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Written by two international experts in the field, this book is the first unified survey of the advances made in the last 15 years on key non-standard and improperly posed problems for partial differential equations.This reference for mathematicians, scientists, and engineers provides an overview of the methodology typically used to study improperly posed problems. It focuses on structural stability--the continuous dependence of solutions on the initial conditions and the modeling equations--and on problems for which data are only prescribed on part of the boundary.The book addresses conti

Differential equations --- Differential equations, Partial --- Approximation theory. --- Improperly posed problems.

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Ill-posed problems are encountered in countless areas of real world science and technology. A variety of processes in science and engineering is commonly modeled by algebraic, differential, integral and other equations. In a more difficult case, it can be systems of equations combined with the associated initial and boundary conditions. Frequently, the study of applied optimization problems is also reduced to solving the corresponding equations. These equations, encountered both in theoretical and applied areas, may naturally be classified as operator equations. The current textbook will focus on iterative methods for operator equations in Hilbert spaces.

Differential equations, Partial --- Iterative methods (Mathematics) --- Iteration (Mathematics) --- Numerical analysis --- Improperly posed problems in partial differential equations --- Improperly posed problems. --- Ill-posed problems --- Hilbert Space. --- Ill-posed Problem. --- Inverse Problem. --- Iterative Method. --- Operator Equation.

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This monograph is a valuable contribution to the highly topical and extremely productive field of regularization methods for inverse and ill-posed problems. The author is an internationally outstanding and accepted mathematician in this field. In his book he offers a well-balanced mixture of basic and innovative aspects. He demonstrates new, differentiated viewpoints, and important examples for applications. The book demonstrates the current developments in the field of regularization theory, such as multi parameter regularization and regularization in learning theory. The book is written for graduate and PhDs

Numerical analysis --- Numerical differentiation. --- Graphic differentiation --- Functions --- Improperly posed problems in numerical analysis --- Improperly posed problems. --- Ill-posed problems --- Balancing Principle. --- Blood Glucose Prediction. --- Convergence Rate. --- Discrepancy Principle. --- Error Bound Estimation. --- Ill-posed Problem. --- Learning Theory, Meta-learning. --- Multi-parameter Regularization. --- Regularization Method.

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Internal boundary value problems deals with the problem of determining the solution of an equation if data are given on two manifolds. One manifold is the domain boundary and the other manifold is situated inside the domain. This monograph studies three essentially ill-posed internal boundary value problems for the biharmonic equation and the Cauchy problem for the abstract biharmonic equation, both qualitatively and quantitatively. In addition, some variants of these problems and the Cauchy problem, as well as the m-dimensional case, are considered. The author introduces some new notions, such as the notion of complete solvability.

Differential equations, Partial --- Boundary value problems. --- Biharmonic equations. --- Equations, Biharmonic --- Boundary conditions (Differential equations) --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Improperly posed problems in partial differential equations --- Improperly posed problems. --- Ill-posed problems --- Biharmonic Equation. --- Cauchy Problem. --- Domain Boundary. --- Ill-posed. --- Internal Boundary Value Problems. --- Manifolds.

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Interest in regularization methods for ill-posed nonlinear operator equations and variational inequalities of monotone type in Hilbert and Banach spaces has grown rapidly over recent years. Results in the field over the last three decades, previously only available in journal articles, are comprehensively explored with particular attention given to applications of regularization methods as well as to practical methods used in computational analysis.

Monotone operators. --- Differential equations, Partial --- Improperly posed problems. --- Improperly posed problems in partial differential equations --- Operator theory --- Ill-posed problems --- Global analysis (Mathematics). --- Computer science --- Operator theory. --- Functional analysis. --- Mathematical optimization. --- Analysis. --- Computational Mathematics and Numerical Analysis. --- Operator Theory. --- Functional Analysis. --- Calculus of Variations and Optimal Control; Optimization. --- Mathematics. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Functional analysis --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematics --- Mathematical analysis. --- Analysis (Mathematics). --- Computer mathematics. --- Calculus of variations. --- Isoperimetrical problems --- Variations, Calculus of --- 517.1 Mathematical analysis