Well-posed problem

ill-posedill-posed problemwell-posedill-posed problemswell posedwell-posednessill posedwell posednessclosureIll-posedness
The mathematical term well-posed problem stems from a definition given by Jacques Hadamard.wikipedia
75 Related Articles

Jacques Hadamard

HadamardHadamard, JacquesHadamard, Jacques Solomon
He introduced the idea of well-posed problem and the method of descent in the theory of partial differential equations, culminating in his seminal book on the subject, based on lectures given at Yale University in 1922.

Heat equation

heat diffusionheatanalytic theory of heat
Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions.

Inverse problem

inverse problemsinverse theoryinversion
Inverse problems are often ill-posed.
Inverse problems are typically ill posed, as opposed to the well-posed problems usually met in mathematical modeling.

Regularization (mathematics)

regularizationregularizedregularize
This process is known as regularization.
In mathematics, statistics, and computer science, particularly in machine learning and inverse problems, regularization is the process of adding information in order to solve an ill-posed problem or to prevent overfitting.

Tikhonov regularization

ridge regressionregularizeda squared regularizing function
Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems.
Tikhonov regularization, named for Andrey Tikhonov, is a method of regularization of ill-posed problems.

Condition number

ill-conditionedwell-conditionedconditioning
An ill-conditioned problem is indicated by a large condition number.
The condition number may also be infinite, but this implies that the problem is ill-posed (does not possess a unique, well-defined solution for each choice of data; that is, the matrix is not invertible), and no algorithm can be expected to reliably find a solution.

Total absorption spectroscopy

total absorption spectrometer
*Total absorption spectroscopy – an example of an inverse problem or ill-posed problem in a real-life situation that is solved by means of the expectation–maximization algorithm
To be able to extract the value of i from the data d the equation has to be inverted (this equation is also called the "inverse problem").

Archetype

archetypalarchetypesarchetypical
Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation with specified initial conditions.

Discretization

discretizeddiscretizingdichotomization
Continuum models must often be discretized in order to obtain a numerical solution.

Complex system

complex systemscomplexity theorycomplexity science
Problems in nonlinear complex systems (so called chaotic systems) provide well-known examples of instability.

Numerical stability

numerically stablenumerical instabilitynumerically unstable
While solutions may be continuous with respect to the initial conditions, they may suffer from numerical instability when solved with finite precision, or with errors in the data.

Expectation–maximization algorithm

expectation-maximization algorithmEM algorithmexpectation maximization
*Total absorption spectroscopy – an example of an inverse problem or ill-posed problem in a real-life situation that is solved by means of the expectation–maximization algorithm

Landweber iteration

Landweberprojected Landweber
The Landweber iteration or Landweber algorithm is an algorithm to solve ill-posed linear inverse problems, and it has been extended to solve non-linear problems that involve constraints.

List of numerical analysis topics

List of eigenvalue algorithmsList of numerical analysis topics. Solving systems of linear equationsNumerical analysis

Regularized least squares

listPenalized regressionregularized least-squares
In such settings, the ordinary least-squares problem is ill-posed and is therefore impossible to fit because the associated optimization problem has infinitely many solutions.

Fritz John

JohnJohn, Fritz
Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems.

Yvonne Choquet-Bruhat

Yvonne BruhatBruhatChoquet-Bruhat
She is one of the pioneers of the study of general relativity, and particularly known as the first to prove the well-posedness of the Einstein equations.

Raymond L. Johnson

Raymond Lewis Johnson
His research concerns non-well-posed problems and harmonic analysis.

Motion estimation

feature trackingMotion-tracking
It is an ill-posed problem as the motion is in three dimensions but the images are a projection of the 3D scene onto a 2D plane.

Electrical impedance tomography

Electrical Resistance Tomographyelectrical impedance tomography & electrical resistance tomographyInverse problems of conductivity
Mathematically, the problem of recovering conductivity from surface measurements of current and potential is a non-linear inverse problem and is severely ill-posed.

Boundary value problem

boundary conditionboundary conditionsboundary-value problem
To be useful in applications, a boundary value problem should be well posed.

Shrinkage estimator

less biased estimatorshrink
In this sense, shrinkage is used to regularize ill-posed inference problems.

Andrey Nikolayevich Tikhonov

Andrey TikhonovAndrey TychonoffAndrey Nikolayevich Tychonoff
Andrey Nikolayevich Tikhonov (Андре́й Никола́евич Ти́хонов; October 17, 1906 – October 7, 1993) was a Soviet and Russian mathematician and geophysicist known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems.

Signorini problem

Signorini asked to determine if the problem is well-posed or not in a physical sense, i.e. if its solution exists and is unique or not: he explicitly invited young analysts to study the problem.