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Workshop on

Large-Scale Inverse Problems and Applications in the Earth Sciences

October 24-28, 2011

as part of the

Radon Special Semester 2011 on

Multiscale Simulation & Analysis in Energy and the Environment

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This workshop will focus on the large inverse problems commonly arising in simulation and forecasting in the earth sciences. For example, operational weather forecasting models have between 107 and 108 degrees of freedom. Even so, these degrees of freedom represent grossly space-time averaged properties of the atmosphere. Accurate forecasts require accurate initial conditions. With recent developments in satellite data, there are between 106 and 107 observations each day. However, while these also represent space-time averaged properties, the averaging implicit in the measurements is quite different from that used in the models. In atmosphere and ocean applications, there is a physically-based model available which can be used to regularise the problem. We assume that there is a set of observations with known error characteristics available over a period of time. The basic deterministic technique is to fit a model trajectory to the observations over a period of time to within the observation error. Since the model is not perfect the model trajectory has to be corrected, which defines the data assimilation problem. The stochastic view can be expressed by using an ensemble of model trajectories, and calculating corrections to both the mean value and the spread which allow the observations to be fitted by each ensemble member.

In other areas of earth science, only the structure of the model formulation itself is known and the aim is to use the past observation history to determine the unknown model parameters.

The workshop will involve experts in the theory of inverse problems together with experts working on both theoretical and practical aspects of the techniques by which large inverse problems arise in the earth sciences.

Workshop Organizers

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Welcome

to Linz and thank you very much for participating in the sixth RICAM Special Semester on Multi- scale Simulation & Analysis in Energy and the Environment, hosted by the Johann Radon Insitute for Computational and Applied Mathematics (RICAM) from October 3 to December 16, 2011.

Technological advances have greatly improved our quality of life. However, they bring with them a huge surge in energy requirements which in turn puts at risk our entire bio-sphere. It is of paramount importance to predict these risks and to develop better solutions for the future. One of the central tasks is the accurate simulation of multiphase flow above and under ground. The risk analysis and uncertainty quantification, as well as the assimilation of data require statistical tools and efficient solvers for stochastic and deterministic PDEs as well as for the associated inverse problems. The key features that make it extremely hard to predict these physical phenomena accurately are the multiple time and length scales that arise, as well as the lack of and uncertainty in data. Because of the highly varying scales involved, the resolution of all scales is currently impossible even on the largest supercomputers. While there is a fairly long history of empirically successful robust computational techniques for certain multiscale problems, the rigorous (numerical) analysis of such methods is of extremely high current interest.

The goal of the special semester is to provide a stimulating environment for civil engineers, hydrologists, meteorologists and other environmental scientists to address together with mathematicians working at the cutting edge of rigorous numerical analysis for multiscale (direct and inverse) problems the emerging challenges in the quantitative assessment of the risks and uncertainties of atmospheric and subsurface flow, focusing in particular on

• Simulation of Flow in Porous Media and Applications in Waste Management and CO2Sequestration

• Large-Scale Inverse Problems and Applications in the Earth Sciences

• Data Assimilation and Multiscale Simulation in Atmospheric Flow

• Wave Propagation and Scattering, Direct and Inverse Problems and Applications in Energy and the Environment

• Multiscale Numerical Methods and their Analysis and Applications in Energy and the Environment

• Stochastic Modelling of Uncertainty and Numerical Methods for Stochastic PDEs Specific activities planned for the Special Semester are

• 4 thematic workshops addressing some of the key topics of the Special Semester;

• Special Lecture Series on ”Multilevel Methods for Multiscale Problems”;

• Graduate Seminar on ”Multiscale Discretization Techniques”;

• Wednesday Research Kitchen;

• Public Lecture byProf. J¨orn Behrens (KlimaCampus, Universit¨at Hamburg) on

“Tsunami Fr¨uh-Warnung: Mathematik und Wissenschaftliches Rechnen im Dienste der Sicherheit”.

We sincerely hope that you enjoy your stay in Linz!

Local Organizing Committee Program Committee

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Contents

Information 2

Workshop Information . . . 2

Social Events . . . 2

Restaurants and Cafes . . . 4

General Information . . . 4

Program 7

Posters 9

Abstracts 10

Abstracts for Posters 18

List of Participants 22

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Information

Workshop Information

Registration. The workshop registration will be on October 24th, 2011 from 8:30 - 9:00 am next to the seminar room SP2 416 on the 4th floor of the Science Park Building 2 (see floor plan). Participants that arrive later in the week can register at the special semester office SP2 456.

Registration Fee. Non-invited participants are kindly asked to pay the registration fee in cash upon registration.

Campus plan and overview map as well as a floor plan of the 4th floor of the workshop venue (Sci- ence Park Building 2) are located on the next pages.

Seminar room. The workshop will take place in seminar room SP2 416 on the 4th floor of the Science Park Building 2 (see floor plan).

Program. A time schedule for the workshop is located on the backside of this booklet.

Coffee breaks. The coffee breaks will be in the corridor of the 4th floor of the Science Park Building 2.

Internet access. There will be an extra information sheet regarding internet access available at regis- tration.

Social Events

Welcome Reception & Poster Session. Monday, October 24th, 2011, 5:00 pm, at the 4th floor of the Science Park Building 2.

Excursion & Conference Dinner. There will be an excursion on Wednesday, October 26th, 2011.

The bus leaves from Sommerhaus hotel at 12:00 pm and it will bring us to the restaurant “Donauschlinge”

(www.donauschlinge.at) in Schl¨ogen at theSchl¨ogener Schlinge of the danube (see left picture in Figure 1). Afterwards, we will go to the Baumkronenweg (www.baumkronenweg.at) in Kopfing (see right picture in Figure 1), a unique experience of walking in the forest canopy. The conference dinner will be held in the restaurant “Gasthof Oachkatzl” at the “Baumkronenweg”. The bus will bring us back after the dinner and will return to the Sommerhaus hotel at around 9:30 pm.

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Restaurants and Cafes

• Mensa Markt (lunch time only) - Main canteen of the University (see campus plan)

• KHG Mensa (lunch time only) - Smaller canteen - good traditional food (see overview map: “KHG Linz”)

• Pizzeria “Bella Casa” - Italian and Greek restaurant (located next to the tram stop)

• Chinese restaurant “Jadegarten” - (located close by the tram stop, adjacent to “Bella Casa”)

• Asia restaurant “A2” - (located behind the Science Park on Altenbergerstrasse)

• “Chat” cafe - coffee, drinks and sandwiches (located in the “H¨orsaaltrakt” - see overview map)

• Cafe “Sassi” - coffee, drinks and small snacks (located in the building “Johannes Kepler Universit¨at”

- see overview map)

• Bakery “Kandur” - bakery and small cafe (located opposite the tram stop)

General Information

Accommodation. The arranged accomodation for invited participants is the “Sommerhaus” hotel. You can find its location in the overview map on page 4.

Special Semester Office: Room SP2 456. The special semester administrator is Susanne Dujardin.

Audiovisual & Computer Support. Room SP2 458, Wolfgang Forsthuber or Florian Tischler.

Orientation/ Local Transport. From the railway station you have to take tram number 1 or 2 in direction “Universit¨at”. It takes about 25 minutes to reach the desired end stop “Universit¨at”.

In order to get to the city center of Linz (“Hauptplatz”) and back you have to take again tram number 1 or 2 (about 20 minutes). For more information seewww.ricam.oeaw.ac.at/location/.

Taxi Numbers.

+43 732 6969 Ober¨osterreichische Taxigenossenschaft +43 732 2244 2244 Linzer Taxi

+43 732 781463 Enzendorfer Taxi & Transport +43 732 2214 Linzer Taxi

+43 732 660217 LINTAX TaxibetriebsgesmbH Further important phone numbers.

+43 (0)732 2468 5222 RICAM & Special Semester Office (Susanne Dujardin) +43 (0)732 2468 5250/5255 RICAM IT Support (Florian Tischler/ Wolfgang Forsthuber) +43 (0)732 2457-0 Reception of Hotel Sommerhaus

133 General emergency number for the police

144 General emergency number for the ambulance

More information about RICAM can be found at www.ricam.oeaw.ac.at. See also the Special Semester webpagewww.ricam.oeaw.ac.at/specsem/specsem2011/for additional information.

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Figure 2: 4th floor of Science Park Building 2.

Figure 3: Campus plan

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Figure 4: Overview map

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Program

Monday, October 24th

08:30 - 09:00 Registration 09:00 - 09:20 Opening

09:20 - 10:10 Olaf A. Cirpka (University of T¨ubingen)

“Geostatistical Approaches for Inverse Modeling in Subsurface Hydrology”

10:10 - 10:40 Coffee Break

10:40 - 11:30 Mike Cullen (Met Office, Exeter, UK)

“4dVar in the Presence of Model Error”

11:30 - 14:00 Lunch Break

14:00 - 14:50 Amos S. Lawless (University of Reading)

“Optimal State Estimation for Numerical Weather Prediction using Reduced Order Models”

14:50 - 15:20 Coffee Break

15:20 - 16:10 Tim Payne (Met Office, Exeter, UK)

“The Construction and Use of Linear Models in Large-Scale Data Assimilation”

16:10 - 17:00 Melina Freitag (University of Bath)

“Resolution of Sharp Fronts in the Presence of Model Error in Variational Data Assimilation”

17:00 Poster Session & Welcome Reception

Tuesday, October 25th

08:30 - 09:20 Stefan Kindermann (Johannes Kepler University Linz)

“Inverse Problems and Regularization - an Introduction”

09:20 - 10:10 Michael Fisher (European Centre for Medium-Range Weather Forecasts)

“Parallel Algorithms for Four-Dimensional Variational Data Assimilation”

10:10 - 10:40 Coffee Break

10:40 - 11:30 Serge Gratton (CERFACS-IRIT, France)

“Dual Methods for Data Assimilation in Geosciences”

11:30 - 14:00 Lunch Break

14:00 - 14:50 Uri Ascher ( University of BC, Vancouver)

“Adaptive and Stochastic Algorithms for piecewise constant EIT and DC Resistiv- ity Problems with many Measurements”

14:50 - 15:20 Coffee Break

15:20 - 16:10 Antonio Leitao (Federal University of St. Catarina)

“On Iterative Regularization Methods for Nonlinear Inverse Problems”

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Wednesday, October 26th

08:30 - 09:20 Tapio Helin (RICAM, Linz)

“On the Bayesian Approach to Inverse Problems”

09:20 - 10:10 Masoumeh Dashti (University of Sussex)

“Besov Priors for Bayesian Inverse Problems”

10:10 - 10:40 Coffee Break

10:40 - 11:30 Barbara Kaltenbacher (Alpen-Adria Universit¨at Klagenfurt)

“Adaptive Discretization Strategies for the Identification of Distributed Parameters in Partial Differential Equations”

12:00 - 21:30 Excursionto the “Schl¨ogener Schlinge” and to the “Baumkronenweg” in Kopfing.

Conference Dinnerat the “Baumkronenweg”.

Thursday, October 27th

08:30 - 09:20 Sebastian Reich (University of Potsdam)

“Ensemble Transform Filters for Geophysical Data Assimilation”

09:20 - 10:10 Martin Burger (Westf¨alische Wilhelms-Universit¨at (WWU) M¨unster)

“4D Variational Models preserving Sharp Edges”

10:10 - 10:40 Coffee Break

10:40 - 11:30 Nancy K. Nichols (University of Reading)

“Conditioning, Preconditioning and Regularization of the Optimal State Estima- tion Problem”

11:30 - 14:00 Lunch Break

14:00 - 14:50 Eldad Haber (The University of British Columbia, Vancouver)

“Numerical Methods for the Design of Large-Scale Nonlinear Discrete Ill-Posed Inverse Problems”

14:50 - 15:20 Coffee Break

15:20 - 16:10 Roland Potthast (Deutscher Wetterdienst, Uni Reading, Uni G¨ottingen)

“On Instability in Data Assimilation”

17:15 - 18:30 Public Lecture:

J¨orn Behrens (KlimaCampus, Universit¨at Hamburg)

“Tsunami Fr¨uh-Warnung: Mathematik und Wissenschaftliches Rechnen im Dien- ste der Sicherheit”

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Friday, October 28th

08:30 - 09:20 Anthony T. Weaver (CERFACS, Toulouse, France)

“Background-Error Correlation Modelling in Variational Assimilation using a Dif- fusion Equation”

09:20 - 10:10 Daniel Wachsmuth (RICAM, Linz)

“Regularization Error Estimates and Discrepancy Principle for Optimal Control Problems with Inequality Constraints”

10:10 - 10:40 Coffee Break

10:40 - 11:30 Olivier Talagrand (Laboratoire de M´et´eorologie Dynamique, Paris)

“Assimilation of Observations and Bayesianity”

11:30 Closing

Posters

The poster session will take place on the 4th floor of the 2nd Science Park Building. It will start at5:00 pmon Monday, October 24th.

Maria Brune (Universit¨at Paderborn)

“Towards Regularization Strategies of Inverse Electromagnetic Scattering Problems”

Siˆan Jenkins (University of Bath)

“The Dissipative and Dispersive Effects of Numerical Models on 4D-Var Data Assimilation”

Ludovic M´etivier (ISTerre, Joseph Fourier University, Grenoble)

“Full Waveform Inversion of Acoustic Density from Well Seismic Data”

Alexander Moodey (University of Reading)

“Instability and Regularization for Data Assimilation”

Valeriya Naumova (Radon Institute for Computational and Applied Mathematics (RICAM), Linz)

“Extrapolation in variable RKHSs with Application to the BG Reading”

Sridhara Nayak (Indian Institute of Technology Kharagpur)

“A Numerical Simulation Study of the Indian Summer Monsoon of 1998 using RegCM3”

Andreas Obereder (MathConsult GmbH, Linz)

“CuRe - A new Wavefront Reconstruction Method for SH-WFS Measurements”

Iuliia Shatokhina (Johannes Kepler University Linz)

“Wavefront Reconstruction for Extreme Adaptive Optics”

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Abstracts

“Adaptive and Stochastic Algorithms for piecewise constant EIT and DC Resistivity Problems with many Measurements”

Uri Ascher

Dept. Computer Science, University of BC, Vancouver, Canada, V6T1Z4

We develop fast numerical methods for the practical solution of the famous EIT and DC-resistivity problems in the presence of discontinuities and potentially many experiments or data.

Based on a Gauss-Newton (GN) approach coupled with preconditioned conjugate gradient (PCG) iterations, we propose two algorithms. One determines adaptively the number of inner PCG iterations required to stably and effectively carry out each GN iteration. The other algorithm, useful especially in the presence of many experiments, employs a randomly chosen subset of experiments at each GN iteration that is controlled using a cross validation approach. Numerical examples demonstrate the efficacy of our algorithms.

“4D Variational Models preserving Sharp Edges”

Martin Burger

Institute for Computational and Applied Mathematics, Westf¨alische Wilhelms-Universit¨at (WWU) M¨unster.

Einsteinstr. 62,

D 48149 M¨unster, Germany.

http://imaging.uni-muenster.de

Reconstructions with sharp edges in two and three dimensions have been an important topic in inverse problems in the last two decades. Recently, motivated by applications in dynamic imaging and data assimilation, extensions to four-dimensional problems (or n spatial plus one time dimension) have gained increasing attention.

In this talk we will focus on reconstruction methods combining transport models with total variation regularization and inverse scale space methods. We will discuss basic issues in the modelling, anal- ysis, numerical solutions. Finally we present an application of the approach to the variational data assimilation setup.

“Geostatistical Approaches for Inverse Modeling in Subsurface Hydrology”

Olaf A. Cirpka University of T¨ubingen, Center for Applied Geoscience, H¨olderlinstr. 12,

72074 T¨ubingen, Germany

Natural formations exhibit tremendous spatial variability in their properties. For subsurface flow, hydraulic conductivity is the most important spatially variable property, followed by specific storativity and porosity. Since the exact distribution of subsurface materials is generally unknown, a statistical description, accounting for both variability and uncertainty, is recommended. Over the past three

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We present approached based on modified Gauss-Newton methods of parameter estimation with geo- statistical regularization. Sensitivities are obtained by adjoint-state methods. A particular emphasis is led on using temporal moments of data streams as a means to condense the information. Temporal- moments generating equations may be used to compute spatial distributions of temporal moments by steady-state calculations rather than postprocessing of transient simulation runs. Successful im- plementations include temporal moments of drawdown during (potentially tomographic) pumping tests, temporal moments of concentration, and temporal moments of geoelectrical signals obtained by monitoring of salt-tracer tests. Special techniques are needed to stabilize inversion of data that show jump-like sensitivity to hydraulic conductivity, such as steady-state concentration measurements.

Application include virtual tests in 2-D and 3-D, bench-scale laboratory experiments, and field surveys.

“4dVar in the Presence of Model Error”

Mike Cullen Met Office, Fitzroy Road,

Exeter, EX1 3PB, UK

We consider formulations of 4dVar in situations where the forecast model is imperfect, the model error cannot be represented by a random forcing and the model error is ’unknowable’. This means that it can only be inferred from current observations. It is shown that the smoothing property of 4dvar allows the model error to be compensated during the assimilation window, and if the model error is correlated in time then this compensation effect extends into the forecast, resulting in more accurate forecasts.

Reductions in forecast error growth can be achieved by matching the model time derivative with that inferred from observations. While this would normally be expected to require a long window, it is shown that a short window coupled with a strong background constraint can achieve the same effect.

This is proved for a simple case amenable to analysis.

“Besov Priors for Bayesian Inverse Problems”

Masoumeh Dashti Mathematics Department, University of Sussex,

Falmer Campus, Brighton BN1 9QH, UK

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).

“Parallel Algorithms for Four-Dimensional Variational Data Assimilation”

Michael Fisher

European Centre for Medium-Range Weather Forecasts

Four-Dimensional Variational Data Assimilation (4D-Var) is used by many national and international numerical weather forecasting centres to provide initial conditions for forecast models. The method solves a very large least-squares problem to determine a statistically optimal balance between the conflicting information provided by a prior estimate and by inhomogeneous, spatially and temporally distributed observations. The solution takes into account knowledge of the dynamical evolution of the system, as embodied in the numerical model, and optionally takes into account statistical information about the error characteristics of the numerical model.

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talk, I will characterise current 4D-Var solution algorithms according to their potential for paralleli- sation, and will propose a “saddle-point” formulation that has the potential to significantly increase the parallelism of 4D-Var.

“Resolution of Sharp Fronts in the Presence of Model Error in Variational Data Assimilation”

Melina A. Freitag

Department of Mathematical Sciences, University of Bath,

Bath BA2 7AY, United Kingdom.

http://people.bath.ac.uk/mamamf/

Data assimilation is an important tool for numerical weather prediction. In this talk we give a short introduction to data assimilation and show that data assimilation using four-dimensional variation (4DVar) can be interpreted as a form of Tikhonov regularisation, a familiar method for solving ill-posed inverse problems. It is known from image restoration problems thatL1-norm penalty regularisation recovers sharp edges in the image better than theL2-norm penalty regularisation. We apply this idea to 4DVar for problems where shocks are present and give some examples where theL1-norm penalty approach performs better than the standardL2-norm regularisation in 4DVar.

“Dual Methods for Data Assimilation in Geosciences”

Serge Gratton

CERFACS-IRIT, France

The problem considered in this talk is the data assimilation problem arising in weather forecasting and oceanography, which consists in estimating the initial condition of a dynamical system whose future behaviour is to be predicted. More specifically, new linear algebra techniques will be discussed for the iterative solution of the particular nonlinear least-squares formulation of this inverse problem known under the name of 4DVAR, for four-dimensional data assimilation. These new methods are designed to decrease the computational cost in applications where the number of variables involved is expected to exceed 109. They involve the exploitation of the problem’s underlying geometrical structure in reformulating standard Krylov methods such as CG, FOM or GMRES into significantly cheaper variants, and also exploit the possibility of computing matrix-vector products inexactly for further computational savings. If time allows, adapted preconditioning issues for the considered systems of equations will be discussed, which also depend on the problem’s geometrical structure and which exploit limited-memory techniques in a novel way.

“Numerical Methods for the Design of Large-Scale Nonlinear Discrete Ill-Posed Inverse Problems”

Eldad Haber

The University of British Columbia, Vancouver, Canada

Design of experiments for discrete ill-posed problems is a relatively new area of research. While there has been some limited work concerning the linear case, little has been done to study design criteria and numerical methods for ill-posed nonlinear problems. We present an algorithmic framework for nonlinear experimental design with an efficient numerical implementation. The data are modeled as

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“On the Bayesian Approach to Inverse Problems”

Tapio Helin

RICAM, Linz, Austria

In this talk the challenges and benefits of the Bayesian approach to inverse problems are discussed.

A short introduction to the topic is given and we illustrate an application of the Bayes paradigm appearing in atmospheric tomography.

“Adaptive Discretization Strategies for the Identification of Distributed Parameters in Partial Differential Equations”

Barbara Kaltenbacher

Alpen-Adria Universit¨at Klagenfurt

Parameter identification in partial differential equation is a class of large scale inverse problems de- manding for highly efficient solution strategies. On the other hand, due to the inherent ill-posedness of such problems, any solution approach needs to contain regularization.

After a short motivation for the use of adaptivity in inverse problems for PDEs, we intend to give an at least partial overview on existing literature and report on our own research, which includes joint work with Hend Benameur, Anke Griesbaum, Alana Kirchner, Jonas Offtermatt, and Boris Vexler.

We will mainly dwell on two approaches. The first one is a method based on refinement and coarsening indicators where sensitivities of the data misfit functional with respect to changes in the discretization are computed as Largange multipliers of appropriately constrained misfit minimization problems.

The second one originates from PDE constrained optimal control and relies on adaptive refinement according to goal oriented error estimators.

In both cases, special care has to be taken due to the ill-posedness of the underlying inverse problem in the sense that either stability has to be additionally incorporated or the stabilizing effect of discretiza- tion has to be approproately exploited. Therefore, key tasks in this context are on one hand to prove regularization properties for the resulting methods and on the other hand to show their efficiency for the solution of large scale inverse problems.

“Inverse Problems and Regularization - an Introduction”

Stefan Kindermann

Industrial Mathematics Insitute, Johannes Kepler University Linz

This talk is an introduction into the theory of inverse problems, regularization and the corresponding convergence theory. We discuss the notion of ill-posed problems, the idea of regularization, the role of parameter choice and the corresponding convergence theory with a focus on the deterministic –in opposite to the stochastic or Bayesian – viewpoint. Furthermore, we would like to bring up some recent methods and ideas that might be relevant for the scope of this workshop.

“Optimal State Estimation for Numerical Weather Prediction using Reduced Order Models”

Amos S. Lawless

Department of Mathematics and Statistics, University of Reading,

Reading, RG6 6AX, U.K.

Joint work with C. Boess, A. Bunse-Gerstner (University of Bremen, Germany) and N.K. Nichols (University of Reading).

Data assimilation techniques are used to combine observational data with numerical models in order to estimate the state of a dynamical system. A common approach to data assimilation is the method of four-dimensional variational data assimilation (4D-Var), which treats the assimilation problem as a

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107−108variables, and so methods must be found to solve the linear least squares problem efficiently when applying 4D-Var to these systems. One possible solution is to reduce the space in which the optimization problem is solved, using techniques of model order reduction. However, most current methods of model order reduction rely on the dynamical system being stable or having a relatively small unstable subspace. In practice this is not the case and many numerical prediction models are unstable or close to unstable. Recently a new method of model reduction has been developed, based on balanced truncation, which is able to treat unstable systems. In this talk we show how this new approach can be incorporated into the 4D-Var algorithm. Numerical experiments using the two- dimensional Eady model, a simple model of atmospheric instability, are used to illustrate the benefit that may be obtained from using this new method in the state estimation problem.

“On Iterative Regularization Methods for Nonlinear Inverse Problems”

Antonio Leitao

Department of Mathematics, Federal University of St. Catarina

This talk is devoted to the investigation of two distinct iterative regularization methods for solving nonlinear operator equations in Hilbert spaces. First we address iterated Tikhonov type methods. We show that the proposed method is a convergent regularization method. In the case of noisy data we propose a modification, the so called loping iterated Tikhonov-Kaczmarz method, where a sequence of relaxation parameters is introduced and a different stopping rule is used. Convergence analysis for this method is also provided. The second method addressed in this talk is a modified Levenberg-Marquardt method coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill- posed operator equations. 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.

“Conditioning, Preconditioning and Regularization of the Optimal State Estimation Problem”

Nancy K. Nichols University of Reading,

Department of Mathematics & Statistics, PO Box 220, Whiteknights,

Reading RG6 6AX, UK

This is joint work with S.A. Haben and A. S. Lawless.

Data assimilation is a technique for determining an ’optimal’ estimate of the current and future states of a dynamical system from a prior estimate, or model forecast, together with observations of the system. Applications arise in very large environmental problems where the number of state variables is O(107−108) and the number of observations is O(104−106). The errors in the prior estimate and in the observations are assumed to be random with known distributions and the solution to the optimization problem is taken to be the maximum a posteriori likelihood estimate. With the aid of Bayes Theorem, the problem reduces to a very large nonlinear least squares problem, subject to the dynamical system equations. The problem is treated in practice using an approximate Gauss-Newton iterative method, where a linearized least squares problem is solved at each step of the procedure by an ’inner’ gradient iteration procedure.

The convergence of the variational scheme and the sensitivity of the solution to perturbations are dependent on the conditioning of the least-squares variational equation. The problem is generally ill-conditioned and hence is difficult to solve quickly and accurately. In this study we examine how

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“The Construction and Use of Linear Models in Large-Scale Data Assimilation”

Tim Payne Met Office, FitzRoy Road,

Exeter, EX1 3PB, United Kingdom

Both the Extended Kalman Filter and incremental 4D-Var make use of a linear model, to propagate covariances explicitly in the first case and implicitly in the second. Conventionally this linear model is taken to be the first derivative of the forecast model, which is appropriate for infinitesimal increments but is a poor predictor of the true evolution of finite-sized increments.

We show that if the pdf of the increments is known we may construct better linearisations, and show how the use of this type of linearisation can improve the assimilation of data and thereby the model forecast.

Furthermore, since the full model is available to us, we have unlimited information on the errors in the linear model compared with the full model. We consider how this information may be used, however the linear model is chosen, to improve the Extended Kalman Filter and incremental 4D-Var.

“On Instability in Data Assimilation”

Roland Potthast Deutscher Wetterdienst, Uni Reading,

Uni G¨ottingen

Instability is one of the key features of many inverse problems in geophysics. Instability is a property which can arise from the underlying dynamics systems, in particular when they are chaotic or effec- tively have branching points. Instabilities of data assimilation algorithms can also arise from compact or ill-posed observation operators. Often, data assimilation algorithms regularize the inversion for every time-step or cycle of the assimilation method, such that each individual reconstruction is sta- ble. But over time instabilities are likely to occur in basically all practically applicable realizations of such systems. We will describe a mathematical framework which is suitable to study and anlyse such instabilities. In particular, we work out the behaviour of cycled data assimilation systems such as 3dVar or 4dVar explicitly for simple linear systems. Also, we show how assimilation systems can be stabilized, a general theory for self-adjoint linear system dynamics is presented. Numerical examples are provided.

“Ensemble Transform Filters for Geophysical Data Assimilation”

Sebastian Reich University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany

Data assimilation is the task to combine model based simulations with measurements to provide optimal state and/or parameter estimates. Data assimilation requires that one can quantify the uncertainty in the mathematical model forecasts. Bayes’ theorem is used to assimilate observations into the model forecast and to reduce uncertainties. I will discuss the data assimilation filtering problem in the context of a McKean-Vlasov system for the time evolution of the probability density function characterizing model uncertainty. Specific algorithms are obtained by using ensemble/particle methods for the time evolution of the probability density functions under the model dynamics and by fitting statistical models to the ensemble of particles such as Gaussians and Gaussian mixture models prior to a data assimilation step. Contrary to particle filters (or sequential Monte Carlo methods, the data assimilation step itself is implemented as a transport of the ensemble/particles under continuous transformations. The resulting filters can be viewed as generalizations of the popular ensemble Kalman filter to non-Gaussian probability density functions.

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“Assimilation of Observations and Bayesianity”

Olivier Talagrand

Laboratoire de M´et´eorologie Dynamique,

´Ecole Normale Sup´erieure, Paris

This is joint work with Mohamed Jardak (LMD/ENS, Paris).

The purpose of assimilation of observations can be described as to determine as accurately as possible the state of the observed system, using all available relevant information. If that information can be described in terms of probability distributions, assimilation can be stated as a problem in bayesian estimation. Namely, determine the probability distribution for the state of the system, conditioned by the available information.

In the standard linear and gaussian case, the sought-for conditional probability distribution is explicitly known, and a simple numerical procedure is available for obtaining a sample of independent realizations of that distribution. Perturb the data (observations, background, dynamical model) according to their own error probability distribution, and perform a standard least-variance assimilation for each set of perturbed data. That procedure, although it will not in general produce a bayesian sample, can easily be implemented, with any assimilation algorithm, in nonlinaer and/or non-gaussian situations. The present work is a numerical study of the degree to which nonlinearity and/or non-gaussianity degrades the bayesian character of the sample produced by that procedure. The particular assimilation method that is used is variational assimilation.

There is not, and cannot be, a general test of bayesianity. A weaker, objectively verifiable property is reliability, namely the property that the verifying observation is statistically consistent with the predicted probabilities (it rains 40% of the time when rain is predicted to occur with probability 40%).

Several non-equivalent measures of reliability (reliability diagramme, reliability component of the Brier and Brier-like scores, rank histograms, measures of the spread-skill relationship) are commonly used for evaluating ensemble prediction systems. A number of those measures are used here.

Diagnostics are performed with two small one-dimensional chaotic systems, namely the Kuramoto- Sivashinsky equation and the Lorenz 96 model, in the condition of perfect model. Least-variance estimation is implemented with standard variational assimilation. In the linear and gaussian case (linearity being obtained by simple linearization about a particular model solution), exact reliabil- ity is achieved. Non-gaussianity in observation errors does not significantly affect reliability. But nonlinearity, even weak, does significantly degrades reliability, and therefore bayesianity.

For the purpose of comparison, assimilations are also peformed with a particle filter, which is meant to achieve bayesianity. Results show that reliability is almost as much degraded as in the case of variational assimilation, showing that, at least in the cases under consideration, a particle filter does not achieve bayesianity.

“Regularization Error Estimates and Discrepancy Principle for Optimal Control Problems with Inequality Constraints”

Daniel Wachsmuth

Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy ofSciences,

Altenbergerstraße 69, A–4040 Linz, Austria

In this article we study the regularization of optimization problems by Tikhonov regularization. The optimization problems are subject to pointwise inequality constraints in L2(Ω). We derive a-priori

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“Background-Error Correlation Modelling in Variational Assimilation using a Diffusion Equation”

Anthony T. Weaver CERFACS,

42 avenue Coriolis, 31057 Toulouse, France

Variational data assimilation problems require a regularization or background term that penalizes statistically-weighted departures from a background state. The statistical weights are derived from estimates of (the inverse of) the background-error covariance matrix (B). The fundamental role of B in determining the quality of analyses and forecasts is well known in atmospheric and ocean data assimilation. The number of independent elements that need to be specified in B is enormous (typically on the order of 1012−1013elements). This makes the problem of specifyingBparticularly challenging, both from a computational and estimation point of view.

In variational assimilation it is common to transform model variables to a new set of variables whose cross-covariances are much weaker than those of the original variables and can be neglected for practical purposes. Differential operators derived from the explicit or implicit solution of a pseudo-diffusion equation can be used to model the background-error correlationsof the transformed variables. The diffusion model avoids the explicit computation of a costly integral equation by transforming the problem into differential form. It is particularly convenient for problems involving complex boundaries and is flexible to allow the specification of general correlation structures (anisotropic, inhomogeneous).

These features make the model well suited for applications in ocean data assimilation. Some of the practical issues involved in constructing a diffusion-based B model for a global ocean variational assimilation system will be discussed.

A review of key theoretical results underpinning diffusion as a method for representing correlation functions will be given first. Solutions to the isotropic diffusion problems on the spherical space S2 and the general d-dimensional Euclidean space Rd will be considered. The correlation kernels implied by both explicit and implicit diffusion on these spaces will be identified. Next, solutions to theanisotropicdiffusion problem will be presented. Statistical methods for estimating the elements of the diffusion tensor, which is the fundamental parameter controlling the spatial response of the correlation model, from a sample of simulated errors will be described and illustrated. Since the number of independent parameters needed to specify the diffusion tensor at all model grid points is of the order of the total number of grid pointsN, sampling errors are inherently much smaller than those involved in the orderN2estimation problem of the full correlation matrix. The potential of the method for defining flow-dependent background-error correlations in a hybrid ensemble-variational assimilation system will be discussed.

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Abstracts for Posters

“Towards Regularization Strategies of Inverse Electromagnetic Scattering Problems”

Maria Brune

Universit¨at Paderborn, 33098 Paderborn

The subject of this poster is a large-scaled inverse electromagnetic scattering problem where a 3D re- construction of dielectric properties in a large cubic area should be performed. Since there is only one source available, the corresponding forward problem comprises an instationary three-dimensional sim- ulation. Hence, for a chosen permittivityan electromagnetic problem is solved by a FDTD-method.

Subsequently, the difference between the simulated E- and H-fields and the provided measured fields has to be minimized. The arising optimization problem is solved by a limited-memory quasi-Newton algorithm. In this context ill-posedness is a serious issue. To cope with this problem, regularization strategies are required. For this purpose, different well-established approaches have been analyzed and tested. As the problem has many local minima and the optimal material parameters only corre- spond to the global minimum, a suitable globalization strategy is indispensable. Therefore a detailed analysis of the properties of the problem has been performed. Basic aspects of the problem, the employed regularization strategies and a globalization strategy will be explained. Several theoretical and numerical results will be presented and discussed. This is joint work with Andrea Walther.

“The Dissipative and Dispersive Effects of Numerical Models on 4D-Var Data Assimilation”

Siˆan Jenkins University of Bath, Bath, UK

Variational data assimilation is a method used to solve a particular kind of inverse problem.

Given a set of observations and a numerical model for a physical system, what initial condition for the numerical model will provide the best approximation to the set of observations?

This is achieved by finding the initial condition for the model that provides the least squares solution between the observations and a simulation of the model constrained by prior information. The results of the simulation beyond the time of the observations, when using this initial condition, is a forecast for the system. This method is used in operational weather forecast centres.

Given a linear PDE and a mesh with a particular grid size, a finite difference method can be constructed to simulate the PDE over the mesh. Different finite difference approximations create different methods which simulate the PDE in slightly different ways, giving the results different properties over time.

The initial condition for a numerical method can be decomposed into a sum of eigenfunctions. In this linear problem, the eigenfunctions are sinusoidal with different wave numbers. The finite difference methods propagate these eigenfunctions to simulate the PDE over time. Given a mesh withN spatial grid points,N different eigenfunctions can be resolved. The eigenfunctions with lower wave numbers are propagated similarly by each method, however the propagation of eigenfunctions with higher wave numbers is method dependent. This results in numerical model error and leads to the differing solutions for the same PDE when using different numerical methods.

The numerical modelling of small-scale structures is a very challenging problem. On a coarse grid, these structures can only be resolved using high wave number eigenfunctions. However this is problematic

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“Full Waveform Inversion of Acoustic Density from Well Seismic Data”

Ludovic M´etivier

ISTerre, Joseph Fourier University, Grenoble, France

Assuming an acoustic description of the wave propagation beneath the subsurface, we design a method for the estimation of the density ρ, given an initial estimation of the pressure wave velocity Vp, and considering seismic data acquired inside a well with sources located at the ground surface. The aim of the method is to build a high resolution estimation of the densityρin a vincinity of the well, that could be used either for CO2storage monitoring or reservoir monitoring.

To this purpose, we extend an existing 1D method, based on the restricted assumption of plane wave propagation, to a multi-dimensional framework. One interesting feature of this method is that no estimation of the sources is required, which is conserved within the multi-dimensional extension.

However, the multi-dimensional problem is highly underdetermined, and involves a huge number of discrete data and unknowns. Adapted regularization technics are used [1], and a suitable numerical method is developed to overcome these difficulties. This involves an interlocked optimization method that couples the conjugate gradient algorithm with the l-BFGS method, and a parallelized computation of the misfit gradient using the adjoint-state technic and domain decomposition methods [2].

In the context of a 2D implementation, the results we obtain show satisfactory convergence of the numerical method toward a high resolution estimation of the subsurface density around the well.

References

[1] L.M´etivier, L.Halpern, F.Delprat-Jannaud, P.Lailly, A 2D Nonlinear inversion of well seismic data, Inverse Problems,27, 055005, 2011.

[2] L.M´etivier, Interlocked optimization and fast gradient algorithm for a seismic inverse problem, Journal of Computational Physics,230, 7502-7518, 2011.

“Instability and Regularization for Data Assimilation”

Alexander Moodey

Department of Mathematics and Statistics, Whiteknights, PO Box 220,

Reading, RG6 6AX

Data assimilation algorithms are a crucial part of operational systems in numerical weather prediction, hydrology and climate science. Usually, a variety of diverse measurement data are employed to determine the state of the atmosphere or wider system. Modern data assimilation systems use more and more remote sensing data; in particular radiances measured by satellites, radar data and integrated water vapour measurements via GPS/GNSS signals. The inversion of some of these measurements are ill-posed in the classical sense, i.e. the inverse of the operator H which maps the state onto the data is unbounded. In this case, the use of such data can lead to significant instabilities of data assimilation algorithms.

The goal of this work is to provide a rigorous mathematical analysis of the instability of well-known data assimilation algorithms. Here, we will restrict our attention to particular linear systems, in which the instability can be explicitly analysed. We investigate three dimensional variational data assimilation. A theory for the instability is developed using the classical theory of ill-posed problems in a Banach space framework. Further, we demonstrate with a numerical example that instabilities can and will occur.

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“Extrapolation in variable RKHSs with Application to the BG Reading”

Valeriya Naumova

Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences ( ¨OAW)

Altenbergerstrae 69, A-4040 Linz

This is the joint work with Sergei V. Pereverzyev and S. Sivananthan.

In this talk we present a new scheme of a kernel adaptive regularization algorithm, where the kernel and the regularization parameter are adaptively chosen within regularization procedure. The construction of such fully adaptive regularization algorithm is motivated by the problem of reading the blood glucose concentration of diabetic patients. We describe how proposed scheme can be used for this purpose and report the results of numerical experiments with real clinical data.

“A Numerical Simulation Study of the Indian Summer Monsoon of 1998 using RegCM3”

Sridhara Nayak

Center for Oceans, Rivers, Atmosphere and Land Sciences;

Indian Institute of Technology Kharagpur;

Kharagpur 721 302; West Bengal; INDIA

Joint work with S. Nayak, M. Mandal and S. Maity.

In this study, the International Centre for Theoretical Physics (ICTP) Regional Climate Model (RegCM3) has been used to examine its suitability in simulating the Indian summer monsoon. The model is implemented over Indian region for the months of June, July, August and September of the year 1998 with horizontal grid resolution of 60 Km, and 23 vertical levels. The initial and boundary conditions are taken from NCEP/NCAR reanalysis (NNRP1) data at 2.5 degree grid interval, and in- terpolated to the model domain. Terrain heights and land use data are obtained from a global data set produced by the United States Geographical Survey (USGS) at 30 minute resolution. The simulated monsoon features by RegCM3 are compared with those of the NCEP/NCAR reanalysis and simulated rainfall against observations from the Global Precipitation Climatology Centre (GPCC) to evaluate the model performance. The results indicate that RegCM3 successfully simulates some important characteristics of Indian summer monsoon and the simulated model values are close to GPCC values.

In sum, the study indicates that the RegCM3 can be effectively used to study the summer monsoon processes over the Indian region.

“CuRe - A new Wavefront Reconstruction Method for SH-WFS Measurements”

Andreas Obereder MathConsult GmbH Altenbergerstrasse 69 A - 4040 Linz

In order to fulfill the real-time requirements for AO on ELTs one has to either invest in (very) high performance hardware or spend some effort on the development of highly efficient reconstruction algorithms for wave front sensors. The AAO (Austrian Adaptive Optics) team is involved in deriving reconstructors for SH- and Pyramid WFS measurements utilizing the mathematical properties of the

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“Wavefront Reconstruction for Extreme Adaptive Optics”

Iuliia Shatokhina

Industrial Mathematics Institute, Johannes Kepler University, Altenbergerstrasse 69 A-4040 Linz, Austria

It is a joint work with Mariya Zhariy and Ronny Ramlau.

We present our work done within the Austrian InKind project ’Mathematical algorithms and software for E-ELT Adaptive Optics’. In particular, we are concerned with eXtreme Adaptive Optics (XAO).

Our aim is the development of fast algorithms for reconstructing the incoming wavefrontφfrom the wavefront sensor (WFS) measurementss. Possible choices of sensors for the XAO system are Pyramid WFS (P-WFS) and Roof WFS (R-WFS).

First, we concentrate on the non-modulated Pyramid WFS case. Under the infinite telescope assump- tion, we derive a simplified version of the existing analytical forward linearized model of the WFS.

Based on this simplified model, we suggest an inversion method called the Hilbert Transform with Mean Restoration (HTMR), which involves mainly the Hilbert transform of the datas. We present the algorithm itself and consider a numerical example.

For the full linearized model of the non-modulated Pyramid WFS, we suggest a Fixed-Point Iteration (FPI) algorithm which includes the HTMR as one step. We describe the algorithm and show a numer- ical example. The two methods suggested for the non-modulated Pyramid WFS case are compared numerically.

Then, we consider a Roof WFS with sinusoidal modulation. We suggest a fast numerical approach for the implementation of the forward WFS model. Based on this fast implementation, we consider two iterative inversion methods, FPI and CGNE. Numerical study of the algorithms’ behavior is our current work.

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List of Participants

Alyaev Sergey Universitetet i Bergen [email protected]

Andreev Roman SAM, ETH Z¨urich [email protected]

Ascher Uri UBC [email protected]

Borges Carlos Worcester Polytechnic Institute [email protected]

Brune Maria Universti¨at Paderborn [email protected] Buck Marco Fraunhofer Institute for

Industrial Mathematics [email protected]

Burger Martin WWU Muenster [email protected]

Challa Durga Prasad RICAM, Linz [email protected] Cirpka Olaf Arie University of Tuebingen [email protected]

Cullen Mike UK Met Office [email protected]

Dashti Masoumeh University of Warwick,

Mathematics Institute [email protected] Engl Heinz RICAM & University of Vienna [email protected]

Fisher Michael ECMWF [email protected]

Freitag Melina University of Bath [email protected] Gaudio Loredana University of Basel [email protected]

Georgiev Ivan RICAM, Linz [email protected]

Gratton Serge Joint laboratory

CERFACS-IRIT [email protected]

Haber Eldad UBC [email protected]

Heged¨us G´abor RICAM, Linz [email protected]

Helin Tapio RICAM, Linz [email protected]

Hrtus Rostislav Institute of Geonics of the AS

CR, v.v.i., Ostrava, CZ [email protected]

Husain Akhlaq LNMIIT, Jaipur [email protected]

Jenkins Sian University of Bath [email protected]

Kaltenbacher Barbara University of Graz/University of

Klagenfurt [email protected]

Kar Manas RICAM, Linz [email protected]

Karer Erwin RICAM, Linz [email protected]

Kollmann Markus Doctoral Program

Computational Mathematics, Linz

[email protected]

Kolmbauer Michael Johannes Kepler University

Linz [email protected]

Kraus Johannes RICAM, Linz [email protected]

Langer Ulrich Johannes Kepler University

Linz [email protected]

Lawless Amos University of Reading [email protected] Leitao Antonio Federal University of Santa

Catarina [email protected]

Livshits Irene Ball State University [email protected]

Melenk Jens Markus TU Wien [email protected]

M´etivier Ludovic ISTerre, Joseph Fourier

University, Grenoble [email protected] Migliorati Giovanni Department of Mathematics,

Politecnico di Milano [email protected]

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Payne Tim UK Met Office [email protected] Pechstein Clemens Johannes Kepler University

Linz [email protected]

Potthast Roland Deutscher Wetterdienst, Uni

Reading, Uni G¨ottingen [email protected] Reich Sebastian University of Potsdam [email protected] Sarkis Marcus Mathematical Sciences Dept./

Worcester Polytechnic Institute [email protected] Scheichl Robert University of Bath [email protected]

Schicho Josef RICAM, Linz [email protected]

Shatokhina Iuliia Johannes Kepler University

Linz [email protected]

Sini Mourad RICAM, Linz [email protected]

Sokol Vojt˘ech Institute of Geonics of the AS

CR, v.v.i., Ostrava, CZ [email protected] Talagrand Olivier Laboratoire de M´et´eorologie

Dynamique, ´Ecole Normale Sup´erieure, Paris

[email protected]

Teckentrup Aretha University of Bath, Department

of Mathematical Sciences [email protected] Tomar Satyendra RICAM, Linz [email protected] Wachsmuth Daniel RICAM, Linz [email protected] Weaver Anthony CERFACS, Toulouse [email protected]

Willems J¨org RICAM, Linz [email protected]

Wohlmuth Barbara Technische Universit¨at

M¨unchen [email protected]

Wolfmayr Monika Johannes Kepler University

Linz [email protected]

Yang Huidong RICAM, Linz [email protected]

Zikatanov Ludmil The Pennsylvania State

University [email protected]

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MondayTuesdayWednesdayThursdayFriday Opening(9:10)StefanKindermannTapioHelinSebastianReichAnthonyT.Weaver InverseProblemsand Regularization-anIntroductionOntheBayesianApproachto InverseProblemsEnsembleTransformFiltersfor GeophysicalDataAssimilation

Background-ErrorCorrelation ModellinginVariational AssimilationusingaDiffusion Equation OlafA.CirpkaMichaelFisherMasoumehDashtiMartinBurgerDanielWachsmuth GeostatisticalApproachesfor InverseModelinginSubsurface Hydrology

ParallelAlgorithmsfor Four-DimensionalVariational DataAssimilation BesovPriorsforBayesian Inverseproblems4DVariationalModelspreserving SharpEdges

RegularizationErrorEstimates andDiscrepancyPrinciplefor OptimalControlProblemswith InequalityConstraints CoffeeCoffeeCoffeeCoffeeCoffee MikeCullenSergeGrattonBarbaraKaltenbacherNancyK.NicholsOlivierTalagrand 4dVarinthePresenceofModel ErrorDualMethodsforData AssimilationinGeosciences

AdaptiveDiscretizationStrategies fortheIdentificationof DistributedParametersinPartial DifferentialEquations Conditioning,Preconditioning andRegularizationoftheOptimal StateEstimationProblem

AssimilationofObservationsand Bayesianity LunchLunchExcursion&Conference Dinner Thebusleavesat12:00pmfrom Sommerhaushotel. Lunchatrestaurant “Donauschlinge”atthe “Schl¨ogenerSchlingeofthe danube. Visitofthe“Baumkronenweg”in Kopfing,followedbythe ConferenceDinneratthe restaurant“GasthofOachkatzl“ atthe“Baumkronenweg”. Thebuswillreturntohotel Sommerhausaround9:30pm.

LunchClosing AmosS.LawlessUriAscherEldadHaber OptimalStateEstimationfor NumericalWeatherPrediction usingReducedOrderModels

AdaptiveandStochastic Algorithmsforpiecewiseconstant EIT&DCResistivityProblems withmanyMeasurements

NumericalMethodsforthe DesignofLarge-ScaleNonlinear DiscreteIll-PosedInverse Problems CoffeeCoffeeCoffee TimPayneAntonioLeitaoR.Potthast TheConstructionandUseof LinearModelsinLarge-Scale DataAssimilation

OnIterativeRegularization MethodsforNonlinearInverse Problems

OnInstabilityinData Assimilation MelinaA.Freitag ResolutionofSharpFrontsinthe PresenceofModelErrorin VariationalDataAssimilation Poster&ReceptionPublicLecture Reception WednesdayisapublicholidayinAustria. minuteslong.Aftertheendofeachtalk,thereare10minutesfordiscussion.

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