permits, we will also consider Stochastic equations, where we add to the Schrödinger equation a probabilistic term related to the Brownian motion.

Time/place: Friday 25th of November at 13:15–14:00 in room 734

Speaker: Aingeru Fernández (University of Bordeaux)
Title: The Hardy Uncertainty Principle and Schrödinger evolutions.

Abstract:In Mathematics, there are different properties that relate the behavior of a function to the behavior of its Fourier transform. In this talk, we will focus on Uncertainty Principles, which state that a function and its Fourier transform cannot decay simultaneously too fast at infinity. Among the different known uncertainty principles, the one which will be of interest was given by Hardy in 1933, with a proof based on complex analysis arguments. However, in the last years, Escauriaza, Kenig, Ponce and Vega developed a proof of this principle, in terms of solutions to Schrödinger equations, by using real analysis arguments. The aim of the talk is to review this theory and adapt it to other settings. On the one hand, we will discuss discrete versions of the principle, assuming that our space variable is not in Rn, but it is a point of the lattice Zn. On the other hand, if time permits, we will also consider Stochastic equations, where we add to the Schrödinger equation a probabilistic term related to the Brownian motion.

Time/place: Thursday 3rd of November at 13:15–14:00 in room 734
Speaker: Kundan Kumar (UiB) Title: Iterative methods for coupled flow and geomechanics problems in porous media
Abstract: Coupling of geomechanics and flow in a poroelastic porous media has several energy and environmental applications including subsidence events and ground water remediation. The geomechanical effects account for the influence of deformations in the porous media caused due to the pore pressure whereas the changes in the pore structure due to mechanical stresses affect the flow field. Single phase quasi-static Biot model is typically used to model these coupled flow and deformation processes. We report here some of the developments in suitable iterative schemes for such models and their extensions. Our work has two components: 1. Developing suitable iterative schemes for the extensions of the Biot model to include more physics such as fractures and non-linearities, 2. Developing multirate schemes by exploiting the different time scales of mechanics and flow solve by taking coarser time step for mechanics and smaller time steps for flow. The iterative multirate schemes combine the advantages of both implicit and explicit approaches. They are efficient, allow larger time steps, are robust, and the decoupling allows us to solve the linear systems efficiently. We rigorously analyse the convergence and stability properties of some of the iterative and explicit multirate schemes. This work is done in collaboration with Tameem Almani (Austin), Mary F. Wheeler (Austin), Vivette Girault (Paris), Florin Radu (Bergen), and Jan Nordbotten (Bergen).

Time/place: Thursday 3rd of November at 13:15–14:00 in room 734

Speaker: Kundan Kumar (UiB)
Title: Iterative methods for coupled flow and geomechanics problems in porous media

Abstract: Coupling of geomechanics and flow in a poroelastic porous media has several energy and environmental applications including subsidence events and ground water remediation. The geomechanical effects account for the influence of deformations in the porous media caused due to the pore pressure whereas the changes in the pore structure due to mechanical stresses affect the flow field. Single phase quasi-static Biot model is typically used to model these coupled flow and deformation processes.
We report here some of the developments in suitable iterative schemes for such models and their extensions. Our work has two components: 1. Developing suitable iterative schemes for the extensions of the Biot model to include more physics such as fractures and non-linearities, 2. Developing multirate schemes by exploiting the different time scales of mechanics and flow solve by taking coarser time step for mechanics and smaller time steps for flow. The iterative multirate schemes combine the advantages of both implicit and explicit approaches. They are efficient, allow larger time steps, are robust, and the decoupling allows us to solve the linear systems efficiently. We rigorously analyse the convergence and stability properties of some of the iterative and explicit multirate schemes.
This work is done in collaboration with Tameem Almani (Austin), Mary F. Wheeler (Austin), Vivette Girault (Paris), Florin Radu (Bergen), and Jan Nordbotten (Bergen).

Time/place: Tuesday 25 October at 13:15–14:00 in room 656
Speaker: Timo Klock (Simula) Title: Heuristic regularisation parameter choice rule in multi-penalty functionals for exact support recovery
Abstract: Inverse problems of unmixing type arise in many real-life applications such as audio processing or medical image analysis. In such problems additive noise directly affects a sparse signal before being measured through a sampling matrix. Consequently, the noise in the measurement is amplified through the sampling process and the so-called noise folding phenomenon occurs. This amplification worsens the results on support identification by means of ”classical” sparse recovery techniques based on the l1-penalised Lasso functional. Several recent works suggest to apply a multi-penalty framework for a correct modeling and separation of the original signal in such type of problems. Theoretical results and numerical experiments with oracle-given regularisation parameters have shown that this approach allows to recover the correct support in a larger number of problems than its single-penalty counterpart. Admittedly, the parameter choice in such multi-penalty functionals becomes more involved compared to the Lasso or other single-penalty methods. In this talk, we start by introducing the multi-penalty regularisation for the unmixing problem. We show that the resulting functional can be seen as a parameterised Lasso such that we can solve the multi-penalty minimisation with known techniques for the Lasso functional. Moreover, we derive a heuristic method, that does neither require knowledge on the noise level nor an assumption on the solution, to automatically choose regularisation parameters. Finally, we present numerical studies that indicate an improved performance of the multi-penalty functional with automatically calculated parameters, when compared to common techniques from sparse recovery.

Time/place: Tuesday 25 October at 13:15–14:00 in room 656

Speaker: Timo Klock (Simula)
Title: Heuristic regularisation parameter choice rule in multi-penalty functionals for exact support recovery

Abstract: Inverse problems of unmixing type arise in many real-life applications such as audio processing or medical image analysis. In such problems additive noise directly affects a sparse signal before being measured through a sampling matrix. Consequently, the noise in the measurement is amplified through the sampling process and the so-called noise folding phenomenon occurs. This amplification worsens the results on support identification by means of ”classical” sparse recovery techniques based on the l1-penalised Lasso functional. Several recent works suggest to apply a multi-penalty framework for a correct modeling and separation of the original signal in such type of problems. Theoretical results and numerical experiments with oracle-given regularisation parameters have shown that this approach allows to recover the correct support in a larger number of problems than its single-penalty counterpart. Admittedly, the parameter choice in such multi-penalty functionals becomes more involved compared to the Lasso or other single-penalty methods. In this talk, we start by introducing the multi-penalty regularisation for the unmixing problem. We show that the resulting functional can be seen as a parameterised Lasso such that we can solve the multi-penalty minimisation with known techniques for the Lasso functional. Moreover, we derive a heuristic method, that does neither require knowledge on the noise level nor an assumption on the solution, to automatically choose regularisation parameters. Finally, we present numerical studies that indicate an improved performance of the multi-penalty functional with automatically calculated parameters, when compared to common techniques from sparse recovery.

Time/place: Monday 24 October at 09:15–10:00 in room 734
Speaker: Erwan Faou (INRIA Rennes) Title: On travelling wave for the discrete nonlinear Schrödinger equation
Abstract: I will discuss the possible existence of travelling wave solutions in discrete nonlinear Schrödinger equations on a grid. I will show the influence of the nonlinearity in this problem and give some partial results for the long time existence and stability. This is joint work with Dario Bambusi, Joackim Bernier, Benoît Grébert and Alberto Maspero.

Time/place: Thursday 6th of October at 13:15–14:00 in room 734
Speaker: Dag Nilsson, Lund University. Title: Solitary waves of a class of Green–Naghdi type systems
Abstract: We consider a class of Green–Naghdi type systems and prove the existence of solitary wave solutions. The solutions are identified as critical points of a scalar functional and we are able to show that there exist minimizers of this functional, under certain constraints. A key component of the proof is the use of the concentration compactness principle. The talk is based upon a work in progress with Erik Wahlén (Lund University) and Vincent Duchene (University of Rennes).

Time/place: Thursday 6th of October at 13:15–14:00 in room 734

Speaker: Dag Nilsson, Lund University.
Title: Solitary waves of a class of Green–Naghdi type systems

Abstract: We consider a class of Green–Naghdi type systems and prove the existence of solitary wave solutions. The solutions are identified as critical points of a scalar functional and we are able to show that there exist minimizers of this functional, under certain constraints. A key component of the proof is the use of the concentration compactness principle. The talk is based upon a work in progress with Erik Wahlén (Lund University) and Vincent Duchene (University of Rennes).

Time/place: Wednesday 21st of September at 14:15–15:00 in room 656
Speaker: Chris Budd (University of Bath) Title: Monge Ampere based methods for Mesh Generation on the Plane and on the Sphere
Abstract: TBA