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In this text, we introduce the basic concepts for the numerical modelling of partial differential equations. We consider the classical elliptic, parabolic and hyperbolic linear equations, but also the diffusion, transport, and Navier-Stokes equations, as well as equations representing conservation laws, saddle-point problems and optimal control problems. Furthermore, we provide numerous physical examples which underline such equations. We then analyze numerical solution methods based on finite elements, finite differences, finite volumes, spectral methods and domain decomposition methods, and reduced basis methods. In particular, we discuss the algorithmic and computer implementation aspects and provide a number of easy-to-use programs. The text does not require any previous advanced mathematical knowledge of partial differential equations: the absolutely essential concepts are reported in a preliminary chapter. It is therefore suitable for students of bachelor and master courses in scientific disciplines, and recommendable to those researchers in the academic and extra-academic domain who want to approach this interesting branch of applied mathematics.

Mathematics. --- Mathematics, general. --- Analysis. --- Numerical Analysis. --- Mathematical Modeling and Industrial Mathematics. --- Applications of Mathematics. --- Computational Mathematics and Numerical Analysis. --- Global analysis (Mathematics). --- Computer science --- Numerical analysis. --- Mathématiques --- Analyse globale (Mathématiques) --- Informatique --- Analyse numérique --- Differential equations, Partial --- Numerical analysis --- Numerical solutions --- Numerical solutions. --- Differential equations, Partial -- Numerical solutions. --- Electronic books. -- local. --- Mathematics --- Mathematical Theory --- Calculus --- Physical Sciences & Mathematics --- Mathematical analysis. --- Analysis (Mathematics). --- Applied mathematics. --- Engineering mathematics. --- Computer mathematics. --- Mathematical models. --- Mathematical analysis --- Models, Mathematical --- Simulation methods --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Engineering --- Engineering analysis --- 517.1 Mathematical analysis --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Differential equations, Partial - Numerical solutions

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This up-to-date book is on algorithms for large-scale unconstrained and bound constrained optimization. Optimization techniques are shown from a conjugate gradient algorithm perspective. Large part of the book is devoted to preconditioned conjugate gradient algorithms. In particular memoryless and limited memory quasi-Newton algorithms are presented and numerically compared to standard conjugate gradient algorithms. The special attention is paid to the methods of shortest residuals developed by the author. Several effective optimization techniques based on these methods are presented. Because of the emphasis on practical methods, as well as rigorous mathematical treatment of their convergence analysis, the book is aimed at a wide audience. It can be used by researches in optimization, graduate students in operations research, engineering, mathematics and computer science. Practitioners can benefit from numerous numerical comparisons of professional optimization codes discussed in the book.

Algorithms. --- Conjugate gradient methods. --- Mathematical optimization. --- Conjugate gradient methods --- Algorithms --- Mathematical optimization --- Mathematics --- Civil & Environmental Engineering --- Physical Sciences & Mathematics --- Engineering & Applied Sciences --- Mathematical Theory --- Operations Research --- Algebra --- Calculus --- Algorism --- Gradient methods, Conjugate --- Mathematics. --- Operations research. --- Decision making. --- Calculus of variations. --- Quality control. --- Reliability. --- Industrial safety. --- Calculus of Variations and Optimal Control; Optimization. --- Operation Research/Decision Theory. --- Quality Control, Reliability, Safety and Risk. --- Arithmetic --- Approximation theory --- Equations --- Iterative methods (Mathematics) --- Foundations --- Numerical solutions --- System safety. --- Operations Research/Decision Theory. --- Safety, System --- Safety of systems --- Systems safety --- Accidents --- Industrial safety --- Systems engineering --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Prevention --- Industrial accidents --- Industries --- Job safety --- Occupational hazards, Prevention of --- Occupational health and safety --- Occupational safety and health --- Prevention of industrial accidents --- Prevention of occupational hazards --- Safety, Industrial --- Safety engineering --- Safety measures --- Safety of workers --- System safety --- Dependability --- Trustworthiness --- Conduct of life --- Factory management --- Reliability (Engineering) --- Sampling (Statistics) --- Standardization --- Quality assurance --- Quality of products --- Deciding --- Decision (Psychology) --- Decision analysis --- Decision processes --- Making decisions --- Management --- Management decisions --- Choice (Psychology) --- Problem solving --- Isoperimetrical problems --- Variations, Calculus of --- Decision making