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mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
term well-posed problem stems from a definition given by 20th-century French mathematician
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...
. He believed that
mathematical model A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
s of
physical phenomena Physical may refer to: *Physical examination, a regular overall check-up with a doctor *Physical (Olivia Newton-John album), ''Physical'' (Olivia Newton-John album), 1981 **Physical (Olivia Newton-John song), "Physical" (Olivia Newton-John song) *P ...
should have the properties that: # a
solution Solution may refer to: * Solution (chemistry), a mixture where one substance is dissolved in another * Solution (equation), in mathematics ** Numerical solution, in numerical analysis, approximate solutions within specified error bounds * Soluti ...
exists, # the solution is unique, # the solution's behaviour changes continuously with the
initial condition In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
s. Examples of
archetypal The concept of an archetype (; ) appears in areas relating to behavior, historical psychology, and literary analysis. An archetype can be any of the following: # a statement, pattern of behavior, prototype, "first" form, or a main model that ...
well-posed problems include the Dirichlet problem for Laplace's equation, and the
heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for t ...
with specified initial conditions. These might be regarded as 'natural' problems in that there are physical processes modelled by these problems. Problems that are not well-posed in the sense of Hadamard are termed ill-posed.
Inverse problem An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, source reconstruction in acoustics, or calculating the ...
s are often ill-posed. For example, the inverse heat equation, deducing a previous distribution of temperature from final data, is not well-posed in that the solution is highly sensitive to changes in the final data. Continuum models must often be discretized in order to obtain a numerical solution. While solutions may be continuous with respect to the initial conditions, they may suffer from
numerical instability In the mathematics, mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the oth ...
when solved with finite
precision Precision, precise or precisely may refer to: Science, and technology, and mathematics Mathematics and computing (general) * Accuracy and precision, measurement deviation from true value and its scatter * Significant figures, the number of digit ...
, or with
error An error (from the Latin ''error'', meaning "wandering") is an action which is inaccurate or incorrect. In some usages, an error is synonymous with a mistake. The etymology derives from the Latin term 'errare', meaning 'to stray'. In statistics ...
s in the data. Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers. Problems in nonlinear
complex systems A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication s ...
(so-called
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kid ...
systems) provide well-known examples of instability. An ill-conditioned problem is indicated by a large
condition number In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input ...
. If the problem is well-posed, then it stands a good chance of solution on a computer using a stable algorithm. If it is not well-posed, it needs to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. This process is known as ''
regularization Regularization may refer to: * Regularization (linguistics) * Regularization (mathematics) * Regularization (physics) In physics, especially quantum field theory, regularization is a method of modifying observables which have singularities in ...
''.
Tikhonov regularization Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also ...
is one of the most commonly used for regularization of linear ill-posed problems.


Energy method

A method to determine the well-posedness of a problem is the energy method. The method is based upon deriving an energy estimate for a given problem. Example: Consider the linear advection equation with homogeneous
Dirichlet boundary conditions In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differential ...
and suitable initial data f(x). \begin u_t+\alpha u_x=0, 0 0,\\ u(x,0)=f(x),\\ u(0,t)=0,\\ u(1,t)=0,\\ \end Then carrying out the energy method for this problem, one would multiply the equation by u and integrate in space over the given interval. \partial_t \int_0^1 \frac u^2 dx=-\alpha\int_0^1uu_xdx\Rightarrow \frac \partial_t\, u\, _2^2=-\alpha\frac\Big, _0^1=0 Then one would integrate in time and one would obtain the energy estimate \, u(\cdot,t)\, _2\leq \, f(\cdot)\, _2 (
p-norm In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki ...
) From this energy estimate one can conclude that the problem is well-posed.


See also

*
Total absorption spectroscopy Total absorption spectroscopy is a measurement technique that allows the measurement of the gamma radiation emitted in the different nuclear gamma transitions that may take place in the daughter nucleus after its unstable parent has decayed by mean ...
– 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 In statistics, an expectation–maximization (EM) algorithm is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variabl ...


References

* * * {{Authority control Numerical analysis Partial differential equations