By Alexander G Ramm
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Extra resources for Mathematical And Analytical Techniques With Applications To Engineering
Example text
Let Then, and Au = p. , u is not the minimal-norm solution to the equation Au = p. 20 is proved. 22. 19 hold and and if where then where and is the minimal-norm solution to the equation A(u) = f. In particular, if 0 < a < l, then Indeed, where is the unique solution to the equation It is well known that provided that and, clearly, one multiplies the identity by and uses the monotonicity of A and the inequality The result similar to the one in the above remark can be found in [ARy]. 9 Regularizers for ill-posed problems must depend on the noise level In this Section we prove the following simple claim: Claim 2.
In this case the minimization problem is ill-posed. Such problems were studied [Vas]. 7 The Cauchy problem for Laplace’s equation Claim 1. The Cauchy problem for Laplace’s equation is an ill-posed problem. Consider the problem: in the half-plane It is clear that solves the above problem, and this solution is unique (by the uniqueness of the solution to the Cauchy problem for elliptic equations). This example belongs to J. Hadamard, and it shows that the Cauchy data may be arbitrarily small (take while the solution tends to infinity, as at any point Thus, the claim is verified (cf.
Let be a linear closed operator, D(A) and R(A) be its domain and range, and be Hilbert spaces, N(A) := {u : Au = 0}, is the adjoint operator, the bar stands for the closure, and is the orthogonal sum. e. and then the inverse operator is defined on I is the identity operator. , for some then the problem has a solution every element is also a solution, and there is a unique solution with minimal norm, namely the solution such that If then the infimum of is not attained. If A is bounded and then the element solves the equation and is the minimal norm solution to this equation, 20 2.