# wave equation

## Top PDF wave equation:

### CONCENTRATION OF LAPLACE EIGENFUNCTIONS AND STABILIZATION OF WEAKLY DAMPED WAVE EQUATION

[1] N. Anantharaman and M. L´eautaud. Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE, 7(1):159–214, 2014. With an appendix by St´ephane Nonnenmacher. [2] C. Bardos, G. Lebeau, and J. Rauch. Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. S.I.A.M. Journal of Control and Optimization, 305:1024–1065, 1992.

### EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS

properties of (1) have been obtained by various authors. For instance, in a very general context, Lebeau related the geometry of the undamped geodesics, the spectral asymptotics of the τ n and the energy decay of the damped wave equation [22]. Related results were also proved in several geometric contexts where the family of undamped geodesics was in some sense not too big: closed elliptic geodesic [19], closed hyperbolic geodesic [11, 9], subsets satisfying a condition of negative pressure [28, 29, 23]. Concerning the distribution of the τ n , Sj¨ ostrand gave a precise asymptotic
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### Exponential decay for the damped wave equation in unbounded domains

If Ω is a bounded manifold, the uniform positivity of hγi T (x, ξ) in Σ for some T > 0 implies that the exponential decay (1.5) holds, as shown in the celebrated articles [30], [3] and [4] of Bardos, Lebeau, Rauch and Taylor. The assumption that there exists T > 0 such that hγi T (x, ξ) > 0 in Σ is called the geometric control condition. The article [20] underlines in addition the importance of the value of min (x,ξ)∈Σ hγi T (x, ξ) in order to control the rate of decay of the high frequencies. In the case of an unbounded manifold, two situations have been investigated. First, some authors have considered the free wave equation (1.1) in an exterior domain (with γ ≡ 0 or γ > 0 only on a compact subset the exterior domain). They have shown that the local energy decays to zero in the sense that, under suitable assumptions, the energy of any solution escapes away from any compact set, see [18], [26] and [2] and the references therein. Secondly, several works have studied the damped wave equation in an unbounded manifold and with a non-linearity, but assuming that the damping satisfies γ(x) ≥ α > 0 outside a compact set, see [33], [10], [9] and [16].
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### Control of a Wave Equation with a Dynamic Boundary Condition

[4] Rhouma Mlayeh, Samir Toumi, and Lotfi Beji. Backstepping boundary observer based-control for hyperbolic PDE in rotary drilling system. Applied Mathematics and Computation, 322:66–78, 2018. [5] Christophe Prieur, Sophie Tarbouriech, and Joao M. Gomes da Silva. Wave Equation With Cone-Bounded Control Laws. IEEE Transactions on Automatic Control, 61(11):3452–3463, 2016.

### Approximate inversion of the wave-equation Hessian via randomized matrix probing

INTRODUCTION Much effort and progress has been made over the past sev- eral years to cast reflection seismic imaging as a waveform inversion problem, and solve it via all-purpose optimization tools. This shift of agenda has generated two major, as-yet un- solved challenges of a computational nature, namely (1) solv- ing the 3D Helmholtz equation with a limited memory imprint in the high frequency regime, and (2) inverting, or precondi- tioning the wave-equation Hessian in order to perform high- quality Gauss-Newton iterative steps to minimize the output least-squares objective. This paper is concerned with investi- gating new ideas to solve the second problem.
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### Approximate inversion of the wave-equation Hessian via randomized matrix probing

2 , where the waveform data d is a function of source position, receiver position, and time; the model m is a function of x, y, z that stands for isotropic wave speed (in this note), or other parameters such as elastic moduli and density; and the (nonlinear) forward modeling op- erator F results from simulating wave equations forward in time and sampling the wavefields at the receivers. We denote by F the linearization of F about a background model veloc- ity, and by F ∗ the corresponding migration (imaging) operator. A gradient descent step for minimizing J yields a model update of the form δ m = αF ∗ (d −F [m]) (for some scalar α), while a Gauss-Newton step would give δ m = (F ∗ F) −1 F ∗ (d − F [m]). The Gauss-Newton step solves the linearized problem exactly in a least-squares sense, but it is of course much harder to com- pute than a gradient step. The product F ∗ F is called normal operator: it is the leading-order approximation to the wave- equation Hessian H = ∂ 2 J
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### A minimum effort optimal control problem for the wave equation.

Received: date / Accepted: date Abstract A minimum effort optimal control problem for the undamped wave equation is considered which involves L ∞ –control costs. Since the problem is non-differentiable a regularized problem is introduced. Uniqueness of the solu- tion of the regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi-smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Thereby special attention has to be paid to the well-posedness of the Newton iteration. Numerical examples confirm the theoretical results.
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### Convergence analysis of hybrid high-order methods for the wave equation

Abstract We prove error estimates for the wave equation semi-discretized in space by the hybrid high-order (HHO) method. These estimates lead to optimal convergence rates for smooth solutions. We consider first the second-order formulation in time, for which we establish H 1 and L 2 -error estimates, and then the first-order formulation, for which we establish H 1 -error estimates. For both formulations, the space semi-discrete HHO scheme has close links with hybridizable discontinuous Galerkin schemes from the literature. Numerical ex- periments using either the Newmark scheme or diagonally-implicit Runge–Kutta schemes for the time discretization illustrate the theoretical findings and show that the proposed numerical schemes can be used to simulate accurately the propagation of elastic waves in heterogeneous media.
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### Numerical resolution of the wave equation on a network of slots

[2] Katrin Boxberger. Propagation d’ ondes dans un réseau de fentes minces: Étude math- ématique et simulation. Master’s thesis, Université Paris-Sud, 2009. [3] Gary Cohen, Patrick Joly, and Nathalie Tordjman. Higher-order finite elements with mass-lumping for the 1D wave equation. Finite Elem. Anal. Des., 16(3-4):329–336, 1994. ICOSAHOM ’92 (Montpellier, 1992).

### Asymptotics of the solutions of the stochastic lattice wave equation

Abstract We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy, and in the unpinned case also conserves the momentum. We obtain a time-inhomogeneous Ornstein- Uhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit is understood in the point-wise sense in the former case, and in the weak sense in the latter. On the other hand, the weak limit for square integrable initial data is deterministic.
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### PI regulation control of a 1-D semilinear wave equation

Adjacently, this discussion raises the problem of discussing the numerical efficiency of vari- ous control design approaches. It would be interesting to compare our approach, developed in the present paper, with other possible approaches and to compare their efficiency. One different approach that may come to our mind is backtepping design, which is well known to promote robustness properties (see, e.g., [ 26 ]), at least, for parabolic equations. It is not clear whether backstepping could be performed for the 1-D wave equation investigated in the present article.

### Determining the potential in a wave equation without a geometric condition. Extension to the heat equation

 . The proof of Theorem 1.1 we present here follows the method initiated by the first and second authors in [2]. This method is mainly based on a spectral decomposition combined with an observability inequality. Due to the fact that we do not assume any geometric condition on the domain, the classical observability inequality is no longer valid in our case. We substitute it by an interpolation inequality established by Robbiano in [9]. It is worthwhile to mention that a spectral decomposition combined with an observability inequality was also used in [3] to establish a logarithmic stability estimate for the problem of determining a boundary coefficient in a wave equation from boundary measurements.
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### Optimal design of boundary observers for the wave equation

[12] P. Jounieaux, Y. Privat, E. Tr´ elat, Optimal boundary observation domain for the wave equation, ongoing work. [13] C. Morawetz, Notes on time decay and scatteringfor some hyperbolic problems, Regional conference series in applied mathe- matics 19, SIAM, Philadelphia (1975). [14] C. B. Morrey, On the Analyticity of the Solutions of Analytic Non-Linear Elliptic Systems of Partial Differntial Equations, American Journal of Mathematics, Vol. 80, No. 1 (Jan., 1958),, pp. 198-237.

### Low frequency dispersive estimates for the wave equation in higher dimensions

) satisfying V (x) = O hxi −(n+1)/2−ǫ  , ǫ > 0. 1 Introduction and statement of results High frequency dispersive estimates with loss of (n − 3)/2 have been recently proved in [9] for the wave equation with a real-valued potential V ∈ L ∞ (R n ), n ≥ 4, satisfying

### Energy decay for the damped wave equation under a pressure condition

1. Introduction One of the standard questions in geometric control theory concerns the so-called sta- bilization problem: given a dissipative wave equation on a manifold, one is interested in the behaviour of the solutions and their energies for long times. The answers that can be given to this problem are closely related to the underlying manifold and the geometry of the control (or damping) region.

### 2-D non-periodic homogenization of the elastic wave equation: SH case

For Peer Review 3.2 Set up of the homogenization problem in the nonperiodic case In 1D, Capdeville et al. (2009a) applied the previous filtering operation to physical quantities that were explicitly known a priori. For wave propagation in higher dimensions, we do not have access to this kind of information, and do not know how to separate scales for density and elastic constants, in order to proceed to an homogenization of these quantities and of the wave equation. In other words, we do not know, for a given distribution of material properties, how to construct the x and y contributions of ρ and µ, from ρ 0 and µ 0 . To that purpose, we then present here, an original but heuristic procedure. Classically a small parameter ε is introduced to solve the so-called two-scale homogenization prob- lems:
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### AN ANALYSIS OF HIGHER ORDER BOUNDARY CONDITIONS FOR THE WAVE EQUATION

1. Introduction. The design of accurate absorbing boundary conditions (ABCs) for the numerical calculation of waves in the time domain is already an old subject since the major work of Engquist and Majda [11], [12] in the late 1970s. Their main contribution was the construction and analysis of a hierarchy of local boundary condi- tions for the wave equation. Let us concentrate on the two-dimensional (2D) acoustic wave equation:

### DECAY FOR THE KELVIN-VOIGT DAMPED WAVE EQUATION: PIECEWISE SMOOTH DAMPING

PIECEWISE SMOOTH DAMPING NICOLAS BURQ AND CHENMIN SUN Abstract. We study the energy decay rate of the Kelvin-Voigt damped wave equation with piecewise smooth damping on the multi-dimensional domain. Under suitable geometric as- sumptions on the support of the damping, we obtain the optimal polynomial decay rate which turns out to be different from the one-dimensional case studied in [LR05]. This optimal de- cay rate is saturated by high energy quasi-modes localised on geometric optics rays which hit the interface along non orthogonal neither tangential directions. The proof uses semi-classical analysis of boundary value problems.
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### Efficient solution of a wave equation with fractional-order dissipative terms

0. Introduction The Webster–Lokshin system is a dissipative model that describes acoustic waves traveling in a duct with visco-thermal losses at the lateral walls. This system couples a wave equation with spatially-varying coefficients with absorbing terms involving fractional-order integrals and derivatives. The main goal of this article is to propose an efficient numerical discretization of this type of model that, in particular, would avoid storing the solution from all the past time steps, because that would be too computationally penalizing in long time simulations.

### LAPLACE EIGENFUNCTIONS AND DAMPED WAVE EQUATION II: PRODUCT MANIFOLDS

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