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Convex Analysisの二,三の進展について

Recent research in inverse problems seeks to develop a mathematically coherent foundation for combining data-driven models, and in particular those based on deep learning, with domain-specific knowledge contained in physical–analytical models. The focus is on solving ill-posed inverse problems that are at the core of many challenging applications in the natural sciences, medicine and life sciences, as well as in engineering and industrial applications. This survey paper aims to give an account of some of the main contributions in data-driven inverse problems.

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Solving inverse problems using data-driven models

This textbook provides an exciting new addition to the area of network science featuring a stronger and more methodical link of models to their mathematical origin and explains how these relate to each other with special focus on epidemic spread on networks. The content of the book is at the interface of graph theory, stochastic processes and dynamical systems. The authors set out to make a significant contribution to closing the gap between model development and the supporting mathematics. 

This is done by:

-Summarising and presenting the state-of-the-art in modeling epidemics on networks with results and readily usable models signposted throughout the book;

-Presenting different mathematical approaches to formulate exact and solvable models;

-Identifying the concrete links between approximate models and their rigorous mathematical representation;

-Presenting a model hierarchy and clearly highlighting the links between model assumptions and model complexity;

-Providing a reference source for advanced undergraduate students, as well as doctoral students, postdoctoral researchers and academic experts who are engaged in modeling stochastic processes on networks;

-Providing software that can solve differential equation models or directly simulate epidemics on networks.

Replete with numerous diagrams, examples, instructive exercises, and online access to simulation algorithms and readily usable code, this book will appeal to a wide spectrum of readers from different backgrounds and academic levels. Appropriate for students with or without a strong background in mathematics, this textbook can form the basis of an advanced undergraduate or graduate course in both mathematics and other departments alike.

Mathematics of Epidemics on Networks: From Exact to Approximate Models

Numerical methods for unilateral problems in solid mechanics

The Projection Theorem. Theorems on Extension and Separation. Dual Spaces and Transposed Operators. The Banach Theorem and the Banach-Steinhaus Theorem. Construction of Hilbert Spaces. L ~ 2 Spaces and Convolution Operators. Sobolev Spaces of Functions of One Variable. Some Approximation Procedures in Spaces of Functions. Sobolev Spaces of Functions of Several Variables and the Fourier Transform. Introduction to Set-Valued Analysis and Convex Analysis. Elementary Spectral Theory. Hilbert-Schmidt Operators and Tensor Products. Boundary Value Problems. Differential-Operational Equations and Semigroups of Operators. Viability Kernels and Capture Basins. First-Order Partial Differential Equations. Selection of Results. Exercises. Bibliography. Index.

Applied functional analysis, by Jean-Pierre Aubin; translated by Carole Labrousse. Pp 423. £16·50. 1980. ISBN 0-471-02149-0 (Wiley)

The problem of minimizing a possibly non-convex and non-coercive functional is studied. Either necessary or sufficient conditions for the existence of solutions are given, involving a generalized recession functional, whose properties are discussed thoroughly. The abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region.

General existence theorems for unilateral problems in continuum mechanics

We state a weak formulation for the Blake & Zissermann model of image segmentation and prove the existence of a minimizer for such a problem.

A second order model in image segmentation: Blake & Zisserman functional

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Franco Tomarelli

Papers

General existence theorems for unilateral problems in continuum mechanics

A second order model in image segmentation: Blake & Zisserman functional

Some existence results on noncoercive variational inequalities

Strong minimizers of Blake & Zisserman functional

Special Bounded Hessian and elastic-plastic plate