10 research outputs found
Quantitative Convergence and Stability of Seismic Inverse Problems.
International audienc
The output least squares identifiability of the diffusion coefficient from an -observation in a 2-D elliptic equation
Output least squares stability for the diffusion coefficient in an elliptic equation in dimension
two is analyzed. This guarantees Lipschitz stability of the solution of the least squares
formulation with respect to perturbations in the data independently of their attainability.
The analysis shows the influence of the flow direction on the parameter to be estimated.
A scale analysis for multi-scale resolution of the unknown parameter is provided
Convergence guarantees for coefficient reconstruction in PDEs from boundary measurements by variational and Newton type methods via range invariance
A key observation underlying this paper is the fact that the range invariance
condition for convergence of regularization methods for nonlinear ill-posed
operator equations -- such as coefficient identification in partial
differential equiations (PDE)s from boundary observations -- can often be
achieved by extending the seached for parameter in the sense of allowing it to
depend on additional variables. This clearly counteracts unique identifiability
of the parameter, though. The second key idea of this paper is now to restore
the original restricted dependency of the parameter by penalization. This is
shown to lead to convergence of variational (Tikhonov type) and iterative
(Newton type) regularization methods. We concretize the abstract convergence
analysis in a framework typical of parameter identification in PDEs in a
reduced and an all-at-once setting. This is further illustrated by three
examples of coefficient identification from boundary observations in elliptic
and time-dependent PDEs
Well-Behavior, Well-Posedness and Nonsmooth Analysis
AMS subject classification: 90C30, 90C33.We survey the relationships between well-posedness and well-behavior. The latter
notion means that any critical sequence (xn) of a lower semicontinuous function
f on a Banach space is minimizing. Here “critical” means that the remoteness of
the subdifferential ∂f(xn) of f at xn (i.e. the distance of 0 to ∂f(xn)) converges
to 0. The objective function f is not supposed to be convex or smooth and the
subdifferential ∂ is not necessarily the usual Fenchel subdifferential. We are thus
led to deal with conditions ensuring that a growth property of the subdifferential
(or the derivative) of a function implies a growth property of the function itself.
Both qualitative questions and quantitative results are considered
Recommended from our members
Computational Inverse Problems for Partial Differential Equations
The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges
A priori estimates of attraction basins for nonlinear least squares, with application to Helmholtz seismic inverse problem
International audienceIn this paper, we provide an a priori optimizability analysis of nonlinear least squares problems that are solved by local optimization algorithms. We define attraction (convergence) basins where the misfit functional is guaranteed to have only one local-and hence global-stationary point, provided the data error is below some tolerable error level. We use geometry in the data space (strictly quasiconvex sets) in order to compute the size of the attraction basin (in the parameter space) and the associated tolerable error level (in the data space). These estimates are defined a priori, i.e., they do not involve any least squares minimization problem, and only depend on the forward map. The methodology is applied to the comparison of the optimizability properties of two methods for the seismic inverse problem for a time-harmonic wave equation: the Full Waveform Inversion (FWI) and its Migration Based Travel Time (MBTT) reformulation. Computation of the size of attraction basins for the two approaches allows to quantify the benefits of the latter, which can alleviate the requirement of low-frequency data for the reconstruction of the background velocity model