71 research outputs found
Heat Kernel Bounds on Metric Measure Spaces and Some Applications
Let be a space with and . For , we derive the upper and lower bounds of the
heat kernel on by applying the parabolic Harnack inequality and the
comparison principle, and then sharp bounds for its gradient, which are also
sharp in time. When , we also establish a sharp upper bound of the
heat kernel by using the dimension free Harnack inequality. For applications,
we study the large time behavior of the heat kernel, the stability of solutions
to the heat equation, and show the boundedness of (local) Riesz
transforms.Comment: 27pp,Section 6 was removed, to appear in Potential Ana
Harnack Inequalities for SDEs with Multiplicative Noise and Non-regular Drift
The log-Harnack inequality and Harnack inequality with powers for semigroups
associated to SDEs with non-degenerate diffusion coefficient and non-regular
time-dependent drift coefficient are established, based on the recent papers
\cite{Flandoli, Zhang11}. We consider two cases in this work: (1) the drift
fulfills the LPS-type integrability, and (2) the drift is uniformly H\"older
continuous with respect to the spatial variable. Finally, by using explicit
heat kernel estimates for the stable process with drift, the Harnack inequality
for the stochastic differential equation driven by symmetric stable process is
also proved.Comment: All comments are welcom
A neutron noise solver based on a discrete ordinates method
A neutron noise transport modelling tool is presented in this thesis. The simulator allows to determine the static solution of a critical system and the neutron noise induced by a prescribed perturbation of the critical system. The simulator is based on the neutron balance equations in the frequency domain and for two-dimensional systems. The discrete ordinates method is used for the angular discretization and the diamond finite difference method for the treatment of the spatial variable. The energy dependence is modelled with two neutron energy groups. The conventional inner-outer iterative scheme is employed for solving the discretized neutron transport equations. For the acceleration of the iterative scheme, the diffusion synthetic acceleration is implemented.The convergence rate of the accelerated and unaccelerated versions of the simulator is studied for the case of a perturbed infinite homogeneous system. The theoretical behavior predicted by the Fourier convergence analysis agrees well with the numerical performance of the simulator. The diffusion synthetic acceleration decreases significantly the number of numerical iterations, but its convergence rate is still slow, especially for perturbations at low frequencies.The simulator is further tested on neutron noise problems in more realistic, heterogeneous systems and compared with the diffusion-based solver. The diffusion synthetic acceleration leads to a reduction of the computational burden by a factor of 20. In addition, the simulator shows results that are consistent with the diffusion-based approximation. However, discrepancies are found because of the local effects of the neutron noise source and the strong variations of material properties in the system, which are expected to be better reproduced by a higher-order transport method such as the one used in the new solver
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