# 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

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### 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.