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Differential equations. --- 517.91 Differential equations --- Differential equations --- Asymptotic theory.

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This volume gathers contributions from participants of the Introductory School and the IHP thematic quarter on Numerical Methods for PDE, held in 2016 in Cargese (Corsica) and Paris, providing an opportunity to disseminate the latest results and envisage fresh challenges in traditional and new application fields. Numerical analysis applied to the approximate solution of PDEs is a key discipline in applied mathematics, and over the last few years, several new paradigms have appeared, leading to entire new families of discretization methods and solution algorithms. This book is intended for researchers in the field.

Differential equations, Partial --- Asymptotic theory in partial differential equations --- Asymptotic expansions --- Asymptotic theory. --- Numerical analysis. --- Differential equations, partial. --- Computer science. --- Numerical Analysis. --- Partial Differential Equations. --- Computational Science and Engineering. --- Informatics --- Science --- Partial differential equations --- Mathematical analysis --- Partial differential equations. --- Computer mathematics. --- Computer mathematics --- Electronic data processing --- Mathematics

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This book, which is a continuation of Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, presents recent trends and developments upon fractional, first, and second order semilinear difference and differential equations, including degenerate ones. Various stability, uniqueness, and existence results are established using various tools from nonlinear functional analysis and operator theory (such as semigroup methods). Various applications to partial differential equations and the dynamic of populations are amply discussed. This self-contained volume is primarily intended for advanced undergraduate and graduate students, post-graduates and researchers, but may also be of interest to non-mathematicians such as physicists and theoretically oriented engineers. It can also be used as a graduate text on evolution equations and difference equations and their applications to partial differential equations and practical problems arising in population dynamics. For completeness, detailed preliminary background on Banach and Hilbert spaces, operator theory, semigroups of operators, and almost periodic functions and their spectral theory are included as well.

Evolution equations. --- Evolution equations --- Asymptotic theory in evolution equations --- Asymptotic expansions --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- Differential equations --- Asymptotic theory. --- Functional analysis. --- Operator theory. --- Differential equations, partial. --- Functional equations. --- Functional Analysis. --- Operator Theory. --- Partial Differential Equations. --- Difference and Functional Equations. --- Equations, Functional --- Functional analysis --- Partial differential equations --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Partial differential equations. --- Difference equations. --- Calculus of differences --- Differences, Calculus of --- Equations, Difference

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This book provides a direct and comprehensive introduction to theoretical and numerical concepts in the emerging field of optimal control of partial differential equations (PDEs) under uncertainty. The main objective of the book is to offer graduate students and researchers a smooth transition from optimal control of deterministic PDEs to optimal control of random PDEs. Coverage includes uncertainty modelling in control problems, variational formulation of PDEs with random inputs, robust and risk-averse formulations of optimal control problems, existence theory and numerical resolution methods. The exposition focusses on the entire path, starting from uncertainty modelling and ending in the practical implementation of numerical schemes for the numerical approximation of the considered problems. To this end, a selected number of illustrative examples are analysed in detail throughout the book. Computer codes, written in MatLab, are provided for all these examples. This book is adressed to graduate students and researches in Engineering, Physics and Mathematics who are interested in optimal control and optimal design for random partial differential equations.

Differential equations, Partial --- Asymptotic theory in partial differential equations --- Asymptotic expansions --- Asymptotic theory. --- Differential equations, partial. --- Engineering mathematics. --- Vibration. --- Mechanics. --- Mechanics, Applied. --- Mathematical optimization. --- Distribution (Probability theory. --- Partial Differential Equations. --- Mathematical and Computational Engineering. --- Vibration, Dynamical Systems, Control. --- Solid Mechanics. --- Calculus of Variations and Optimal Control; Optimization. --- Probability Theory and Stochastic Processes. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Classical mechanics --- Newtonian mechanics --- Physics --- Dynamics --- Quantum theory --- Cycles --- Mechanics --- Sound --- Engineering --- Engineering analysis --- Partial differential equations --- Mathematics --- Partial differential equations. --- Applied mathematics. --- Dynamical systems. --- Dynamics. --- Calculus of variations. --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Isoperimetrical problems --- Variations, Calculus of --- Dynamical systems --- Kinetics --- Mechanics, Analytic --- Force and energy --- Statics

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This book summarizes recent advances in applying saddlepoint approximation methods to financial engineering. It addresses pricing exotic financial derivatives and calculating risk contributions to Value-at-Risk and Expected Shortfall in credit portfolios under various default correlation models. These standard problems involve the computation of tail probabilities and tail expectations of the corresponding underlying state variables.  The text offers in a single source most of the saddlepoint approximation results in financial engineering, with different sets of ready-to-use approximation formulas. Much of this material may otherwise only be found in original research publications. The exposition and style are made rigorous by providing formal proofs of most of the results. Starting with a presentation of the derivation of a variety of saddlepoint approximation formulas in different contexts, this book will help new researchers to learn the fine technicalities of the topic. It will also be valuable to quantitative analysts in financial institutions who strive for effective valuation of prices of exotic financial derivatives and risk positions of portfolios of risky instruments.  .

Financial engineering. --- Method of steepest descent (Numerical analysis) --- Mathematics. --- Economics, Mathematical. --- Quantitative Finance. --- Approximations, Saddlepoint --- Descent, Method of steepest (Numerical analysis) --- Method of saddle points (Numerical analysis) --- Saddle point method (Numerical analysis) --- Saddle points, Method of (Numerical analysis) --- Saddlepoint approximations --- Saddlepoint method (Numerical analysis) --- Steepest descent method (Numerical analysis) --- Approximation theory --- Asymptotic expansions --- Computational finance --- Engineering, Financial --- Finance --- Finance. --- Funding --- Funds --- Economics --- Currency question --- Economics, Mathematical . --- Mathematical economics --- Econometrics --- Mathematics --- Methodology