Well-posed problem

In mathematics, a well-posed problem is one for which the following properties hold:

1. The problem has a solution
2. The solution is unique
3. The solution's behavior changes continuously with the initial conditions

Examples of archetypal well-posed problems include the Dirichlet problem for Laplace's equation, and the heat equation 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 above are termed ill-posed. Inverse problems 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 when solved with finite precision, or with errors in the data.

Conditioning

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 (so-called chaotic systems) provide well-known examples of instability. An ill-conditioned problem is indicated by a large condition number.

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. Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems.

Energy method

The energy method is useful for establishing both uniqueness and continuity with respect to initial conditions (i.e. it does not establish existence). The method is based upon deriving an upper bound of an energy-like functional for a given problem.

Example: Consider the diffusion equation on the unit interval with homogeneous Dirichlet boundary conditions and suitable initial data $f(x)$ (e.g. for which $f(0)=f(1)=0$).

$${\begin{aligned}u_{t}&=Du_{xx},&&0<x<1,\,t>0,\,D>0,\\u(x,0)&=f(x),\\u(0,t)&=0,\\u(1,t)&=0,\\\end{aligned}}$$

Multiply the equation $u_{t}=Du_{xx}$ by $u$ and integrate in space over the unit interval to obtain

$${\begin{aligned}&&\int _{0}^{1}uu_{t}dx&=D\int _{0}^{1}uu_{xx}dx\\\Longrightarrow &&\int _{0}^{1}{\frac {1}{2}}\partial _{t}u^{2}dx&=Duu_{x}{\Big |}_{0}^{1}-D\int _{0}^{1}(u_{x})^{2}dx\\\Longrightarrow &&{\frac {1}{2}}\partial _{t}\|u\|_{2}^{2}&=0-D\int _{0}^{1}(u_{x})^{2}dx\leq 0\end{aligned}}$$

This tells us that $\|u\|_{2}$ (p-norm) cannot grow in time. By multiplying by two and integrating in time, from $0$ up to $t$, one finds

$$\|u(\cdot ,t)\|_{2}^{2}\leq \|f(\cdot )\|_{2}^{2}$$

This result is the energy estimate for this problem.

To show uniqueness of solutions, assume there are two distinct solutions to the problem, call them $u$ and $v$, each satisfying the same initial data. Upon defining $w=u-v$ then, via the linearity of the equations, one finds that $w$ satisfies

$${\begin{aligned}w_{t}&=Dw_{xx},&&0<x<1,\,t>0,\,D>0,\\w(x,0)&=0,\\w(0,t)&=0,\\w(1,t)&=0,\\\end{aligned}}$$

Applying the energy estimate tells us $\|w(\cdot ,t)\|_{2}^{2}\leq 0$ which implies $u=v$ (almost everywhere).

Similarly, to show continuity with respect to initial conditions, assume that $u$ and $v$ are solutions corresponding to different initial data $u(x,0)=f(x)$ and $v(x,0)=g(x)$. Considering $w=u-v$ once more, one finds that $w$ satisfies the same equations as above but with $w(x,0)=f(x)-g(x)$. This leads to the energy estimate $\|w(\cdot ,t)\|_{2}^{2}\leq D\|f(\cdot )-g(\cdot )\|_{2}^{2}$ which establishes continuity (i.e. as $f$ and $g$ become closer, as measured by the $L^{2}$ norm of their difference, then $\|w(\cdot ,t)\|_{2}\to 0$).

The maximum principle is an alternative approach to establish uniqueness and continuity of solutions with respect to initial conditions for this example. The existence of solutions to this problem can be established using Fourier series.

See also

• 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

References

• Hadamard, Jacques (1902). Sur les problèmes aux dérivées partielles et leur signification physique. Princeton University Bulletin. pp. 49–52.
• Parker, Sybil B., ed. (1989) [1974]. McGraw-Hill Dictionary of Scientific and Technical Terms (4th ed.). New York: McGraw-Hill. ISBN 0-07-045270-9.
• Tikhonov, A. N.; Arsenin, V. Y. (1977). Solutions of ill-Posed Problems. New York: Winston. ISBN 0-470-99124-0.
• Strauss, Walter A. (2008). Partial differential equations; An introduction (2nd ed.). Hoboken: Wiley. ISBN 978-0470-05456-7.