variables to reduce to the case above. Equivalently, one can seek an to minimize where we have used to stand for the weighted norm squared (compare with the Mahalanobis distance). In the Bayesian interpretation is the inverse covariance matrix of , is the expected value of , and is the inverse covariance matrix of . The Tikhonov matrix is then given as a factorization of the matrix (e.g. the Cholesky factorization) and is considered a whitening filter. This generalized problem has an optimal solution which can be written explicitly using the formula or equivalently Lavrentyev regularization In some situations, one can avoid using the transpose , as proposed by Mikhail Lavrentyev. For example, if is symmetric positive definite, i.e. , so is its inverse , which can thus be used to set up the weighted norm squared in the generalized Tikhonov regularization, leading to minimizing or, equivalently up to a constant term, . This minimization problem has an optimal solution which can be written explicitly using the formula , which is nothing but the solution of the generalized Tikhonov problem where The Lavrentyev regularization, if applicable, is advantageous to the original Tikhonov regularization, since the Lavrentyev matrix can be better conditioned, i.e., have a smaller condition number, compared to the Tikhonov matrix Regularization in Hilbert space Typically discrete linear ill-conditioned problems result from discretization of integral equations, and one can formulate a Tikhonov regularization in the original infinite-dimensional context. In the above we can interpret as a compact operator on Hilbert spaces, and and as elements in the domain and range of . The operator is then a self-adjoint bounded invertible operator. Relation to singular-value decomposition and Wiener filter With , this least-squares solution can be analyzed in a special way using the singular-value decomposition. Given the singular value decomposition with singular values , the Tikhonov regularized solution can be expressed as where has diagonal values and is zero elsewhere. This demonstrates the effect of the Tikhonov parameter on the condition number of the regularized problem. For the generalized case, a similar representation can be derived using a generalized singular-value decomposition. Finally, it is related to the Wiener filter: where the Wiener weights are and is the rank of . Determination of the Tikhonov factor The optimal regularization parameter is usually unknown and often in practical problems is determined by an ad hoc method. A possible approach relies on the