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The book examines theoretical and computational aspects of least-squares finite element methods(LSFEMs) for partial differential equations (PDEs) arising in key science and engineering applications. It is intended for mathematicians, scientists, and engineers interested in either or both the theory and practice associated with the numerical solution of PDEs. The first part looks at strengths and weaknesses of classical variational principles, reviews alternative variational formulations, and offers a glimpse at the main concepts that enter into the formulation of LSFEMs. Subsequent parts introduce mathematical frameworks for LSFEMs and their analysis, apply the frameworks to concrete PDEs, and discuss computational properties of resulting LSFEMs. Also included are recent advances such as compatible LSFEMs, negative-norm LSFEMs, and LSFEMs for optimal control and design problems. Numerical examples illustrate key aspects of the theory ranging from the importance of norm-equivalence to connections between compatible LSFEMs and classical-Galerkin and mixed-Galerkin methods. Pavel Bochev is a Distinguished Member of the Technical Staff at Sandia National Laboratories with research interests in compatible discretizations for PDEs, multiphysics problems, and scientific computing. Max Gunzburger is Frances Eppes Professor of Scientific Computing and Mathematics at Florida State University and recipient of the W.T. and Idelia Reid Prize in Mathematics from the Society for Industrial and Applied Mathematics. .

Differential equations, Partial. --- Finite element method. --- Least squares. --- Least squares --- Finite element method --- Differential equations, Partial --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Mathematical Theory --- Partial differential equations --- FEA (Numerical analysis) --- FEM (Numerical analysis) --- Finite element analysis --- Method of least squares --- Squares, Least --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Computer mathematics. --- Numerical analysis. --- Calculus of variations. --- Applied mathematics. --- Engineering mathematics. --- Fluid mechanics. --- Numerical Analysis. --- Analysis. --- Appl.Mathematics/Computational Methods of Engineering. --- Computational Mathematics and Numerical Analysis. --- Calculus of Variations and Optimal Control; Optimization. --- Engineering Fluid Dynamics. --- Hydromechanics --- Continuum mechanics --- Engineering --- Engineering analysis --- Mathematical analysis --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- 517.1 Mathematical analysis --- Math --- Science --- Numerical analysis --- Isogeometric analysis --- Curve fitting --- Geodesy --- Mathematical statistics --- Probabilities --- Triangulation --- Global analysis (Mathematics). --- Computer science --- Mathematical optimization. --- Hydraulic engineering. --- Mathematical and Computational Engineering. --- Engineering, Hydraulic --- Fluid mechanics --- Hydraulics --- Shore protection --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Operations research --- Simulation methods --- System analysis --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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This volume is focused on the review of recent algorithmic and mathematical advances and the development of new research directions for Mathematical Model Approximations via RAMSES (Reduced order models, Approximation theory, Machine learning, Surrogates, Emulators, Simulators) in the setting of parametrized partial differential equations also with sparse and noisy data in high-dimensional parameter spaces. The book is a valuable resource for researchers, as well as masters and Ph.D students.

Numerical analysis. --- Machine learning. --- Numerical Analysis. --- Machine Learning.

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