A refinement and coarsening indicator algorithm for finding sparse solutions of inverse problems
Barbara Kaltenbacher Jonas Offtermatt
Inverse Problems & Imaging 2011, 5(2): 391-406 doi: 10.3934/ipi.2011.5.391
In this paper we extend the idea of adaptive discretization by using refinement and coarsening indicators from papers by Chavent, Bissell, Benameur and Jaffré (cf., e.g., [5], [9]) to a general setting. This allows to make use of the relation between adaptive discretization and sparse paramerization in order to construct an algorithm for finding sparse solutions of inverse problems. We provide some first steps in the analysis of the proposed method and apply it to an inverse problem in systems biology, namely the reconstruction of gene networks in an ordinary differential equation (ODE) model. Here due to the fact that not all genes interact with each other, reconstruction of a sparse connectivity matrix is a key issue.
keywords: adaptive discretization sparsity systems biology. Nonlinear inverse problems
Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions
Barbara Kaltenbacher Irena Lasiecka
Conference Publications 2011, 2011(Special): 763-773 doi: 10.3934/proc.2011.2011.763
In this paper we show wellposedness of two equations of nonlinear acoustics, as relevant e.g. in applications of high intensity ultrasound. After having studied the Dirichlet problem in previous papers, we here consider Neumann boundary conditions which are of particular practical interest in applications. The Westervelt and the Kuznetsov equation are quasilinear evolutionary wave equations with potential degeneration and strong damping. We prove local in time well-posedness as well as global existence and exponential decay for a slightly modi ed model. A key step of the proof is an appropriate extension of the Neumann boundary data to the interior along with exploitation of singular estimates associated with the analytic semigroup generated by the strongly damped wave equation.
keywords: nonlinear acoustics local and global well- posedness Kuznetsov's equation
Well-posedness and asymptotic behavior of solutions for the Blackstock-Crighton-Westervelt equation
Rainer Brunnhuber Barbara Kaltenbacher
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4515-4535 doi: 10.3934/dcds.2014.34.4515
We consider a nonlinear fourth order in space partial differential equation arising in the context of the modeling of nonlinear acoustic wave propagation in thermally relaxing viscous fluids.
    We use the theory of operator semigroups in order to investigate the linearization of the underlying model and see that the underlying semigroup is analytic. This leads to exponential decay results for the linear homogeneous equation.
    Moreover, we prove local in time well-posedness of the model under the assumption that initial data are sufficiently small by employing a fixed point argument. Global in time well-posedness is obtained by performing energy estimates and using the classical barrier method, again for sufficiently small initial data.
    Additionally, we provide results concerning exponential decay of solutions of the nonlinear equation.
keywords: asymptotic behavior. well-posedness Nonlinear acoustics
Global existence and exponential decay rates for the Westervelt equation
Barbara Kaltenbacher Irena Lasiecka
Discrete & Continuous Dynamical Systems - S 2009, 2(3): 503-523 doi: 10.3934/dcdss.2009.2.503
We consider the Westervelt equation which models propagation of sound in a fluid medium. This is an accepted in nonlinear acoustics model which finds a multitude of applications in medical imaging and therapy. The PDE model consists of the second order in time evolution which is both quasilinear and degenerate. Degeneracy depends on the fluctuations of the acoustic pressure.
    Our main results are : (1) global well-posedness, (2) exponential decay rates for the energy function corresponding to both weak and strong solutions. The proof is based on (i) application of a suitable fixed point theorem applied to an appropriate formulation of the PDE which exhibits analyticity properties of the underlying linearised semigroup, (ii) exploitation of decay rates associated with the dissipative mechanism along with barrier's method leading to global wellposedness. The obtained result holds for all times, provided that the initial data are taken from a suitably small ball characterized by the parameters of the equation.
keywords: Westervelt equation exponential decay. global existence
Avoiding degeneracy in the Westervelt equation by state constrained optimal control
Christian Clason Barbara Kaltenbacher
Evolution Equations & Control Theory 2013, 2(2): 281-300 doi: 10.3934/eect.2013.2.281
The Westervelt equation, which describes nonlinear acoustic wave propagation in high intensity ultrasound applications, exhibits potential degeneracy for large acoustic pressure values. While well-posedness results on this PDE have so far been based on smallness of the solution in a higher order spatial norm, non-degeneracy can be enforced explicitly by a pointwise state constraint in a minimization problem, thus allowing for pressures with large gradients and higher-order derivatives, as is required in the mentioned applications. Using regularity results on the linearized state equation, well-posedness and necessary optimality conditions for the PDE constrained optimization problem can be shown via a relaxation approach by Alibert and Raymond [2].
keywords: singular PDEs nonlinear acoustics. Westervelt equation Optimal control of PDEs state constraints
On Levenberg-Marquardt-Kaczmarz iterative methods for solving systems of nonlinear ill-posed equations
Johann Baumeister Barbara Kaltenbacher Antonio Leitão
Inverse Problems & Imaging 2010, 4(3): 335-350 doi: 10.3934/ipi.2010.4.335
In this article a modified Levenberg-Marquardt method coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations is investigated. We show that the proposed method is a convergent regularization method. Numerical tests are presented for a non-linear inverse doping problem based on a bipolar model.
keywords: Kaczmarz method. Levenberg-Marquardt method Regularization Nonlinear systems Ill-posed equations
Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling
Rainer Brunnhuber Barbara Kaltenbacher Petronela Radu
Evolution Equations & Control Theory 2014, 3(4): 595-626 doi: 10.3934/eect.2014.3.595
In this paper we show local (and partially global) in time existence for the Westervelt equation with several versions of nonlinear damping. This enables us to prove well-posedness with spatially varying $L_\infty$-coefficients, which includes the situation of interface coupling between linear and nonlinear acoustics as well as between linear elasticity and nonlinear acoustics, as relevant, e.g., in high intensity focused ultrasound (HIFU) applications.
keywords: nonlinear damping quasilinear wave equation local existence. Nonlinear acoustics
Preface: Introduction to the Special Volume on Nonlinear PDEs and Control Theory with Applications
Lorena Bociu Barbara Kaltenbacher Petronela Radu
Evolution Equations & Control Theory 2013, 2(2): i-ii doi: 10.3934/eect.2013.2.2i
This volume collects a number of contributions in the fields of partial differential equations and control theory, following the Special Session Nonlinear PDEs and Control Theory with Applications held at the 9th AIMS conference on Dynamical Systems, Differential Equations and Applications in Orlando, July 1--5, 2012.

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