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  "body": "\n## Summary\n\nA problem is ill-posed in the sense of Hadamard when it is not a [[well-posed problem|W09B9NZ9B5]].\nThis means that a solution does not exist, is not unique, or does not depend continuously on the data.\nRegularization replaces an ill-posed problem by a family of nearby well-posed problems, indexed by a parameter.\nThe parameter is chosen from the noise level so that the regularized solutions converge to the true solution as the noise tends to zero.\nIt is the basic tool for [[inverse problems|PCMW7KN5P4]] such as image deblurring, tomography and parameter identification.\n\n## Explanations\n\n### L1 human\n\nSome problems amplify small errors in the data enormously, so that the naive answer is useless.\nUndoing a blur in a photograph is an example: tiny noise gets magnified into wild patterns.\nRegularization adds a mild preference, such as \"the answer should not be too large or too rough\", which tames this amplification.\nThe strength of the preference is tuned to the amount of noise.\nToo little regularization leaves the noise in, and too much smooths away the real details.\n",
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