The local limits of imperial and ecclesiastical power

This chapter looks at the limits that the aristocracy in general and the landowning elites at the local level set to imperial and ecclesiastical power. In late Roman society, aristocrats had remarkable power, economic resources, and prestige, especially on the local level. The wishes of the central administration and the local realities were often in tension with each other. In some cases, imperial decrees could be ignored in the local setting if they were not well received. Local authorities could even decide not to enforce a law. In the local realities of dissident groups, the patronage relationships were decisive: a powerful landowner could influence his tenants to either embrace Christianity or retain old practices. There was often a conflict of interests between local landowners and bishops in their struggle for hegemony at the regional level. Bishops expected Christian landowners to put an end to pagan practices on their estates, and they complained about the laxity if they did not.

Efficient Local Level Set Method without Reinitialization and Its Appliance to Topology Optimization

The local level set method (LLSM) is higher than the LSMs with global models in computational efficiency, because of the use of narrow-band model. The computational efficiency of the LLSM can be further increased by avoiding the reinitialization procedure by introducing a distance regularized equation (DRE). The numerical stability of the DRE can be ensured by a proposed conditionally stable difference scheme under reverse diffusion constraints. Nevertheless, the proposed method possesses no mechanism to nucleate new holes in the material domain for two-dimensional structures, so that a bidirectional evolutionary algorithm based on discrete level set functions is combined with the LLSM to replace the numerical process of hole nucleation. Numerical examples are given to show high computational efficiency and numerical stability of this algorithm for topology optimization.

A local level-set method for 3D inversion of gravity-gradient data

We have developed a local level-set method for inverting 3D gravity-gradient data. To alleviate the inherent nonuniqueness of the inverse gradiometry problem, we assumed that a homogeneous density contrast distribution with the value of the density contrast specified a priori was supported on an unknown bounded domain [Formula: see text] so that we may convert the original inverse problem into a domain inverse problem. Because the unknown domain [Formula: see text] may take a variety of shapes, we parametrized the domain [Formula: see text] by a level-set function implicitly so that the domain inverse problem was reduced to a nonlinear optimization problem for the level-set function. Because the convergence of the level-set algorithm relied heavily on initializing the level-set function to enclose the gravity center of a source body, we applied a weighted [Formula: see text]-regularization method to locate such a gravity center so that the level-set function can be properly initialized. To rapidly compute the gradient of the nonlinear functional arising in the level-set formulation, we made use of the fact that the Laplacian kernel in the gravity force relation decayed rapidly off the diagonal so that matrix-vector multiplications for evaluating the gradient can be accelerated significantly. We conducted extensive numerical experiments to test the performance and effectiveness of the new method.