‎In this paper‎, ‎we find matrix representation of a class of sixth order Sturm-Liouville problem (SLP) with separated‎, ‎self-adjoint boundary conditions and we show that such SLP have finite spectrum‎. ‎Also for a given matrix eigenvalue problem $HX=lambda VX$‎, ‎where $H$ is a block tridiagonal matrix and $V$ is a block diagonal matrix‎, ‎we find a sixth order boundary value problem of Atkin...

We discuss the numerical computation of sensitivities via the adjoint approach in optimization problems governed by differential equations. We focus on the adjoint problem in its weak form. We show how one can avoid some of the problems with the adjoint approach, such as deriving suitable boundary conditions for the adjoint equation. We discuss the convergence of numerical approximations of the...

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...

‎In this paper‎, ‎we study spectral element approximation for a constrained‎ ‎optimal control problem in one dimension‎. ‎The equivalent a posteriori error estimators are derived for‎ ‎the control‎, ‎the state and the adjoint state approximation‎. ‎Such estimators can be used to‎ ‎construct adaptive spectral elements for the control problems.

Generalized adjoint with modified discretization and generalized coarse-grain adjoint are derived for a vector system of equations that contains parameterized on/off switches. With vector examples, it is shown that the conventional adjoint minimization may have a convergence problem in multidimensional space. The problem can be solved by the generalized adjoint with modified discretization or b...

In this investigation we address the problem of adjoint–based optimization of PDE systems in moving domains. As an example we consider the one–dimensional heat equation with prescribed boundary temperatures and heat fluxes. We discuss two methods of deriving an adjoint system necessary to obtain a gradient of a cost functional. In the first approach we derive the adjoint system after mapping th...

Common velocity analysis is based on the invariance of migrated images with respect to the experiment index (shot number, plane-wave take-off angle, etc.) or extension parameters (reflection angle, correlation lags in extended images, etc.). All the information available, i.e. the entire survey, is used for assessing the quality of the model used for imaging. This approach is effective but forc...

we investigate a class of fourth-order differential operators with eigenparameter dependent boundary conditions and transmission conditions. a self-adjoint linear operator a is defined in a suitable hilbert space h such that the eigenvalues of such a problem coincide with those of a . we discuss asymptotic behavior of its eigenvalues and completeness of its eigenfunctions. finally, we obtain th...

a mathematical model is presented in the present study to control‎ ‎the light propagation in an inhomogeneous media‎. ‎the method is ‎based on the identification of the optimal materials distribution‎ ‎in the media such that the trajectories of light rays follow the‎ ‎desired path‎. ‎the problem is formulated as a distributed parameter ‎identification problem and it is solved by a numerical met...

In this manuscript, we study the inverse problem for non self-adjoint Sturm--Liouville operator $-D^2+q$ with eigenparameter dependent boundary and discontinuity conditions inside a finite closed interval. By defining  a new Hilbert space and  using its spectral data of a kind, it is shown that the potential function can be uniquely determined by part of a set of values of eigenfunctions at som...