Wave propagation in discrete heterogeneous media
|
|
- Noah Nash
- 5 years ago
- Views:
Transcription
1 Wave propagation in discrete heterogeneous media Aurora MARICA BCAM - Basque Center for Applied Mathematics Derio, Basque Country, Spain Summer school & workshop: PDEs, optimal design and numerics - IV edition Benasque, Centro de Ciencias Pedro Pascual August 3, 20 Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 /
2 Problem formulation Finite difference approximation of the homogeneous wave equation on a non-uniform grid y = (y j 0 j N+ of (0, : u j+ (t u j (t u j u j (t u j (t y (t j+ y j y j y j y j+ y j = 0, ( 2 u 0 (t = u N+ (t = 0, u j (0 = uj 0, u j (0 = u j, j N. Aim: analyze the observability inequality for (: where h n uh (t = u N (t/( y N and E h (u h,0, u h, = 2 j= T E h (u h,0, u h, C h (T n h uh (t 2 dt, 0 N y j+ y j u j 2 (t N (y j+ y j u j+ (t u j (t 2 y j=0 j+ y. j Pathologies and remedies for numerical approximations of waves on uniform media y = x = [0 : h : ], h = /(N + : Ervedoza, Zuazua, The wave equation: control and numerics, CIME Subseries, Springer. Rays of Geometric Optics: Bardos, Lebeau, Rauch, Sharp sufficient conditions for observation, control and stabilization of waves from the boundary, SIAM J. Cont. Optim., 992. Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 2 /
3 Rays of GO - continuous and discrete on uniform meshes cases Continuous rays: x(t = x ± t Discrete rays on uniform meshes: x(t = x ± tω h (ξ 0, where ω h (ξ := 2 sin(ξh/2/h. (a continuous (b discrete, ξ 0 = π/2h (c discrete, ξ 0 = 2π/3h Nπ 0 0 N (d Continuous (blue and discrete (red dispersion relations Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 3 /
4 Known facts on non-uniform meshes I RESOLVENT ESTIMATES Ervedoza, Spectral conditions for admissibility and observability of wave systems, Numer. Mathematik, The observability inequality for general finite element semi-discretizations of the wave equation corresponding to a convergent approximation of order h θ of the Laplacian holds uniformly as h 0 in a class of truncated solutions generated by eigenvalues of order (ɛ/h θ 2, for some ɛ > 0, which does not include the critical scale /h 2 appearing in the numerical approximations on uniform meshes. MIXED FINITE ELEMENTS METHODS Ervedoza, On the mixed finite element method for the d wave equation on non-uniform meshes, ESAIM:COCV, 200. The numerical scheme: y j+ y j (u j+ 4 (t+u j (t+ y j y j 4 (u j (t+u j (t u j+(t u j (t y j+ y j u j (t u j (t y j y j = 0. Eigenvalues λ: kπ N 2 = arctan λ(y j+ y j, k N. 2 j=0 Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 4 /
5 Known facts on non-uniform meshes II SPECTRAL DISTRIBUTION OF LOCALLY TOEPLITZ SEQUENCE MATRICES Beckerman, Serra-Capizzano, On the asymptotic spectrum of finite element matrix sequences, SINUM, Tilli, Locally Toeplitz sequences: spectral properties and applications, Lin. Alg. Appl., 998. Szegö Theorem. Uniform mesh + constant coefficients. T N (ω = Toeplitz matrix whose diagonals are Fourier coefficients ω j (0 j N of ω : ( π, π C and (λ j j N are the eigenvalues of T N (ω. Then F C b c (R: lim N N N F (λ j = π F (ω(ξ dξ. 2π j= π Uniform mesh + variable coefficients. T N (ω, a = locally Toeplitz matrix, a : (0, R, then N lim F (λ j = π F (ω(ξa(x dx dξ. N N 2π j= π 0 Non-uniform mesh + variable coefficients. For (a(xu x x = b(x, x (0, discretized using a scheme generating a Toeplitz matrix whose diagonals are generated by ω and on a non-uniform mesh y : (0, (0,, we obtain the locally Toeplitz matrix T N (ω, a, y s.t. lim N N N F (λ j = π 2π j= π 0 ( a(y(x F y (x 2 ω(ξ dx dξ. Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 5 /
6 Our results based on pseudo-differential calculus The d transport equation: ρ(xu ɛ t (x, t u ɛ x (x, t = 0, x R, t > 0, u ɛ (x, 0 = ϕ ɛ (x, x R. (2 u ɛ (, t L 2 (R, ρ is conserved in time. If ρ L (R, ρ 0 and ϕ ɛ is uniformly bounded in L 2 (R, then ρ(x u ɛ (x, t 2 is uniformly bounded in L (R, so that ρ( u ɛ (, t 2 µ, weakly in M(R. The ρ-weighted Wigner transform of u ɛ : w ɛ (x, t, ξ = ρ(xu ɛ (x + ɛz/2, tu ɛ (x ɛz/2, t exp( iξz dz. (3 2π R f ɛ (x, t, z := (F ξ z w ɛ (x, t, z - the Fourier transform of w ɛ in ξ. f ɛ satisfies the following equation: ft ɛ (x, t, z = ( 2 ρ(x + ɛz/2 + ρ(x ɛz/2 ρ (x ( ρ(x 2 ρ(x + ɛz/2 + ρ(x ɛz/2 + ( ɛ 2 ρ(x + ɛz/2 fz ɛ (x, t, z. ρ(x ɛz/2 f ɛ x (x, t, z (4 f ɛ (x, t, z Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 6 /
7 K ɛ (x, ξ := 2π R K2 ɛ (x, ξ := 2π R ( 2 ( ɛz ρ(x + ɛz/2 + ρ(x ɛz/2 ρ(x + ɛz/2 ρ(x ɛz/2 The Wigner transform w ɛ (x, t, ξ satisfies the following equation: exp( iξz dz (5 exp( iξz dz. (6 w ɛ t (x, t, ξ = K ɛ (x, w ɛ x (x, t, ξ ρ (x ρ(x K ɛ (x, w ɛ (x, t, ξ (7 K2 ɛ (x, w ɛ (x, t, ξ K2 ɛ (x, ξw ξ ɛ (x, t, ξ. Formally, K ɛ (x, ξ /ρ(xδ 0(ξ and K ɛ 2 (x, ξ (/ρ (xδ 0 (ξ, so that w ɛ (x, t, ξ converges to a function w(x, t, ξ which verifies the equation: w t(x, t, ξ = ρ(x wx (x, t, ξ + ρ (x ρ 2 (x ξw ξ(x, t, ξ. (8 Characteristics verifying the Hamiltonian system: x (t = ρ(x(t and ξ (t = ρ (x(t ρ 2 (x(t ξ(t. Uniqueness for the Hamiltonian system iff ρ C, (R. Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 7 /
8 Gérard, Markowich, Mauser, Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 997. Lions P.-L., Paul, Sur les mesures de Wigner, Rev. Matemática Iberoamericana, 993. Schrödinger equation with potential ihψ h t = h2 ψ h xx + V (xψh, (x, t R R, is transformed into Liouville equation w t + ξw x V x w ξ = 0, which needs V C, (R for uniqueness. FINITE DIFFERENCE APPROXIMATION of the transport equation f ɛ (x, t, z verifies the equation: ρ(xu ɛ t (x, t uɛ (x + ɛ, t u ɛ (x ɛ, t 2ɛ = 0, x R, t > 0. (9 f ɛ t (x, t, z = ρ(x f ɛ (x + ɛ/2, t, z + 2ɛρ(x + ɛz/2ρ(x + ɛ/2 ρ(x f ɛ (x ɛ/2, t, z 2ɛρ(x + ɛz/2ρ(x ɛ/2 f ɛ (x + ɛ/2, t, z + ρ(x 2ɛρ(x ɛz/2ρ(x + ɛ/2 ρ(x f ɛ (x ɛ/2, t, z + 2ɛρ(x ɛz/2ρ(x ɛ/2. (0 Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 8 /
9 wt ɛ (x, t, ξ = = ρ(x ( 4 ρ(x + ɛ/2 + ρ(x ɛ/2 + ρ(x ( 8 ρ(x + ɛ/2 + ρ(x ɛ/2 2 cos(ξ ( w ɛ (x + ɛ/2, t, ξ + w ɛ (x ɛ/2, t, ξ + ρ(x ( 4ɛ ρ(x + ɛ/2 ( + ɛ ρ(x 8ɛ ρ(x ɛ/2 ρ(x + ɛ/2 ρ(x ɛ/2 2 cos(ξ ( w ɛ (x + ɛ/2, t, ξ w ɛ (x ɛ/2, t, ξ. The limit equation is K ɛ (x, 2 cos(ξ w ɛ (x + ɛ/2, t, ξ w ɛ (x ɛ/2, t, ξ ɛ ( K 2 ɛ (x, 2 sin(ξ ( wξ ɛ (x + ɛ/2, t, ξ + w ξ ɛ (x ɛ/2, t, ξ K ɛ (x, 2 cos(ξ(w ɛ (x + ɛ/2, t, ξ + w ɛ (x ɛ/2, t, ξ ( K 2 ɛ (x, 2 sin(ξ ( wξ ɛ (x + ɛ/2, t, ξ w ξ ɛ (x ɛ/2, t, ξ w t(x, t, ξ = ρ(x cos(ξwx (x, t, ξ + ρ (x ρ 2 (x sin(ξw ξ(x, t, ξ. For the transport equation on a non-uniform grid y = g(x, we have w t(x, t, ξ = ( ρ(g(xg cos(ξwx (x, t, ξ + (x ρ(g(xg sin(ξw ξ (x, t, ξ. (x Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 9 /
10 Numerical simulations u 0 (y = ϕ γ ξ 0 (y y 0 exp(iξ 0 y 0 Legend: (e h = /200, uniform grid of size h for y (0, /2 and h/2 for y (/2, and y 0 = /4 (f-g h = /200, y = tan(πx/4, y 0 = /4; (h h = /200, y = sin(πx/3 for x (0, /2, y = sin(π( x/3 for x (/2, and y 0 = /2; (i h = /00, uniform grid of size h/8 and h/4 for y (0, /4 and y (3/4,, y = /4 + tan(x/4/2 for y (/4, 3/4, and y 0 = 7/8 We illustrate some phenomena, most of them being pathological and requiring further analysis: reflection-transmission problem at the interface between two piecewise uniform discrete media (see Fig. 4(e torsion of the rays of Geometric Optics, reflecting before touching the boundary of the domain (see Fig. 4(f-i (e ξ 0 = π/2h (f ξ 0 = π/2h min (g ξ 0 = π/h min (h ξ 0 = π/2h min (i ξ 0 = π/2h min Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 0 /
11 Conclusions and open problems In this talk: We give a meaning to the notion of rays of Geometric Optics by constructing appropriate transport equations. Open problems: To solve the Hamiltonian systems, we need ρ(g(xg (x C, (R. Study the case when the non-homogeneity of the grid is less regular. Adapt the multiplier techniques to prove the observability inequality for the non-uniform mesh case (a posteriori error estimates techniques. Adapt the filtering techniques (the bi-grid ones to remedy the pathological effects of the high-frequency spurious solutions. Study more sophisticated methods for the wave equation (DG ones, higher order ones on non-uniform meshes. The multi-d case. Dispersive estimates for the Schrödinger equation on non-uniform meshes. Meshes which are given randomly. Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 /
12 Conclusions and open problems In this talk: We give a meaning to the notion of rays of Geometric Optics by constructing appropriate transport equations. Open problems: To solve the Hamiltonian systems, we need ρ(g(xg (x C, (R. Study the case when the non-homogeneity of the grid is less regular. Adapt the multiplier techniques to prove the observability inequality for the non-uniform mesh case (a posteriori error estimates techniques. Adapt the filtering techniques (the bi-grid ones to remedy the pathological effects of the high-frequency spurious solutions. Study more sophisticated methods for the wave equation (DG ones, higher order ones on non-uniform meshes. The multi-d case. Dispersive estimates for the Schrödinger equation on non-uniform meshes. Meshes which are given randomly. Thank you very much for your attention! Aurora Marica (BCAM Heterogeneous discrete media PDEsODN, Benasque - 3/08/20 /
High frequency wave propagation in non-uniform regular discrete media
www.bcamath.org High frequency wave propagation in non-uniform regular discrete media Aurora MARICA marica@bcamath.org BCAM - Basque Center for Applied Mathematics Derio, Basque Country, Spain Workshop
More informationThe effect of Group Velocity in the numerical analysis of control problems for the wave equation
The effect of Group Velocity in the numerical analysis of control problems for the wave equation Fabricio Macià École Normale Supérieure, D.M.A., 45 rue d Ulm, 753 Paris cedex 5, France. Abstract. In this
More informationNumerical dispersion and Linearized Saint-Venant Equations
Numerical dispersion and Linearized Saint-Venant Equations M. Ersoy Basque Center for Applied Mathematics 11 November 2010 Outline of the talk Outline of the talk 1 Introduction 2 The Saint-Venant equations
More informationHilbert Uniqueness Method and regularity
Hilbert Uniqueness Method and regularity Sylvain Ervedoza 1 Joint work with Enrique Zuazua 2 1 Institut de Mathématiques de Toulouse & CNRS 2 Basque Center for Applied Mathematics Institut Henri Poincaré
More informationControl of Waves: Theory and Numerics
BCAM, October, 2010 Control of Waves: Theory and Numerics Enrique Zuazua BCAM Basque Center for Applied Mathematics E-48160 Derio - Basque Country - Spain zuazua@bcamath.org www.bcamath.org/zuazua THE
More informationThird part: finite difference schemes and numerical dispersion
Tird part: finite difference scemes and numerical dispersion BCAM - Basque Center for Applied Matematics Bilbao, Basque Country, Spain BCAM and UPV/EHU courses 2011-2012: Advanced aspects in applied matematics
More informationDispersive numerical schemes for Schrödinger equations
Dispersive numerical schemes for Schrödinger equations Enrique Zuazua joint work with L. Ignat zuazua@bcamath.org Basque Center for Applied Mathematics (BCAM), Bilbao, Basque Country, Spain IMA Workshop:
More informationNumerical control of waves
Numerical control of waves Convergence issues and some applications Mark Asch U. Amiens, LAMFA UMR-CNRS 7352 June 15th, 2012 Mark Asch (GT Contrôle, LJLL, Parris-VI) Numerical control of waves June 15th,
More informationDispersive numerical schemes for Schrödinger equations
Dispersive numerical schemes for Schrödinger equations Enrique Zuazua joint work with L. Ignat & A. Marica zuazua@bcamath.org Basque Center for Applied Mathematics (BCAM), Bilbao, Basque Country, Spain
More informationPropagation of Monokinetic Measures with Rough Momentum Profiles I
with Rough Momentum Profiles I Ecole Polytechnique Centre de Mathématiques Laurent Schwartz Quantum Systems: A Mathematical Journey from Few to Many Particles May 16th 2013 CSCAMM, University of Maryland.
More informationA Survey of Computational High Frequency Wave Propagation II. Olof Runborg NADA, KTH
A Survey of Computational High Frequency Wave Propagation II Olof Runborg NADA, KTH High Frequency Wave Propagation CSCAMM, September 19-22, 2005 Numerical methods Direct methods Wave equation (time domain)
More informationZ. Zhou On the classical limit of a time-dependent self-consistent field system: analysis. computation
On the classical limit of a time-dependent self-consistent field system: analysis and computation Zhennan Zhou 1 joint work with Prof. Shi Jin and Prof. Christof Sparber. 1 Department of Mathematics Duke
More informationEvolution problems involving the fractional Laplace operator: HUM control and Fourier analysis
Evolution problems involving the fractional Laplace operator: HUM control and Fourier analysis Umberto Biccari joint work with Enrique Zuazua BCAM - Basque Center for Applied Mathematics NUMERIWAVES group
More informationNumerics for the Control of Partial Differential
Springer-Verlag Berlin Heidelberg 2015 Björn Engquist Encyclopedia of Applied and Computational Mathematics 10.1007/978-3-540-70529-1_362 Numerics for the Control of Partial Differential Equations Enrique
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationNumerical Analysis of Differential Equations Numerical Solution of Parabolic Equations
Numerical Analysis of Differential Equations 215 6 Numerical Solution of Parabolic Equations 6 Numerical Solution of Parabolic Equations TU Bergakademie Freiberg, SS 2012 Numerical Analysis of Differential
More informationMATH 126 FINAL EXAM. Name:
MATH 126 FINAL EXAM Name: Exam policies: Closed book, closed notes, no external resources, individual work. Please write your name on the exam and on each page you detach. Unless stated otherwise, you
More informationCONTROLLABILITY OF FAST DIFFUSION COUPLED PARABOLIC SYSTEMS
CONTROLLABILITY OF FAST DIFFUSION COUPLED PARABOLIC SYSTEMS FELIPE WALLISON CHAVES-SILVA, SERGIO GUERRERO, AND JEAN PIERRE PUEL Abstract. In this work we are concerned with the null controllability of
More informationThe heat equation for the Hermite operator on the Heisenberg group
Hokkaido Mathematical Journal Vol. 34 (2005) p. 393 404 The heat equation for the Hermite operator on the Heisenberg group M. W. Wong (Received August 5, 2003) Abstract. We give a formula for the one-parameter
More informationSemiclassical computational methods for quantum dynamics with bandcrossings. Shi Jin University of Wisconsin-Madison
Semiclassical computational methods for quantum dynamics with bandcrossings and uncertainty Shi Jin University of Wisconsin-Madison collaborators Nicolas Crouseilles, Rennes Mohammed Lemou, Rennes Liu
More informationASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS
Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC
More informationOptimal and Approximate Control of Finite-Difference Approximation Schemes for the 1D Wave Equation
Optimal and Approximate Control of Finite-Difference Approximation Schemes for the 1D Wave Equation May 21, 2004 Enrique Zuazua 1 Departmento de Matemáticas Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es
More informationThe PML Method: Continuous and Semidiscrete Waves.
Intro Continuous Model. Finite difference. Remedies. The PML Method: Continuous and Semidiscrete Waves. 1 Enrique Zuazua 2 1 Laboratoire de Mathématiques de Versailles. 2 Universidad Autónoma, Madrid.
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More informationNumerical methods for hyperbolic systems with singular coefficients: well-balanced scheme, Hamiltonian preservation, and beyond
Proceedings of Symposia in Applied Mathematics Numerical methods for hyperbolic systems with singular coefficients: well-balanced scheme, Hamiltonian preservation, and beyond Shi Jin Abstract. This paper
More informationSpectrum and Exact Controllability of a Hybrid System of Elasticity.
Spectrum and Exact Controllability of a Hybrid System of Elasticity. D. Mercier, January 16, 28 Abstract We consider the exact controllability of a hybrid system consisting of an elastic beam, clamped
More informationEigenvalues and eigenfunctions of the Laplacian. Andrew Hassell
Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean
More informationAsymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction
International Series of Numerical Mathematics, Vol. 154, 445 455 c 2006 Birkhäuser Verlag Basel/Switzerland Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction
More informationThe heat equation. Paris-Sud, Orsay, December 06
Paris-Sud, Orsay, December 06 The heat equation Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua Plan: 3.- The heat equation: 3.1 Preliminaries
More informationThe wave equation. Paris-Sud, Orsay, December 06
Paris-Sud, Orsay, December 06 The wave equation Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua Work in collaboration with: C. Castro, M.
More informationGlobal well-posedness for semi-linear Wave and Schrödinger equations. Slim Ibrahim
Global well-posedness for semi-linear Wave and Schrödinger equations Slim Ibrahim McMaster University, Hamilton ON University of Calgary, April 27th, 2006 1 1 Introduction Nonlinear Wave equation: ( 2
More informationHardy inequalities, heat kernels and wave propagation
Outline Hardy inequalities, heat kernels and wave propagation Basque Center for Applied Mathematics (BCAM) Bilbao, Basque Country, Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ Third Brazilian
More informationDiscontinuous Galerkin and Finite Difference Methods for the Acoustic Equations with Smooth Coefficients. Mario Bencomo TRIP Review Meeting 2013
About Me Mario Bencomo Currently 2 nd year graduate student in CAAM department at Rice University. B.S. in Physics and Applied Mathematics (Dec. 2010). Undergraduate University: University of Texas at
More informationFinite difference method for heat equation
Finite difference method for heat equation Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
More informationDiagonalization of the Coupled-Mode System.
Diagonalization of the Coupled-Mode System. Marina Chugunova joint work with Dmitry Pelinovsky Department of Mathematics, McMaster University, Canada Collaborators: Mason A. Porter, California Institute
More informationNumerical methods for a fractional diffusion/anti-diffusion equation
Numerical methods for a fractional diffusion/anti-diffusion equation Afaf Bouharguane Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux 1, France Berlin, November 2012 Afaf Bouharguane Numerical
More informationEvolution of semiclassical Wigner function (the higher dimensio
Evolution of semiclassical Wigner function (the higher dimensional case) Workshop on Fast Computations in Phase Space, WPI-Vienna, November 2008 Dept. Appl. Math., Univ. Crete & IACM-FORTH 1 2 3 4 5 6
More informationNonlinear stabilization via a linear observability
via a linear observability Kaïs Ammari Department of Mathematics University of Monastir Joint work with Fathia Alabau-Boussouira Collocated feedback stabilization Outline 1 Introduction and main result
More informationFourier analysis for discontinuous Galerkin and related methods. Abstract
Fourier analysis for discontinuous Galerkin and related methods Mengping Zhang and Chi-Wang Shu Abstract In this paper we review a series of recent work on using a Fourier analysis technique to study the
More informationSwitching, sparse and averaged control
Switching, sparse and averaged control Enrique Zuazua Ikerbasque & BCAM Basque Center for Applied Mathematics Bilbao - Basque Country- Spain zuazua@bcamath.org http://www.bcamath.org/zuazua/ WG-BCAM, February
More informationComputational High Frequency Wave Propagation
Computational High Frequency Wave Propagation Olof Runborg CSC, KTH Isaac Newton Institute Cambridge, February 2007 Olof Runborg (KTH) High-Frequency Waves INI, 2007 1 / 52 Outline 1 Introduction, background
More informationWave operators with non-lipschitz coefficients: energy and observability estimates
Wave operators with non-lipschitz coefficients: energy and observability estimates Institut de Mathématiques de Jussieu-Paris Rive Gauche UNIVERSITÉ PARIS DIDEROT PARIS 7 JOURNÉE JEUNES CONTRÔLEURS 2014
More informationProceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005
Proceedings of the 5th International Conference on Inverse Problems in Engineering: Theory and Practice, Cambridge, UK, 11-15th July 2005 SOME INVERSE SCATTERING PROBLEMS FOR TWO-DIMENSIONAL SCHRÖDINGER
More informationOptimal shape and location of sensors. actuators in PDE models
Modeling Optimal shape and location of sensors or actuators in PDE models Emmanuel Tre lat1 1 Sorbonne Universite (Paris 6), Laboratoire J.-L. Lions Fondation Sciences Mathe matiques de Paris Works with
More informationThe Wave Equation: Control and Numerics
The Wave Equation: Control and Numerics Sylvain Ervedoza and Enrique Zuazua Abstract In these Notes we make a self-contained presentation of the theory that has been developed recently for the numerical
More informationCLASSICAL LIMIT FOR SEMI-RELATIVISTIC HARTREE SYSTEMS
CLASSICAL LIMIT FOR SEMI-RELATIVISTIC HARTREE SYSTEMS GONCA L. AKI, PETER A. MARKOWICH, AND CHRISTOF SPARBER Abstract. We consider the three-dimensional semi-relativistic Hartree model for fast quantum
More informationThe Vlasov-Poisson Equations as the Semiclassical Limit of the Schrödinger-Poisson Equations: A Numerical Study
The Vlasov-Poisson quations as the Semiclassical Limit of the Schrödinger-Poisson quations: A Numerical Study Shi Jin, Xiaomei Liao and Xu Yang August 3, 7 Abstract In this paper, we numerically study
More informationComputing High Frequency Waves By the Level Set Method
Computing High Frequency Waves By the Level Set Method Hailiang Liu Department of Mathematics Iowa State University Collaborators: Li-Tien Cheng (UCSD), Stanley Osher (UCLA) Shi Jin (UW-Madison), Richard
More informationControl, Stabilization and Numerics for Partial Differential Equations
Paris-Sud, Orsay, December 06 Control, Stabilization and Numerics for Partial Differential Equations Enrique Zuazua Universidad Autónoma 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua
More informationMATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems -3 and one of Problems 4 and 5. Write your solutions to problems and
More informationControllability of linear PDEs (I): The wave equation
Controllability of linear PDEs (I): The wave equation M. González-Burgos IMUS, Universidad de Sevilla Doc Course, Course 2, Sevilla, 2018 Contents 1 Introduction. Statement of the problem 2 Distributed
More informationWave Equation With Homogeneous Boundary Conditions
Wave Equation With Homogeneous Boundary Conditions MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 018 Objectives In this lesson we will learn: how to solve the
More informationInégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur.
Inégalités spectrales pour le contrôle des EDP linéaires : groupe de Schrödinger contre semigroupe de la chaleur. Luc Miller Université Paris Ouest Nanterre La Défense, France Pde s, Dispersion, Scattering
More informationComputation of Nonlinear Schrödinger Equation on an Open Waveguide Terminated by a PML
Copyright 20 Tech Science Press CMES, vol.7, no.4, pp.347-362, 20 Computation of Nonlinear Schrödinger Equation on an Open Waveguide Terminated by a PML Jianxin Zhu, Zheqi Shen Abstract: It is known that
More informationASYMPTOTIC ANALYSIS OF THE QUANTUM DYNAMICS IN CRYSTALS: THE BLOCH-WIGNER TRANSFORM AND BLOCH DYNAMICS. 1. Introduction
ASYMPTOTIC ANALYSIS OF THE QUANTUM DYNAMICS IN CRYSTALS: THE BLOCH-WIGNER TRANSFORM AND BLOCH DYNAMICS WEINAN E, JIANFENG LU, AND XU YANG Abstract. We study the semi-classical limit of the Schrödinger
More informationOptimal shape and location of sensors or actuators in PDE models
Optimal shape and location of sensors or actuators in PDE models Y. Privat, E. Trélat 1, E. Zuazua 1 Univ. Paris 6 (Labo. J.-L. Lions) et Institut Universitaire de France SIAM Conference on Analysis of
More informationDispersive numerical schemes for linear and nonlinear Schrödinger equations
Collège de France, December 2006 Dispersive numerical schemes for linear and nonlinear Schrödinger equations Enrique Zuazua Universidad Autónoma, 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua
More informationQualitative Properties of Numerical Approximations of the Heat Equation
Qualitative Properties of Numerical Approximations of the Heat Equation Liviu Ignat Universidad Autónoma de Madrid, Spain Santiago de Compostela, 21 July 2005 The Heat Equation { ut u = 0 x R, t > 0, u(0,
More informationNew Identities for Weak KAM Theory
New Identities for Weak KAM Theory Lawrence C. Evans Department of Mathematics University of California, Berkeley Abstract This paper records for the Hamiltonian H = p + W (x) some old and new identities
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationA regularity property for Schrödinger equations on bounded domains
A regularity property for Schrödinger equations on bounded domains Jean-Pierre Puel October 8, 11 Abstract We give a regularity result for the free Schrödinger equations set in a bounded domain of R N
More informationInvariant measures and the soliton resolution conjecture
Invariant measures and the soliton resolution conjecture Stanford University The focusing nonlinear Schrödinger equation A complex-valued function u of two variables x and t, where x R d is the space variable
More informationSINC PACK, and Separation of Variables
SINC PACK, and Separation of Variables Frank Stenger Abstract This talk consists of a proof of part of Stenger s SINC-PACK computer package (an approx. 400-page tutorial + about 250 Matlab programs) that
More informationOBSERVABILITY INEQUALITY AND DECAY RATE FOR WAVE EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 27 (27, No. 6, pp. 2. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu OBSERVABILITY INEQUALITY AND DECAY RATE FOR WAVE EQUATIONS
More informationMATH 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, x < 1, 1 + x, 1 < x < 0, ψ(x) = 1 x, 0 < x < 1, 0, x > 1, so that it verifies ψ 0, ψ(x) = 0 if x 1 and ψ(x)dx = 1. Consider (ψ j )
More informationSpace-time Finite Element Methods for Parabolic Evolution Problems
Space-time Finite Element Methods for Parabolic Evolution Problems with Variable Coefficients Ulrich Langer, Martin Neumüller, Andreas Schafelner Johannes Kepler University, Linz Doctoral Program Computational
More informationWaves: propagation, dispersion and numerical simulation
ENUMATH, Santiago de Compostela, July 2005 Waves: propagation, dispersion and numerical simulation Enrique Zuazua Universidad Autónoma, 28049 Madrid, Spain enrique.zuazua@uam.es http://www.uam.es/enrique.zuazua
More informationNumerical Methods for Modern Traffic Flow Models. Alexander Kurganov
Numerical Methods for Modern Traffic Flow Models Alexander Kurganov Tulane University Mathematics Department www.math.tulane.edu/ kurganov joint work with Pierre Degond, Université Paul Sabatier, Toulouse
More information7 Hyperbolic Differential Equations
Numerical Analysis of Differential Equations 243 7 Hyperbolic Differential Equations While parabolic equations model diffusion processes, hyperbolic equations model wave propagation and transport phenomena.
More informationPath integrals for classical Markov processes
Path integrals for classical Markov processes Hugo Touchette National Institute for Theoretical Physics (NITheP) Stellenbosch, South Africa Chris Engelbrecht Summer School on Non-Linear Phenomena in Field
More informationRadiative transfer equations with varying refractive index: a mathematical perspective
Radiative transfer equations with varying refractive index: a mathematical perspective Guillaume Bal November 23, 2005 Abstract This paper reviews established mathematical techniques to model the energy
More informationStrichartz Estimates in Domains
Department of Mathematics Johns Hopkins University April 15, 2010 Wave equation on Riemannian manifold (M, g) Cauchy problem: 2 t u(t, x) gu(t, x) =0 u(0, x) =f (x), t u(0, x) =g(x) Strichartz estimates:
More informationThe Density Matrix for the Ground State of 1-d Impenetrable Bosons in a Harmonic Trap
The Density Matrix for the Ground State of 1-d Impenetrable Bosons in a Harmonic Trap Institute of Fundamental Sciences Massey University New Zealand 29 August 2017 A. A. Kapaev Memorial Workshop Michigan
More informationRecent result on porous medium equations with nonlocal pressure
Recent result on porous medium equations with nonlocal pressure Diana Stan Basque Center of Applied Mathematics joint work with Félix del Teso and Juan Luis Vázquez November 2016 4 th workshop on Fractional
More informationNodal lines of Laplace eigenfunctions
Nodal lines of Laplace eigenfunctions Spectral Analysis in Geometry and Number Theory on the occasion of Toshikazu Sunada s 60th birthday Friday, August 10, 2007 Steve Zelditch Department of Mathematics
More informationA LEVEL SET FRAMEWORK FOR TRACKING MULTI-VALUED SOLUTIONS OF NONLINEAR FIRST-ORDER EQUATIONS
A LEVEL SET FRAMEWORK FOR TRACKING MULTI-VALUED SOLUTIONS OF NONLINEAR FIRST-ORDER EQUATIONS HAILIANG LIU, LI-TIEN CHENG, AND STANLEY OSHER Abstract. We introduce a level set method for the computation
More informationNull-controllability of the heat equation in unbounded domains
Chapter 1 Null-controllability of the heat equation in unbounded domains Sorin Micu Facultatea de Matematică-Informatică, Universitatea din Craiova Al. I. Cuza 13, Craiova, 1100 Romania sd micu@yahoo.com
More information1 Introduction. Controllability and observability
Matemática Contemporânea, Vol 31, 00-00 c 2006, Sociedade Brasileira de Matemática REMARKS ON THE CONTROLLABILITY OF SOME PARABOLIC EQUATIONS AND SYSTEMS E. Fernández-Cara Abstract This paper is devoted
More informationRiesz bases of Floquet modes in semi-infinite periodic waveguides and implications
Riesz bases of Floquet modes in semi-infinite periodic waveguides and implications Thorsten Hohage joint work with Sofiane Soussi Institut für Numerische und Angewandte Mathematik Georg-August-Universität
More informationAsymptotic Analysis of the Quantum Dynamics in Crystals: The Bloch-Wigner Transform, Bloch Dynamics and Berry Phase
Asymptotic Analysis of the Quantum Dynamics in Crystals: The Bloch-Wigner Transform, Bloch Dynamics and Berry Phase Weinan E, Jianfeng Lu and Xu Yang Department of Mathematics, Princeton University, Princeton,
More informationPropagation of longitudinal waves in a random binary rod
Downloaded By: [University of North Carolina, Charlotte At: 7:3 2 May 28 Waves in Random and Complex Media Vol. 6, No. 4, November 26, 49 46 Propagation of longitudinal waves in a random binary rod YURI
More informationNonlinear and Nonlocal Degenerate Diffusions on Bounded Domains
Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains Matteo Bonforte Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco 28049 Madrid, Spain matteo.bonforte@uam.es
More informationFinite Volume Schemes: an introduction
Finite Volume Schemes: an introduction First lecture Annamaria Mazzia Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di Padova mazzia@dmsa.unipd.it Scuola di dottorato
More informationResolvent estimates for high-contrast elliptic problems with periodic coefficients
Resolvent estimates for high-contrast elliptic problems with periodic coefficients Joint work with Shane Cooper (University of Bath) 25 August 2015, Centro de Ciencias de Benasque Pedro Pascual Partial
More informationVelocity averaging a general framework
Outline Velocity averaging a general framework Martin Lazar BCAM ERC-NUMERIWAVES Seminar May 15, 2013 Joint work with D. Mitrović, University of Montenegro, Montenegro Outline Outline 1 2 L p, p >= 2 setting
More informationMathematical and computational methods for semiclassical Schrödinger equations
Acta Numerica (2012), pp. 1 89 c Cambridge University Press, 2012 DOI: 10.1017/S0962492902 Printed in the United Kingdom Mathematical and computational methods for semiclassical Schrödinger equations Shi
More informationDiffusion for a Markov, Divergence-form Generator
Diffusion for a Markov, Divergence-form Generator Clark Department of Mathematics Michigan State University Arizona School of Analysis and Mathematical Physics March 16, 2012 Abstract We consider the long-time
More informationLecture No 1 Introduction to Diffusion equations The heat equat
Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009 Outline of the lectures We will discuss some basic models of diffusion equations and
More informationOptimal shape and position of the support for the internal exact control of a string
Optimal shape and position of the support for the internal exact control of a string Francisco Periago Abstract In this paper, we consider the problem of optimizing the shape and position of the support
More informationCURRICULUM VITAE. 1322B, New main building, Beijing Normal University, Haidian District, Beijing, China.
CURRICULUM VITAE Name Chuang ZHENG Sex Male Birth (date and place) April, 1982, Sichuan, China Marital Status Married, 2 children. Present Address School of Mathematics, 1322B, New main building, Beijing
More informationSpectral theory for magnetic Schrödinger operators and applicatio. (after Bauman-Calderer-Liu-Phillips, Pan, Helffer-Pan)
Spectral theory for magnetic Schrödinger operators and applications to liquid crystals (after Bauman-Calderer-Liu-Phillips, Pan, Helffer-Pan) Ryukoku (June 2008) In [P2], based on the de Gennes analogy
More informationMcGill University Department of Mathematics and Statistics. Ph.D. preliminary examination, PART A. PURE AND APPLIED MATHEMATICS Paper BETA
McGill University Department of Mathematics and Statistics Ph.D. preliminary examination, PART A PURE AND APPLIED MATHEMATICS Paper BETA 17 August, 2018 1:00 p.m. - 5:00 p.m. INSTRUCTIONS: (i) This paper
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationFunctional Analysis, Math 7320 Lecture Notes from August taken by Yaofeng Su
Functional Analysis, Math 7320 Lecture Notes from August 30 2016 taken by Yaofeng Su 1 Essentials of Topology 1.1 Continuity Next we recall a stronger notion of continuity: 1.1.1 Definition. Let (X, d
More informationQuintic deficient spline wavelets
Quintic deficient spline wavelets F. Bastin and P. Laubin January 19, 4 Abstract We show explicitely how to construct scaling functions and wavelets which are quintic deficient splines with compact support
More informationRecent computational methods for high frequency waves in heterogeneous media ABSTRACT. Shi Jin. 1. Introduction
Recent computational methods for high frequency waves in heterogeneous media Shi Jin ABSTRACT In this note, we review our recent results on the Eulerian computation of high frequency waves in heterogeneous
More informationA Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets
A Minimal Uncertainty Product for One Dimensional Semiclassical Wave Packets George A. Hagedorn Happy 60 th birthday, Mr. Fritz! Abstract. Although real, normalized Gaussian wave packets minimize the product
More informationSome new results related to the null controllability of the 1 d heat equation
Some new results related to the null controllability of the 1 d heat equation Antonio LÓPEZ and Enrique ZUAZUA Departamento de Matemática Aplicada Universidad Complutense 284 Madrid. Spain bantonio@sunma4.mat.ucm.es
More informationHeat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control
Outline Heat equations with singular potentials: Hardy & Carleman inequalities, well-posedness & control IMDEA-Matemáticas & Universidad Autónoma de Madrid Spain enrique.zuazua@uam.es Analysis and control
More informationA remark on the observability of conservative linear systems
A remark on the observability of conservative linear systems Enrique Zuazua Abstract. We consider abstract conservative evolution equations of the form ż = Az, where A is a skew-adjoint operator. We analyze
More information