Skip to main content
×
Home
    • Aa
    • Aa

Inverse problems: A Bayesian perspective

  • A. M. Stuart (a1)
Abstract

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

A. Alekseev and I. Navon (2001), ‘The analysis of an ill-posed problem using multiscale resolution and second order adjoint techniques’, Comput. Meth. Appl. Mech. Engrg 190, 19371953.

A. Apte , M. Hairer , A. Stuart and J. Voss (2007), ‘Sampling the posterior: An approach to non-Gaussian data assimilation’, Physica D 230, 5064.

A. Apte , C. Jones and A. Stuart (2008 a), ‘A Bayesian approach to Lagrangian data assimilation’, Tellus 60, 336347.

A. Apte , C. Jones , A. Stuart and J. Voss (2008 b), ‘Data assimilation: Mathematical and statistical perspectives’, Internat. J. Numer. Methods Fluids 56, 10331046.

G. Backus (1970 a), ‘Inference from inadequate and inaccurate data I’, Proc. Nat. Acad. Sci. 65, 17.

G. Backus (1970 b), ‘Inference from inadequate and inaccurate data II’, Proc. Nat. Acad. Sci. 65, 281287.

G. Backus (1970 c), ‘Inference from inadequate and inaccurate data III’, Proc. Nat. Acad. Sci. 67, 282289.

A. Bain and D. Crisan (2009), Fundamentals of Stochastic Filtering, Springer.

R. Bannister , D. Katz , M. Cullen , A. Lawless and N. Nichols (2008), ‘Modelling of forecast errors in geophysical fluid flows’, Internat. J. Numer. Methods Fluids 56, 11471153.

M. Bell , M. Martin and N. Nichols (2004), ‘Assimilation of data into an ocean model with systematic errors near the equator’, Quart. J. Royal Met. Soc. 130, 873894.

T. Bengtsson , C. Snyder and D. Nychka (2003), ‘Toward a nonlinear ensemble filter for high-dimensional systems’, J. Geophys. Res. 108, 8775.

A. Bennett (2002), Inverse Modeling of the Ocean and Atmosphere, Cambridge University Press.

A. Bennett and W. Budgell (1987), ‘Ocean data assimilation and the Kalman filter: Spatial regularity’, J. Phys. Oceanography 17, 15831601.

A. Bennett and B. Chua (1994), ‘Open ocean modelling as an inverse problem’, Monthly Weather Review 122, 13261336.

A. Bennett and R. Miller (1990), ‘Weighting initial conditions in variational assimilation schemes’, Monthly Weather Review 119, 10981102.

L. Berliner (2001), ‘Monte Carlo based ensemble forecasting’, Statist. Comput. 11, 269275.

J. Bernardo and A. Smith (1994), Bayesian Theory, Wiley.

A. Beskos , G. O. Roberts and A. M. Stuart (2009), ‘Optimal scalings for local Metropolis-Hastings chains on non-product targets in high dimensions’, Ann. Appl. Probab. 19, 863898.

A. Beskos , G. O. Roberts , A. M. Stuart and J. Voss (2008), ‘MCMC methods for diffusion bridges’, Stochastic Dynamics 8, 319350.

V. Bogachev (1998), Gaussian Measures, AMS.

P. Bolhuis , D. Chandler , D. Dellago and P. Geissler (2002), ‘Transition path sampling: Throwing ropes over rough mountain passes’, Ann. Rev. Phys. Chem. 53, 291318.

L. Borcea (2002), ‘Electrical impedence tomography’, Inverse Problems 18, R99–R136.

P. Brasseur , P. Bahurel , L. Bertino , F. Birol , J.-M. Brankart , N. Ferry , S. Losa , E. Remy , J. Schroeter , S. Skachko , C.-E. Testut , B. Tranchat , P. Van Leeuwen and J. Verron (2005), ‘Data assimilation for marine monitoring and prediction: The Mercator operational assimilation systems and the Mersea developments’, Quart. J. Royal Met. Soc. 131, 35613582.

G. Burgers , P. Van Leeuwen and G. Evensen (1998), ‘On the analysis scheme in the ensemble Kalman filter’, Monthly Weather Review 126, 17191724.

D. Calvetti (2007), ‘Preconditioned iterative methods for linear discrete ill-posed problems from a Bayesian inversion perspective’, J. Comput. Appl. Math. 198, 378395.

D. Calvetti and E. Somersalo (2005 a), ‘Priorconditioners for linear systems’, Inverse Problems 21, 13971418.

D. Calvetti and E. Somersalo (2005 b), ‘Statistical elimination of boundary artefacts in image deblurring’, Inverse Problems 21, 16971714.

D. Calvetti and E. Somersalo (2006), ‘Large-scale statistical parameter estimation in complex systems with an application to metabolic models’, Multiscale Modeling and Simulation 5, 13331366.

D. Calvetti and E. Somersalo (2007 a), ‘Gaussian hypermodel to recover blocky objects’, Inverse Problems 23, 733754.

D. Calvetti and E. Somersalo (2007 b), Introduction to Bayesian Scientific Computing, Vol. 2 of Surveys and Tutorials in the Applied Mathematical Sciences, Springer.

D. Calvetti and E. Somersalo (2008), ‘Hypermodels in the Bayesian imaging framework’, Inverse Problems 24, #034013.

D. Calvetti , H. Hakula , S. Pursiainen and E. Somersalo (2009), ‘Conditionally Gaussian hypermodels for cerebral source location’, SIAM J. Imag. Sci. 2, 879909.

D. Calvetti , A. Kuceyeski and E. Somersalo (2008), ‘Sampling based analysis of a spatially distributed model for liver metabolism at steady state’, Multiscale Modeling and Simulation 7, 407431.

E. Candès and M. Wakin (2008), ‘An introduction to compressive sampling’, IEEE Signal Processing Magazine, March 2008, 2130.

J.-Y. Chemin and N. Lerner (1995), ‘Flot de champs de veceurs non lipschitziens et équations de Navier-Stokes’, J. Diff. Equations 121, 314328.

A. Chorin and P. Krause (2004), ‘Dimensional reduction for a Bayesian filter’, Proc. Nat. Acad. Sci. 101, 1501315017.

A. Chorin and X. Tu (2009), ‘Implicit sampling for particle filters’, Proc. Nat. Acad. Sci. 106, 1724917254.

M. Christie (2010), Solution error modelling and inverse problems. In Simplicity, Complexity and Modelling, Wiley, New York, to appear.

S. Cotter , M. Dashti , J. Robinson and A. Stuart (2009), ‘Bayesian inverse problems for functions and applications to fluid mechanics’, Inverse Problems 25, #115008.

P. Courtier (1997), ‘Dual formulation of variational assimilation’, Quart. J. Royal Met. Soc. 123, 24492461.

P. Courtier and O. Talagrand (1987), ‘Variational assimilation of meteorological observations with the adjoint vorticity equation II: Numerical results’, Quart. J. Royal Met. Soc. 113, 13291347.

P. Courtier , E. Anderson , W. Heckley , J. Pailleux , D. Vasiljevic , M. Hamrud , A. Hollingworth , F. Rabier and M. Fisher (1998), ‘The ECMWF implementation of three-dimensional variational assimilation (3D-Var)’, Quart. J. Royal Met. Soc. 124, 17831808.

B. Dacarogna (1989), Direct Methods in the Calculus of Variations, Springer, New York.

M. Dashti and J. Robinson (2009), ‘Uniqueness of the particle trajectories of the weak solutions of the two-dimensional Navier-Stokes equations’, Nonlinearity 22, 735746.

J. Derber (1989), ‘A variational continuous assimilation technique’, Monthly Weather Review 117, 24372446.

B. DeVolder , J. Glimm , J. Grove , Y. Kang , Y. Lee , K. Pao , D. Sharp and K. Ye (2002), ‘Uncertainty quantification for multiscale simulations’, J. Fluids Engrg 124, 2942.

D. Donoho (2006), ‘Compressed sensing’, IEEE Trans. Inform. Theory 52, 1289– 1306.

P. Dostert , Y. Efendiev , T. Hou and W. Luo (2006), ‘Coarse-grain Langevin algorithms for dynamic data integration’, J. Comput. Phys. 217, 123142.

N. Doucet , A. de Frietas and N. Gordon (2001), Sequential Monte Carlo in Practice, Springer.

R. Dudley (2002), Real Analysis and Probability, Cambridge University Press, Cambridge.

Y. Efendiev , A. Datta-Gupta , X. Ma and B. Mallick (2009), ‘Efficient sampling techniques for uncertainty quantification in history matching using nonlinear error models and ensemble level upscaling techniques’, Water Resources Res. 45, #W11414.

B. Ellerbroek and C. Vogel (2009), ‘Inverse problems in astronomical adaptive optics’, Inverse Problems 25, #063001.

H. Engl , M. Hanke and A. Neubauer (1996), Regularization of Inverse Problems, Kluwer.

H. Engl , A. Hofinger and S. Kindermann (2005), ‘Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems’, Inverse Problems 21, 399412.

G. Evensen and P. Van Leeuwen (2000), ‘An ensemble Kalman smoother for nonlinear dynamics’, Monthly Weather Review 128, 18521867.

F. Fang , C. Pain , I. Navon , M. Piggott , G. Gorman , P. Allison and A. Goddard (2009 a), ‘Reduced order modelling of an adaptive mesh ocean model’, Internat. J. Numer. Methods Fluids 59, 827851.

F. Fang , C. Pain , I. Navon , M. Piggott , G. Gorman , P. Farrell , P. Allison and A. Goddard (2009 b), ‘A POD reduced-order 4D-Var adaptive mesh ocean modelling approach’, Internat. J. Numer. Methods Fluids 60, 709732.

C. Farmer (2005), Geological modelling and reservoir simulation. In Mathematical Methods and Modeling in Hydrocarbon Exploration and Production (A. Iske and T. Randen , eds), Springer, Heidelberg, pp. 119212.

C. Farmer (2007), Bayesian field theory applied to scattered data interpolation and inverse problems. In Algorithms for Approximation (A. Iske and J. Levesley , eds), Springer, pp. 147166.

B. Fitzpatrick (1991), ‘Bayesian analysis in inverse problems’, Inverse Problems 7, 675702.

J. Franklin (1970), ‘Well-posed stochastic extensions of ill-posed linear problems’, J. Math. Anal. Appl. 31, 682716.

M. Freidlin and A. Wentzell (1984), Random Perturbations of Dynamical Systems, Springer, New York.

A. Gelfand and A. Smith (1990), ‘Sampling-based approaches to calculating marginal densities’, J. Amer. Statist. Soc. 85, 398409.

A. Gibbs and F. Su (2002), ‘On choosing and bounding probability metrics’, Internat. Statist. Review 70, 419435.

J. Glimm , S. Hou , Y. Lee , D. Sharp and K. Ye (2003), ‘Solution error models for uncertainty quantification’, Contemporary Mathematics 327, 115140.

S. Gratton , A. Lawless and N. Nichols (2007), ‘Approximate Gauss—Newton methods for nonlinear least squares problems’, SIAM J. Optimization 18, 106132.

C. Gu (2002), Smoothing Spline ANOVA Models, Springer.

C. Gu (2008), ‘Smoothing noisy data via regularization’, Inverse Problems 24, #034002.

C. Hagelberg , A. Bennett and D. Jones (1996), ‘Local existence results for the generalized inverse of the vorticity equation in the plane’, Inverse Problems 12, 437454.

E. Hairer and G. Wanner (1996), Solving Ordinary Differential Equations II, Vol. 14 of Springer Series in Computational Mathematics, Springer, Berlin.

M. Hairer , A. M. Stuart and J. Voss (2007), ‘Analysis of SPDEs arising in path sampling II: The nonlinear case’, Ann. Appl. Probab. 17, 16571706.

M. Hairer , A. M. Stuart and J. Voss (2009), Sampling conditioned diffusions. In Trends in Stochastic Analysis, Vol. 353 of London Mathematical Society Lecture Notes, Cambridge University Press, pp. 159186.

M. Hairer , A. Stuart , J. Voss and P. Wiberg (2005), ‘Analysis of SPDEs arising in path sampling I: The Gaussian case’, Comm. Math. Sci. 3, 587603.

W. K. Hastings (1970), ‘Monte Carlo sampling methods using Markov chains and their applications’, Biometrika 57, 97109.

T. Hein (2009), ‘On Tikhonov regularization in Banach spaces: Optimal convergence rate results’, Applicable Analysis 88, 653667.

J. Heino , K. Tunyan , D. Calvetti and E. Somersalo (2007), ‘Bayesian flux balance analysis applied to a skeletal muscle metabolic model’, J. Theor. Biol. 248, 91110.

R. Herbei and I. McKeague (2009), ‘Geometric ergodicity of hybrid samplers for ill-posed inverse problems’, Scand. J. Statist. 36, 839853.

R. Herbei , I. McKeague and K. Speer (2008), ‘Gyres and jets: Inversion of tracer data for ocean circulation structure’, J. Phys. Oceanography 38, 11801202.

A. Hofinger and H. Pikkarainen (2007), ‘Convergence rates for the Bayesian approach to linear inverse problems’, Inverse Problems 23, 24692484.

A. Hofinger and H. Pikkarainen (2009), ‘Convergence rates for linear inverse problems in the presence of an additive normal noise’, Stoch. Anal. Appl. 27, 240257.

M. Huddleston , M. Bell , M. Martin and N. Nichols (2004), ‘Assessment of wind stress errors using bias corrected ocean data assimilation’, Quart. J. Royal Met. Soc. 130, 853872.

M. Hurzeler and H. Kunsch (2001), Approximating and maximizing the likelihood for a general state space model. In Sequential Monte Carlo Methods in Practice (A. Doucet , N. de Freitas and N. Gordon , eds), Springer, pp. 159175.

J. Huttunen and H. Pikkarainen (2007), ‘Discretization error in dynamical inverse problems: One-dimensional model case’, J. Inverse and Ill-posed Problems 15, 365386.

K. Ide and C. Jones (2007), ‘Data assimilation’, Physica D 230, vii–viii.

K. Ide , L. Kuznetsov and C. Jones (2002), ‘Lagrangian data assimilation for pointvortex system’, J. Turbulence 3, 53.

C. Johnson , B. Hoskins and N. Nichols (2005), ‘A singular vector perspective of 4DVAR: Filtering and interpolation’, Quart. J. Royal Met. Soc. 131, 120.

C. Johnson , B. Hoskins , N. Nichols and S. Ballard (2006), ‘A singular vector perspective of 4DVAR: The spatial structure and evolution of baroclinic weather systems’, Monthly Weather Review 134, 34363455.

J. Kaipio and E. Somersalo (2000), ‘Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography’, Inverse Problems 16, 14871522.

J. Kaipio and E. Somersalo (2007 b), ‘Statistical inverse problems: Discretization, model reduction and inverse crimes’, J. Comput. Appl. Math. 198, 493504.

E. Kalnay , H. Li , S. Miyoshi , S. Yang and J. Ballabrera-Poy (2007), ‘4D-Var or ensemble Kalman filter?’, Tellus 59, 758773.

T. Kolda and B. Bader (2009), ‘Tensor decompositions and applications’, SIAM Review 51, 455500.

L. Kuznetsov , K. Ide and C. Jones (2003), ‘A method for assimilation of Lagrangian data’, Monthly Weather Review 131, 22472260.

M. Lassas and S. Siltanen (2004), ‘Can one use total variation prior for edge-preserving Bayesian inversion?’, Inverse Problems 20, 15371563.

A. Lawless and N. Nichols (2006), ‘Inner loop stopping criteria for incremental four-dimensional variational data assimilation’, Monthly Weather Review 134, 34253435.

A. Lawless , S. Gratton and N. Nichols (2005 a), ‘Approximate iterative methods for variational data assimilation’, Internat. J. Numer. Methods Fluids 47, 11291135.

A. Lawless , S. Gratton and N. Nichols (2005 b), ‘An investigation of incremental 4D-Var using non-tangent linear models’, Quart. J. Royal Met. Soc. 131, 459476.

A. Lawless , N. Nichols , C. Boess and A. Bunse-Gerstner (2008 a), ‘Approximate Gauss—Newton methods for optimal state estimation using reduced order models’, Internat. J. Numer. Methods Fluids 56, 13671373.

A. Lawless , N. Nichols , C. Boess and A. Bunse-Gerstner (2008 b), ‘Using model reduction methods within incremental four-dimensional variational data assimilation’, Monthly Weather Review 136, 15111522.

M. Lehtinen , L. Paivarinta and E. Somersalo (1989), ‘Linear inverse problems for generalized random variables’, Inverse Problems 5, 599612.

M. Lifshits (1995), Gaussian Random Functions, Vol. 322 of Mathematics and its Applications, Kluwer, Dordrecht.

D. Livings , S. Dance and N. Nichols (2008), ‘Unbiased ensemble square root filters’, Physica D: Nonlinear Phenomena 237, 10211028.

A. Lorenc (1986), ‘Analysis methods for numerical weather prediction’, Quart. J. Royal Met. Soc. 112, 11771194.

A. Majda and M. Grote (2007), ‘Explicit off-line criteria for stable accurate filtering of strongly unstable spatially extended systems’, Proc. Nat. Acad. Sci. 104, 11241129.

M. Martin , M. Bell and N. Nichols (2002), ‘Estimation of systematic error in an equatorial ocean model using data assimilation’, Internat. J. Numer. Methods Fluids 40, 435444.

I. McKeague , G. Nicholls , K. Speer and R. Herbei (2005), ‘Statistical inversion of south Atlantic circulation in an abyssal neutral density layer’, J. Marine Res. 63, 683704.

D. McLaughlin and L. Townley (1996), ‘A reassessment of the groundwater inverse problem’, Water Resources Res. 32, 11311161.

N. Metropolis , R. Rosenbluth , M. Teller and E. Teller (1953), ‘Equations of state calculations by fast computing machines’, J. Chem. Phys. 21, 10871092.

S. P. Meyn and R. L. Tweedie (1993), Markov Chains and Stochastic Stability, Communications and Control Engineering Series, Springer, London.

A. Michalak and P. Kitanidis (2003), ‘A method for enforcing parameter nonnegativity in Bayesian inverse problems with an application to contaminant source identification’, Water Resources Res. 39, 1033.

T. Mitchell , B. Buchanan , G. DeJong , T. Dietterich , P. Rosenbloom and A. Waibel (1990), ‘Machine learning’, Annual Review of Computer Science 4, 417433.

K. Mosegaard and A. Tarantola (1995), ‘Monte Carlo sampling of solutions to inverse problems’, J. Geophys. Research 100, 431447.

A. Neubauer (2009), ‘On enhanced convergence rates for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces’, Inverse Problems 25, #065009.

A. Neubauer and H. Pikkarainen (2008), ‘Convergence results for the Bayesian inversion theory’, J. Inverse and Ill-Posed Problems 16, 601613.

N. Nichols (2003 a), Data assimilation: Aims and basic concepts. In Data Assimilation for the Earth System (R. Swinbank , V. Shutyaev and W. A. Lahoz , eds), Kluwer Academic, pp. 920.

N. Nichols (2003 b), Treating model error in 3-D and 4-D data assimilation. In Data Assimilation for the Earth System (R. Swinbank , V. Shutyaev and W. A. Lahoz , eds), Kluwer Academic, pp. 127135.

M. Nodet (2006), ‘Variational assimilation of Lagrangian data in oceanography’, Inverse Problems 22, 245263.

B. Oksendal (2003), Stochastic Differential Equations: An Introduction with Applications, sixth edn, Universitext, Springer.

D. Orrell , L. Smith , J. Barkmeijer and T. Palmer (2001), ‘Model error in weather forecasting’, Non. Proc. in Geo. 8, 357371.

A. O'Sullivan and M. Christie (2006 a), ‘Error models for reducing history match bias’, Comput. Geosci. 10, 405–405.

A. O'Sullivan and M. Christie (2006 b), ‘Simulation error models for improved reservoir prediction’, Reliability Engineering and System Safety 91, 13821389.

E. Ott , B. Hunt , I. Szunyogh , A. Zimin , E. Kostelich , M. Corazza , E. Kalnay , D. Patil and J. Yorke (2004), ‘A local ensemble Kalman filter for atmospheric data assimilation’, Tellus A 56, 273277.

H. Pikkarainen (2006), ‘State estimation approach to nonstationary inverse problems: Discretization error and filtering problem’, Inverse Problems 22, 365379.

S. Pimentel , K. Haines and N. Nichols (2008 b), ‘Modelling the diurnal variability of sea surface temperatures’, J. Geophys. Research: Oceans 113, #C11004.

M. Reznikoff and E. Vanden Eijnden (2005), ‘Invariant measures of SPDEs and conditioned diffusions’, CR Acad. Sci. Paris 340, 305308.

G. Roberts and J. Rosenthal (1998), ‘Optimal scaling of discrete approximations to Langevin diffusions’, J. Royal Statist. Soc. B 60, 255268.

G. Roberts and J. Rosenthal (2001), ‘Optimal scaling for various Metropolis—Hastings algorithms’, Statistical Science 16, 351367.

G. Roberts and R. Tweedie (1996), ‘Exponential convergence of Langevin distributions and their discrete approximations’, Bernoulli 2, 341363.

L. Rudin , S. Osher and E. Fatemi (1992), ‘Nonlinear total variation based noise removal algorithms’, Physica D 60, 259268.

H. Rue and L. Held (2005), Gaussian Markov Random Fields: Theory and Applications, Chapman & Hall.

H. Salman , K. Ide and C. Jones (2008), ‘Using flow geometry for drifter deployment in Lagrangian data assimilation’, Tellus 60, 321335.

H. Salman , L. Kuznetsov , C. Jones and K. Ide (2006), ‘A method for assimilating Lagrangian data into a shallow-water equation ocean model’, Monthly Weather Review 134, 10811101.

J. M. Sanz-Serna and C. Palencia (1985), ‘A general equivalence theorem in the theory of discretization methods’, Math. Comp. 45, 143152.

O. Scherzer , M. Grasmair , H. Grossauer , M. Haltmeier and F. Lenzen (2009), Variational Methods in Imaging, Springer.

C. Schwab and R. Todor (2006), ‘Karhunen-Loeve approximation of random fields in domains by generalized fast multipole methods’, J. Comput. Phys. 217, 100122.

T. Snyder , T. Bengtsson , P. Bickel and J. Anderson (2008), ‘Obstacles to high-dimensional particle filtering’, Monthly Weather Review 136, 46294640.

P. Spanos and R. Ghanem (1989), ‘Stochastic finite element expansion for random media’, J. Engrg Mech. 115, 10351053.

E. Spiller , A. Budhiraja , K. Ide and C. Jones (2008), ‘Modified particle filter methods for assimilating Lagrangian data into a point-vortex model’, Physica D 237, 14981506.

A. Stuart , J. Voss and P. Wiberg (2004), ‘Conditional path sampling of SDEs and the Langevin MCMC method’, Comm. Math. Sci 2, 685697.

P. Talagrand and O. Courtier (1987), ‘Variational assimilation of meteorological observations with the adjoint vorticity equation I: Theory’, Quart. J. Royal Met. Soc. 113, 13111328.

R. Todor and C. Schwab (2007), ‘Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients’, IMA J. Numer. Anal. 27, 232261.

P. Van Leeuwen (2001), ‘An ensemble smoother with error estimates’, Monthly Weather Review 129, 709728.

P. Van Leeuwen (2003), ‘A variance minimizing filter for large-scale applications’, Monthly Weather Review 131, 20712084.

P. Van Leeuwen (2009), ‘Particle filtering in geophysical systems’, Monthly Weather Review 137, 40894114.

C. Vogel (2002), Computational Methods for Inverse Problems, SIAM.

F. Vossepoel and P. Van Leeuwen (2007), ‘Parameter estimation using a particle method: Inferring mixing coefficients from sea-level observations’, Monthly Weather Review 135, 10061020.

G. Wahba (1990), Spline Models for Observational Data, SIAM.

L. Watkinson , A. Lawless , N. Nichols and I. Roulstone (2007), ‘Weak constraints in four dimensional variational data assimilation’, Meteorologische Zeitschrift 16, 767776.

L. White (1993), ‘A study of uniqueness for the initialization problem for Burgers' equation’, J. Math. Anal. Appl. 172, 412431.

D. Williams (1991), Probability with Martingales, Cambridge University Press, Cambridge.

M. Wlasak , N. Nichols and I. Roulstone (2006), ‘Use of potential vorticity for incremental data assimilation’, Quart. J. Royal Met. Soc. 132, 28672886.

L. Yu and J. O'Brien (1991), ‘Variational estimation of the wind stress drag coefficient and the oceanic eddy viscosity profile’, J. Phys. Ocean. 21, 13611364.

O. Zeitouni and A. Dembo (1987), ‘A maximum a posteriori estimator for trajectories of diffusion processes’, Stochastics 20, 221246.

E. Zuazua (2005), ‘Propagation, observation, control and numerical approximation of waves approximated by finite difference method’, SIAM Review 47, 197243.

D. Zupanski (1997), ‘A general weak constraint applicable to operational 4DVAR data assimilation systems’, Monthly Weather Review 125, 22742292.

M. Zupanski , I. Navon and D. Zupanski (2008), ‘The maximum likelihood ensemble filter as a non-differentiable minimization algorithm’, Quart. J. Royal Met. Soc. 134, 10391050.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×