We discuss the division method for subspectra which appears to be one of the key approaches in the study of spectral properties of self-adjoint differential vector-operators, that is operators generated as a direct sum of self-adjoint extensions on an Everitt-Markus-Zettl multi-interval system. In the current work we show how the division method may be applied to obtain the ordered spectral rep...

A number of algorithmdevelopments are presented for adjoint methods using the “discrete” approach in which the discretization of the nonlinear equations is linearized and the resulting matrix is then transposed. With a new iterative procedure for solving the adjoint equations, exact numerical equivalence is maintained between the linear and adjoint discretizations. The incorporation of strong b...

For models described by ordinary differential equation boundary value problems (ODE BVPs), we derive adjoint equations for sensitivity analysis, giving explicit forms for the boundary conditions of the adjoint boundary value problem. The solutions of the adjoint equations are used to efficiently compute gradients of both integral-form and pointwise constraints. Existence and stability results a...

We present a version of the Pontryagin Maximum Principle for control dynamics with a possibly non smooth, nonlipschitz and even discontinuous righthand side. The usual adjoint equation, where state derivatives occur, is replaced by an integrated form, containing only differentials of the reference flow maps. The resulting “integrated adjoint equation” leads to “adjoint vectors” that need not be...

Given the symmetric operator AN obtained by restricting the self-adjoint operator A to N , a linear dense set, closed with respect to the graph norm, we determine a convenient boundary triple for the adjoint A N and the corresponding Weyl function. These objects provide us with the self-adjoint extensions of AN and their resolvents.

Adjoint pairs or adjoint triples defined on lattices have proven to be a useful tool when working in fuzzy formal concept analysis. This paper shows that adjoint pairs and triples can play as well an important role within the framework of multilattices, especially in order to form the Galois connections needed to build concept multilattices.

The norm convergence of the Trotter{Kato product formula is established with ultimate optimal error bound for the self-adjoint semigroup generated by the operator sum of two self-adjoint operators. A generalization is also given to the operator sum of several self-adjoint operators.

The development of the adjoint of the forecast model and of the adjoint of the data assimilation system (adjointDAS) make feasible the evaluation of the derivative-based forecast sensitivity to DAS input parameters in numerical weather prediction (NWP). The adjoint estimation of the forecast sensitivity to the observation error covariance in the DAS is considered as a practical approach to prov...

Hybrid methods represent a classic discretization paradigm for elliptic equations. More recently, hybrid methods have been formulated for convection-diffusion problems, in particular compressible fluid flow. In [25], we have introduced a hybrid mixed method for the compressible Navier-Stokes equations as a combination of a hybridized DG scheme for the convective terms, and an H(div,Ω)-method fo...

This paper investigates the necessary conditions of optimality for uniformly overtaking optimal control on infinite horizon with free right endpoint. Clarke’s form of the Pontryagin Maximum Principle is proved without the assumption on boundedness of total variation of adjoint variable. The transversality condition for adjoint variable is shown to become necessary if the adjoint variable is par...