Space time stabilized finite element methods for a unique continuation problem subject to the wave equation

DOI10.1051/m2an/2020062zbMath1487.65131arXiv1912.00891OpenAlexW3083793324MaRDI QIDQ4958862

Ali Feizmohammadi, Arnaud Münch, Erik Burman, Lauri Oksanen

Publication date: 15 September 2021

Published in: ESAIM: Mathematical Modelling and Numerical Analysis (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1912.00891

zbMATH Keywords

wave equationfinite element methodunique continuationgeometric controldata assimilationobservability estimatecondition

Mathematics Subject Classification ID

Numerical optimization and variational techniques (65K10) Geometric methods (93B27) Inverse problems for PDEs (35R30) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Finite difference methods for initial value and initial-boundary value problems involving PDEs (65M06) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Hydro- and aero-acoustics (76Q05) Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs (65M60) Error bounds for initial value and initial-boundary value problems involving PDEs (65M15) Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs (65M32)

Related Items (8)

The Morozov’s principle applied to data assimilation problems ⋮ Least squares solvers for ill-posed PDEs that are conditionally stable ⋮ The Unique Continuation Problem for the Heat Equation Discretized with a High-Order Space-Time Nonconforming Method ⋮ Controllability of the linear elasticity as a first-order system using a stabilized space-time mixed formulation ⋮ Numerical approximation of the boundary control for the wave equation in a square domain with a spectral collocation method ⋮ Spacetime finite element methods for control problems subject to the wave equation ⋮ A Hybridized High-Order Method for Unique Continuation Subject to the Helmholtz Equation ⋮ Accuracy controlled data assimilation for parabolic problems

Uses Software

• FreeFem++

Cites Work

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• Error estimates for stabilized finite element methods applied to ill-posed problems
• Resolvent conditions for the control of unitary groups and their approximations
• A mixed formulation for the direct approximation of the control of minimal \(L^2\)-norm for linear type wave equations
• Geometric control condition for the wave equation with a time-dependent observation domain
• Recovering the initial state of an infinite-dimensional system using observers
• Reconstructing initial data using observers: Error analysis of the semi-discrete and fully discrete approximations
• Data assimilation for the heat equation using stabilized finite element methods
• An inverse obstacle problem for the wave equation in a finite time domain
• Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods
• Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity
• Theory and practice of finite elements.
• Approximation of controls for linear wave equations: a first order mixed formulation
• Space-time discontinuous Petrov-Galerkin methods for linear wave equations in heterogeneous media
• Back and forth nudging algorithm for data assimilation problems
• Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. (On a variational principle for solving Dirichlet problems less boundary conditions using subspaces)
• A stabilized nonconforming finite element method for the elliptic Cauchy problem
• Time reversal method with stabilizing boundary conditions for Photoacoustic tomography
• A Dissipative Time Reversal Technique for Photoacoustic Tomography in a Cavity
• A Fully Discrete Numerical Control Method for the Wave Equation
• Finite Element Interpolation of Nonsmooth Functions Satisfying Boundary Conditions
• Mathematics of thermoacoustic tomography
• The Quasi-Reversibility Method for Thermoacoustic Tomography in a Heterogeneous Medium
• Basis functions for general Hsieh-Clough-Tocher triangles, complete or reduced
• Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data
• Numerical solution of a time-like Cauchy problem for the wave equation
• Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary
• Boundary observability for the space semi-discretizations of the 1 – d wave equation
• Fully discrete finite element data assimilation method for the heat equation
• Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis
• Numerical Approximation of Exact Controls for Waves
• New development in freefem++
• A finite element data assimilation method for the wave equation
• Thermoacoustic tomography with variable sound speed
• Controllability of the Linear One-dimensional Wave Equation with Inner Moving Forces
• Multiwave tomography in a closed domain: averaged sharp time reversal
• Multiwave imaging in an enclosure with variable wave speed
• A uniformly controllable and implicit scheme for the 1-D wave equation
• Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods
• Stabilized Finite Element Methods for Nonsymmetric, Noncoercive, and Ill-Posed Problems. Part I: Elliptic Equations
• The Mathematical Theory of Finite Element Methods
• Inverse problems for linear hyperbolic equations using mixed formulations

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