We consider one-dimensional steady-state conservation laws with discontinuous solutions. Giles and Pierce [7] realized that a shock leads to a new term in the adjoint error representation for target functionals. This term disappears if and only if the adjoint solution satisfies an internal boundary condition. Curiously, most computer codes implementing adjoint error estimation ignore the new te...

A visibility algorithm for triangulated panoramas is proposed. The algorithm can correctly resolve the visibility without making use of any depth information. It is especially useful when depth information is not available, such as in the case of real-world photographs. Based on the optical flow information and the image intensity, the panorama is subdivided into variable-sized triangles, image...

The Laplacian-energy-like of a simple connected graph G is defined as LEL:=LEL(G)=∑_(i=1)^n√(μ_i ), Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the num...

By the theorem of Mantel (1907) it is known that a graph with n vertices and ⌊n24⌋+1 edges must contain triangle. A Erdős gives strengthening: there are not only one, but at least ⌊n2⌋ triangles. We give further improvement: if no vertex contained by all triangles then n−2 them. There some natural generalizations when (a) complete graphs considered (rather than triangles), (b) has t extra (not ...

had been investigated. As it is noted here, these equations are not adjoint equations and are referred to as quasi-adjoint equations. Equation (E) is a special case of linear nth-order difference equations with quasi-differences. Such equations have been widely studied in the literature, see, for example, [6, 11] and the references therein. The natural question which arises is to find the adjoi...

Adjoint consistency – in addition to consistency – is the key requirement for discontinuous Galerkin discretisations to be of optimal order in L as well as measured in terms of target functionals. We provide a general framework for analysing adjoint consistency and introduce consistent modifications of target functionals. This framework is then used to derive an adjoint consistent discontinuous...

A new scheme for differentiating complex mesh-based numerical models (e.g. finite element models), the Independent Set Perturbation Adjoint method (ISP-Adjoint), is presented. Differentiation of the matrices and source terms making up the discrete forward model is realized by a graph colouring approach (forming independent sets of variables) combined with a perturbation method to obtain gradien...

In this paper we investigate the efficient realization of sampling methods, based on solutions of certain adjoint problems. This adjoint approach does not require the explicit knowledge of a Green’s function for the background medium, and allows to sample for all points and all dipole directions simultaneously; thus several limitations of standard sampling methods are relieved. A detailed deriv...

The analytic properties of adjoint solutions are examined for the quasi-onedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green’s function approach is used to derive the analytic adjoint solutions cor...

The formulation and solution of the adjoint problem for unsteady flow simulations using the Reynolds-averaged Navier-Stokes equations in the presence of dynamically deforming unstructured meshes is demonstrated. A discrete adjoint approach is used, and the full linearization is built up in a systematic and modular fashion. Discrete conservation in the analysis problem is ensured through the geo...