---
tag: W09B9NZ9B5
kind: concept
name: Well-posed problem
msc: [35A01, 35A02, 35B30]
sources:
  - url: https://en.wikipedia.org/wiki/Well-posed_problem
same_as:
  wikidata: Q1455898
  wikipedia: Well-posed problem
---

## Summary

A problem is well posed in the sense of Hadamard if it has a unique solution that depends continuously on the data.
These conditions say that a model of a physical process gives one answer, stable under small errors in the data.
A problem that fails one of them is ill posed; such problems, like the backward heat equation or many inverse problems, need extra care.
See the [[definition|FZ64956KX2]].

## Explanations

### L1 human

Data from the real world always carry small errors.
If a tiny change of the data could change the answer completely, the model would be useless for prediction.
A well-posed problem has exactly one answer, and nearby data give nearby answers.
For example, fixing the temperature on the edge of a disc determines the steady temperature inside.
Changing the edge values by at most $0.1$ degrees changes the inside by at most $0.1$ degrees.