Numerical analysis of elliptic inverse problems with interior data

A number of algorithms have been proposed and analyzed for estimating a coefficient in an elliptic boundary value problem when interior data is available. Most of the analysis has been done for the simple scalar BVP -Δ a Δ u = f in Ω, a (∂ u / ∂ n) g on ∂ Ω However, some methods and the associated analysis have been extended to the problem of estimating the Lamé moduli in the system of linear, isotropic elasticity. Under certain idealized conditions, convergence of estimates to the exact Lame moduli has been proved for two techniques, the output least-squares method and a variational method similar to the equation error approach

Generalizing the GSVD

The generalized singular value decomposition (GSVD) of a pair of matrices is the natural tool for certain problems defined on Euclidean space, such as certain weighted least-squares problems, the result of applying Tikhonov regularization to such problems (sometimes called regularization with seminorms), and equality-constrained least-squares problems. There is an extension of the GSVD to pairs of bounded linear operators defined on Hilbert space that turns out to be a natural representation for analyzing the same problems in the infinite-dimensional setting

On the convergence of a heuristic parameter choice rule for Tikhonov regularization

Multiplicative regularization solves a linear inverse problem by minimizing the product of the norm of the data misfit and the norm of the solution. This technique is related to Tikhonov regularization with the parameter chosen to make the data misfit and regularization terms (of the Tikhonov objective function) equal. This suggests a heuristic parameter choice method, equivalent to the rule previously proposed by Reginska. Reginska\u27s rule is well defined provided the data is sufficiently close to exact data and does not lie in the range of the operator. If a sufficiently large portion of the data error lies outside the range of the operator, then the solution defined by Reginska\u27s rule converges weakly to the exact solution as the data error converges to zero. The regularization parameter converges to zero like the square of the norm of the data noise, leading to under-regularization for small noise levels. Nevertheless, the method performs well on a suite of test problems, as shown by comparison with the L-curve, generalized cross-validation, quasi-optimality, and Hanke--Raus parameter choice methods. A modification of the approach yields a heuristic parameter choice rule that is provably convergent (in the norm topology) under the restrictions on the data error described above, as long as the exact solution has a small amount of additional smoothness. On the test problems considered here, the modified rule outperforms all of the above heuristic methods, although it is only slightly better than the quasi-optimality rule

Approximating the generalized singular value expansion

The generalized singular value expansion (GSVE) simultaneously diagonalizes a pair of operators on Hilbert space. From a theoretical point of view, the GSVE enables a straightforward analysis of, for example, weighted least-squares problems and the method of Tikhonov regularization with seminorms. When the operators are discretized, an approximate GSVE can be computed from the generalized singular value decomposition of a pair of Galerkin matrices. Unless the discretization is carefully chosen, spurious modes can appear, but a natural condition on the discretization guarantees convergence of the approximate GSVE to the exact one. Numerical examples illustrate the pitfalls of a poor discretization and efficacy of the convergence conditions

Article the singular value expansion for arbitrary bounded linear operators

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed

Zuverlässigkeit der elektrischen Energieversorgung

Die Zuverlässigkeit eines technischen Produkts oder Systems ist eine Eigenschaft, die angibt, wie verlässlich eine dem Produkt oder System zugewiesene Funktion in einem Zeitintervall erfüllt wird. Sie unterliegt einem stochastischen Prozess, kann qualitativ oder auch quantitativ (durch die Überlebenswahrscheinlichkeit) beschrieben werden und ist nicht unmittelbar messbar. Eine Anwendung dieser Definition auf die elektrische Energieversorgung bedeutet, dass auch ein Ausfall möglich ist, da kein technisches Produkt frei von der Möglichkeit eines Ausfalles ist. Diese grundsätzliche Bedingung wird leider im Alltag oft nicht beachtet, die sehr hohe Zuverlässigkeit der elektrischen Energieversorgung wird als selbstverständlich angenommen und nur bei einem Ausfall, der in Deutschland im Bereich von ca. 15 Minuten pro Energieabnehmer und Jahr liegt, wird über die elektrische Energieversorgung diskutiert

Today's problems with the evaluation methods of full lightning impulse parameters as described in IEC 60060-1

In this paper the present problems with the evaluation methods for lightning impulse parameters, as defined in IEC 60060-1, are described. Also the current practice of evaluation in many laboratories world-wide, that is obtained by a questionnaire, is presented. Some of the work performed up the present time and the initial conclusions are reported, then some recommendations are made for future work

An Infeasible Point Method for Minimizing the Lennard-Jones Potential

Minimizing the Lennard-Jones potential, the most-studied modelproblem for molecular conformation, is an unconstrained globaloptimization problem with a large number of local minima. In thispaper, the problem is reformulated as an equality constrainednonlinear programming problem with only linear constraints. Thisformulation allows the solution to approached through infeasibleconfigurations, increasing the basin of attraction of the globalsolution. In this way the likelihood of finding a global minimizeris increased. An algorithm for solving this nonlinear program isdiscussed, and results of numerical tests are presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44788/1/10589_2004_Article_140555.pd