"Parameter Estimation and Inverse Problems, 2/e" provides geoscience students and professionals with answers to common questions like how one can derive a physical model from a finite set of observations containing errors, and how one may determine the quality of such a model. This book takes on these fundamental and challenging problems, introducing students and professionals to the broad range of approaches that lie in the realm of inverse theory. The authors present both the underlying theory and practical algorithms for solving inverse problems. The authors' treatment is appropriate for geoscience graduate students and advanced undergraduates with a basic working knowledge of calculus, linear algebra, and statistics. "Parameter Estimation and Inverse Problems, 2/e" introduces readers to both Classical and Bayesian approaches to linear and nonlinear problems with particular attention paid to computational, mathematical, and statistical issues related to their application to geophysical problems. The textbook includes Appendices covering essential linear algebra, statistics, and notation in the context of the subject. This book includes a companion website that features computational examples (including all examples contained in the textbook) and useful subroutines using MATLAB. It: includes appendices for review of needed concepts in linear, statistics, and vector calculus; features a companion website that contains comprehensive MATLAB code for all examples, which readers can reproduce, experiment with, and modify; offers an online instructor's guide that helps professors teach, customize exercises, and select homework problems; and, is accessible to students and professionals without a highly specialized mathematical background.

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Parameter estimation and inverse problems

Boundary Value Problems in Physics and Engineering By Frank Chorlton. Pp. 250. (Van Nostrand: London, July 1969.) 70s

Boundary value problems

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

Gabrio Piola’s scientific papers have been underestimated in mathematical physics literature. Indeed, a careful reading of them proves that they are original, deep and far-reaching. Actually, even ...

https://hal.archives-ouvertes.fr/hal-00869677/document

At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola

One of the research direction of Horst Lippmann during his whole scientific career was devoted to the possibilities to explain complex material behavior by generalized continua models. A representative of such models is the Cosserat continuum. The basic idea of this model is the independence of translations and rotations (and by analogy, the independence of forces and moments). With the help of this model some additional effects in solid and fluid mechanics can be explained in a more satisfying manner. They are established in experiments, but not presented by the classical equations. In this paper the Cosserat-type theories of plates and shells are debated as a special application of the Cosserat theory.

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On generalized Cosserat-type theories of plates and shells: a short review and bibliography

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Foundations of Micropolar Mechanics

Acceleration waves in nonlinear thermoelastic micropolar media are considered. We establish the kinematic and dynamic compatibility relations for a singular surface of order 2 in the media. An analogy to the Fresnel–Hadamard–Duhem theorem and an expression for the acoustic tensor are derived. The condition for acceleration wave’s propagation is formulated as an algebraic spectral problem. It is shown that the condition coincides with the strong ellipticity of equilibrium equations. As an example, a quadratic form for the specific free energy is considered and the solutions of the corresponding spectral problem are presented.

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Acceleration waves and ellipticity in thermoelastic micropolar media

Tensor Analysis: Transformations and Vectors Tensors Tensor Fields Elements of Differential Geometry Applications in Mechanics: Linear Elasticity Linear Elastic Shells Appendices: Formulary Hints and Answers.

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Tensor Analysis With Applications In Mechanics

The mathematical investigation of the initial-boundary and boundary value problems in the linear elasticity considering surface stresses is presented. Weak setup of the problems based on mechanical variational principles is studied. Theorems of uniqueness and existence of the weak solution in energy spaces of static and dynamic problems are formulated and proved. Some properties of the spectrum of the problems under consideration are established. The studies are performed applying the functional analysis techniques. Finally, the Rayleigh principle for eigenfrequencies is constructed.

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On the existence of solution in the linear elasticity with surface stresses

Mathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin–Murdoch model to the Steigmann–Ogden model of surface elasticity using techniques from the theory of Sobolev’s spaces and methods of functional analysis. The Steigmann–Ogden model accounts for the bending stiffness of the surface film; it is a generalization of the Gurtin–Murdoch model. Weak setups of the problems, based on variational principles formulated, are employed. Some uniqueness-existence theorems for weak solutions of static and dynamic problems are proved in energy spaces via functional analytic methods. On the boundary surface, solutions to the problems under consideration are smoother than those for the corresponding problems of classical linear elasticity and those described by the Gurtin–Murdoch model. The weak setups of eigenvalue problems for elastic bodies with surface stresses are based on the Rayleigh and Courant variational principles. For the problems based on the Steigmann–Ogden model, certain spectral properties are established. In particular, bounds are placed on the eigenfrequencies of an elastic body with surface stresses; these demonstrate the increase in the body rigidity and the eigenfrequencies compared with the situation where the surface stresses are neglected.

Leonid P. Lebedev

Papers

Foundations of Micropolar Mechanics

Acceleration waves and ellipticity in thermoelastic micropolar media

Tensor Analysis With Applications In Mechanics

On the existence of solution in the linear elasticity with surface stresses

Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity