We investigate the behavior of adaptive time stepping numerical algorithms under the reverse mode of automatic differentiation (AD). By differentiating the time step controller and the error estimator of the original algorithm, reverse mode AD generates spurious adjoint derivatives of the time steps. The resulting discrete adjoint models become inconsistent with the adjoint ODE, and yield incor...

2 Discretization 9 2.1 The Discontinuous Galerkin Method . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.2 Solution Approximation . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.3 Weak Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.4 Discrete System . . . . . . . . . . . . . . . . . ....

The main purpose of this study is to estimate the step heat fluxes applied to the wall of a two-dimensional symmetric channel with turbulent flow. For inverse analysis, conjugate gradient method with adjoint problem is used. In order to calculate the flow field,   two equation model is used. In this study, adjoint problem is developed to conduct an inverse analysis of heat transfer in a channel...

We present an improved algorithm for finding all solutions to Goehl’s problem A = mP for triangles, i.e., the problem of finding all Heronian triangles whose area (A) is an integer multiple (m) of the perimeter (P ). The new algorithm does not involve elimination of extraneous rational triangles, and is a true extension of Goehl’s original method.

Sommerville [10] and Davies [2] classified the spherical triangles that can tile the sphere in an edge-to-edge fashion. Relaxing this condition yields other triangles, which tile the sphere but have some tiles intersecting in partial edges. This paper determines which right spherical triangles within certain families can tile the sphere.

An acute triangulation is a triangulation whose triangles have all their angles less than π 2 . In this paper we prove that i) every planar pentagon can be triangulated into at most 54 acute triangles, and ii) every double pentagon can be triangulated into at most 76 acute triangles.

An adjoint model for a 2D finite element Galerkin shallow water model is developed using the Independent Set Perturbation (ISP, [46]) sensitivity analysis. Its performance in a full 4-D Var setup with a limited area is assessed by comparing with the adjoint model derived by the automatic differentiation approach, where it is used for optimising the initial conditions. It is shown that the ISP s...

We show that there is no (1 + ε)-approximation algorithm for the problem of covering points in the plane by minimum number of fat triangles of similar size (with the minimum angle of the triangles being close to 45 degrees). Here, the available triangles are prespecified in advance. Since a constant factor approximation algorithm is known for this problem [CV07], this settles the approximabilit...

Given, on the Hilbert space H0, the self-adjoint operator B and the skew-adjoint operators C1 and C2, we consider, on the Hilbert space H ≃ D(B)⊕H0, the skew-adjoint operator