# Workshop Programme

## for period 12 - 16 December 2011

### Inverse Problems in Science and Engineering

12 - 16 December 2011

Timetable

 Tuesday 13 December 09:00-09:30 Betcke, T (University College London) Modulated plane wave methods for Helmholtz problems in heterogeneous media Sem 1 A major challenge in seismic imaging is full waveform inversion in the frequency domain. If an acoustic model is assumed the underlying problem formulation is a Helmholtz equation with varying speed of sound. Typically, in seismic applications the solution has many wavelengths across the computational domain, leading to very large linear systems after discretisation with standard finite element methods. Much progress has been achieved in recent years by the development of better preconditioners for the iterative solution of these linear systems. But the fundamental problem of requiring many degrees of freedom per wavelength for the discretisation remains. For problems in homogeneous media, that is, spatially constant wave velocity, plane wave finite element methods have gained significant attention. The idea is that instead of polynomials on each element we use a linear combination of oscillatory plane wave solutions. These basis functions already oscillate with the right wavelength, leading to a significant reduction in the required number of unknowns. However, higher-order convergence is only achieved for problems with constant or piecewise constant media. In this talk we discuss the use of modulated plane waves in heterogeneous media, products of low-degree polynomials and oscillatory plane wave solutions for a (local) average homogeneous medium. The idea is that high-order convergence in a varying medium is recovered due to the polynomial modulation of the plane waves. Wave directions are chosen based on information from raytracing or other fast solvers for the eikonal equation. This approach is related to the Amplitude FEM originally proposed by Giladi and Keller in 2001. However, for the assembly of the systems we will use a discontinuous Galerkin method, which allows a simple way of incorporating multiple phase information in one element. We will discuss the dependence of the element sizes on the wavelenth and the accuracy of the phase information, and present several examples that demonstrate the properties of modulated plane wave methods for heterogeneous media problems. 09:30-10:00 Stolk, C (University of Amsterdam) Seismic inverse scattering by reverse time migration Sem 1 We will consider the linearized inverse scattering problem from seismic imaging. While the first reverse time migration algorithms were developed some thirty years ago, they have only recently become popular for practical applications. We will analyze a modification of the reverse time migration algorithm that turns it into a method for linearized inversion, in the sense of a parametrix. This is proven using tools from microlocal analysis. We will also discuss the limitations of the method and show some numerical results. 10:00-10:30 de Hoop, M (Purdue University) Local analysis of the inverse problem associated with the Helmholtz equation -- Lipschitz stability and iterative reconstruction Sem 1 We consider the Helmholtz equation on a bounded domain, and the Dirichlet-to-Neumann map as the data. Following the work of Alessandrini and Vessalla, we establish conditions under which the inverse problem defined by the Dirichlet-to-Neumann map is Lipschitz stable. Recent advances in developing structured massively parallel multifrontal direct solvers of the Helmholtz equation have motivated the further study of iterative approaches to solving this inverse problem. We incorporate structure through conormal singularities in the coefficients and consider partial boundary data. Essentially, the coefficients are finite linear combinations of piecewise constant functions. We then establish convergence (radius and rate) of the Landweber iteration in appropriately chosen Banach spaces, avoiding the fact the coefficients originally can be $L^{\infty}$, to obtain a reconstruction. Here, Lipschitz (or possibly Hoelder) stability replaces the so-called source condition. We accommodate the exponential growth of the Lipschitz constant using approximations by finite linear combinations of piecewise constant functions and the frequency dependencies to obtain a convergent projected steepest descent method containing elements of a nonlinear conjugate gradient method. We point out some correspondences with discretization, compression, and multigrid techniques. Joint work with E. Beretta, L. Qiu and O. Scherzer. 10:30-11:00 Olsson, P (Chalmers University of Technology) Re-routing of elastodynamic waves by means of transformation optics in planar, cylindrical, and spherical geometries Sem 1 Transformation optics has proven a powerful tool to achieve cloaking from electromagnetic and acoustic waves. There are still technical issues with applications of transformation optics to elastodynamics, due to the fact that the elastodynamic wave equation does not in general possess suitable invariances under the required transformations. However, for a few types of materials, invariances of the appropriate kind have been shown to exist. In the present talk we consider a few canonical scattering and reflection problems, and show that by coating the planar, cylindrical or spherical reflecting or scattering bodies with a fiber-reinforced layer of a metamaterial with a suitable gradient in material properties, the reflection or scattering of shear waves from the body can be significantly reduced. It has been suggested that constructions inspired by transformation optics could potentially provide protection for infrastructure from seismic waves. Even if waves from earthquakes may have wavelengths making some such suggestions implausible, passive protection from shorter elastic bulk waves from other sources may be achieved by a scheme based on transformation optics. Other suggested applications are in the car and aeronautics industries. The problems considered here, albeit rather special model problems, hopefully may provide some additional insight into protection against mechanical waves by means of transformation elastodynamics. A result of the analysis in the case of a spherical case is, that to maximize number of modes to which the coated spherical body is “invisible,” rigid body rotations of the innermost part of the coating should be allowed. (However, this is only essential in the low frequency range.) It is also worth noting that since the transition matrices of scatterers described here have, as it were, quite well-populated null-spaces, they provide simple examples of cases where complete knowledge of the scatterer and of the scattered field does not even remotely suffice to reconstruct the incident field. 11:00-11:30 Morning Coffee 11:30-12:30 Informal Discussion Time 12:30-13:30 Lunch at Wolfson Court 14:00-14:30 Pietschmann, J-F; Burger, M; Wolfram, M-T (Universität Münster/Vienna) Identification of non-linearities in transport-diffusion models of crowded motion Sem 1 14:30-15:00 van Leeuwen, P (University of Reading) Particle filters in highly nonlinear high-dimensional systems Sem 1 Bayes theorem formulates the data-assimilation problem as a multiplication problem and not an inverse problem. In this talk we exploit that using an extremely efficient particle filter on a highly nonlinear geophysical fluid flow problem of dimension 65,000. We show how collapse of the particles can be avoided, and discuss statistics showing that the particle filter is performing correctly. 15:00-15:30 Afternoon Tea 15:30-16:00 Omre, H (University of Science and Technology, Trondheim) Spatial categorical inversion: Seismic inversion into lithology/fluid classes Sem 1 Modeling of discrete variables in a three-dimensional reference space is a challenging problem. Constraints on the model expressed as invalid local combinations and as indirect measurements of spatial averages add even more complexity. Evaluation of offshore petroleum reservoirs covering many square kilometers and buried at several kilometers depth contain problems of this type. Foc us is on identification of hydrocarbon (gas or oil) pockets in the subsurface - these appear as rare events. The reservoir is classified into lithology (rock) cla sses - shale and sandstone - and the latter contains fluids - either gas, oil or brine (salt water). It is known that these classes are vertically thin with large horizontal continuity. The reservoir is considered to be in equilibrium - hence fixed vertical sequences of fluids - gas/oil/brine - occur due to gravitational sorting. Seismic surveys covering the reservoir is made and through processing of the data, angle-dependent amplitudes of reflections are available. Moreover, a few wells are drilled through the reservoir and exact obse rvations of the reservoir properties are collected along the well trace. The inversion is phrased in a hierarchical Bayesian inversion framework. The prior model, capturing the geometry and ordering of the classes, is of Markov random field type. A particular parameterization coined Profile Markov random field is def ined. The likelihood model linking lithology/fluids and seismic data captures maj or characteristics of rock physics models and the wave equation. Several parameters in this likelihood model are considered to be stochastic and they are inferred from seismic data and observations along the well trace. The posterior model is explored by an extremely efficient MCMC-algorithm. The methodology is defined and demonstrated on observations from a real North Sea reservoir. 16:00-16:30 Farmer, CL (University of Oxford) Practical and principled methods for large-scale data assimilation and parameter estimation Sem 1 Uncertainty quantification can begin by specifying the initial state of a system as a probability measure. Part of the state (the 'parameters') might not evolve, and might not be directly observable. Many inverse problems are generalisations of uncertainty quantification such that one modifies the probability measure to be consistent with measurements, a forward model and the initial measure. The inverse problem, interpreted as computing the posterior probability measure of the states, including the parameters and the variables, from a sequence of noise corrupted observations, is reviewed in the talk. Bayesian statistics provides a natural framework for a solution but leads to very challenging computational problems, particularly when the dimension of the state space is very large, as when arising from the discretisation of a partial differential equation theory. In this talk we show how the Bayesian framework provides a unification of the leading techniques in use today. In particular the framework provides an interpretation and generalisation of Tikhonov regularisation, a method of forecast verification and a way of quantifying and managing uncertainty. A summary overview of the field is provided and some future problems and lines of enquiry are suggested. 16:30-17:00 Oliver, D (University Centre for Integrated Petroleum Research in Bergen, Norway) The ensemble Kalman filter for distributed parameter estimation in porous media flow Sem 1 17:00-17:30 Dashti, M (University of Warwick) Besov Priors for Bayesian Inverse problems Sem 1 We consider the inverse problem of estimating a function $u$ from noisy measurements of a known, possibly nonlinear, function of $u$. We use a Bayesian approach to find a well-posed probabilistic formulation of the solution to the above inverse problem. Motivated by the sparsity promoting features of the wavelet bases for many classes of functions appearing in applications, we study the use of the Besov priors within the Bayesian formalism. This is Joint work with Stephen Harris (Edinburgh) and Andrew Stuart (Warwick). 18:45-19:15 Dinner at Fitzwilliam College (residents only)