Note: Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Acknowledgements -- Part I Fundamental machinery -- 1 Introduction -- 1.1 Differential equations -- 1.2 Partial differential equations -- 1.3 More examples -- A tracer box model -- A tomographic problem -- A second tracer problem -- Vibrating string -- 1.4 Importance of the forward model -- Notes -- 2 Basic machinery -- 2.1 Background -- 2.2 Matrix and vector algebra -- Matrices and vectors -- Gram-Schmidt process -- 2.2.1 Matrix multiplication and identities -- 2.2.2 Linear simultaneous equations -- 2.2.3 Matrix norms -- 2.2.4 Identities: differentiation -- 2.3 Simple statistics: regression -- 2.3.1 Probability densities, moments -- 2.3.2 Sample estimates: bias -- 2.3.3 Functions and sums of random variables -- 2.3.4 Multivariable probability densities: correlation -- 2.3.5 Change of variables -- Vector random processes -- 2.3.6 Sums of random variables -- Degrees-of-Freedom -- Stationarity -- 2.4 Least-squares -- 2.4.1 Basic formulation -- 2.4.2 Weighted and tapered least-squares -- 2.4.3 Underdetermined systems and Lagrange multipliers -- Lagrange multipliers and adjoints -- 2.4.4 Interpretation of discrete adjoints -- 2.5 The singular vector expansion -- 2.5.1 Simple vector expansions -- 2.5.2 Square-symmetric problem: eigenvalues/eigenvectors -- 2.5.3 Arbitrary systems -- The singular vector expansion and singular value decomposition -- 2.5.4 The singular value decomposition -- 2.5.5 Some simple examples: algebraic equations -- 2.5.6 Simple examples: differential and partial differential equations -- 2.5.7 Relation of least-squares to the SVD -- 2.5.8 Pseudo-inverses -- 2.5.9 Row and column scaling -- Column scaling -- 2.5.10 Solution and observation resolution: data ranking -- 2.5.11 Relation to tapered and weighted least-squares. , 2.5.12 Resolution and variance of tapered solutions -- 2.6 Combined least-squares and adjoints -- 2.6.1 Exact constraints -- 2.6.2 Relation to Green functions -- 2.7 Minimum variance estimation and simultaneous equations -- 2.7.1 The fundamental result -- 2.7.2 Linear algebraic equations -- 2.7.3 Testing after the fact -- 2.7.4 Use of basis functions -- 2.7.5 Determining a mean value -- 2.8 Improving recursively -- 2.8.1 Least-squares -- 2.8.2 Minimum variance recursive estimates -- 2.9 Summary -- Appendix 1. Maximum likelihood -- Appendix 2. Differential operators and Green functions -- Appendix 3. Recursive least-squares and Gauss-Markov solutions -- Notes -- 3 Extensions of methods -- 3.1 The general eigenvector/eigenvalue problem -- 3.2 Sampling -- 3.2.1 One-dimensional interpolation -- 3.2.2 Higher-dimensional mapping -- 3.2.3 Mapping derivatives -- 3.3 Inequality constraints: non-negative least-squares -- 3.4 Linear programming -- 3.5 Empirical orthogonal functions -- 3.6 Kriging and other variants of Gauss-Markov estimation -- 3.7 Non-linear problems -- 3.7.1 Total least-squares -- 3.7.2 Method of total inversion -- 3.7.3 Variant non-linear methods, including combinatorial ones -- Notes -- 4 The time-dependent inverse problem: state estimation -- 4.1 Background -- 4.2 Basic ideas and notation -- 4.2.1 Models -- 4.2.2 How to find the matrix A(t) -- 4.2.3 Observations and data -- 4.3 Estimation -- 4.3.1 Model and data consistency -- 4.3.2 The Kalman filter -- 4.3.3 The smoothing problem -- 4.3.4 Other smoothers -- 4.4 Control and estimation problems -- 4.4.1 Lagrange multipliers and adjoints -- 4.4.2 Terminal constraint problem: open-loop control -- 4.4.3 Representers and boundary Green functions -- 4.4.4 The control Riccati equation -- 4.4.5 The initialization problem -- 4.5 Duality and simplification: the steady-state filter and adjoint. , 4.6 Controllability and observability -- Controllability -- Observability -- 4.7 Non-linear models -- 4.7.1 The linearized and extended Kalman filter -- 4.7.2 Parameter estimation and adaptive estimation -- 4.7.3 Non-linear adjoint equations: searching for solutions -- 4.7.4 Automatic differentiation, linearization, and sensitivity -- 4.7.5 Approximate methods -- 4.8 Forward models -- 4.9 A summary -- Appendix. Automatic differentiation and adjoints -- Notes -- 5 Time-dependent methods - 2 -- 5.1 Monte Carlo/ensemble methods -- 5.1.1 Ensemble methods and particle filters -- 5.2 Numerical engineering: the search for practicality -- 5.2.1 Meteorological assimilation -- 5.2.2 Nudging and objective mapping -- 5.2.3 Approximate filter/smoother methods -- Steady-state approximation -- 5.2.4 Reduced state methods -- Other approaches to state reduction -- 5.3 Uncertainty in Lagrange multiplier method -- 5.4 Non-normal systems -- 5.4.1 POPs and optimal modes -- 5.5 Adaptive problems -- Appendix. Doubling -- Notes -- Part II Applications -- 6 Applications to steady problems -- 6.1 Steady-state tracer distributions -- 6.2 The steady ocean circulation inverse problem -- 6.2.1 Equations of motion -- 6.2.2 Geostrophy -- The classical dynamic method -- 6.2.3 Integral version -- 6.2.4 Discrete version -- 6.2.5 A specific example -- 6.2.6 Solution by SVD -- Controlling the solution -- 6.2.7 Solution by Gauss-Markov estimate -- 6.2.8 Adding further properties -- 6.3 Property fluxes -- 6.4 Application to real oceanographic problems -- 6.4.1 Regional applications -- 6.4.2 The columnar result -- 6.4.3 Global-scale applications -- 6.4.4 Error estimates -- 6.4.5 Finite-difference models -- 6.5 Linear programming solutions -- 6.6 The β-spiral and variant methods -- 6.6.1 The β-spiral -- 6.7 Alleged failure of inverse methods. , 6.8 Applications of empirical orthogonal functions (EOFs) (singular vectors) -- 6.9 Non-linear problems -- Notes -- 7 Applications to time-dependent fluid problems -- 7.1 Time-dependent tracers -- 7.2 Global ocean states by Lagrange multiplier methods -- 7.3 Global ocean states by sequential methods -- 7.4 Miscellaneous approximations and applications -- 7.5 Meteorological applications -- References -- Index.

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Note: Includes bibliographical references and index

Note: Literaturverzeichnis: Seiten [447]-476

Note: Includes bibliographical references and indexes