An ergodic theorem of a parabolic Anderson model driven by Lévy noise

In this paper, we study an ergodic theorem of a parabolic Andersen model driven by Lévy noise. Under the assumption that A = (a(i, j))i,j∈S is symmetric with respect to a σ-finite measure gp, we obtain the long-time convergence to an invariant probability measure νh starting from a bounded nonnegative A-harmonic function h based on self-duality property. Furthermore, under some mild conditions, we obtain the one to one correspondence between the bounded nonnegative A-harmonic functions and the extremal invariant probability measures with finite second moment of the nonnegative solution of the parabolic Anderson model driven by Lévy noise, which is an extension of the result of Y. Liu and F. X. Yang

PEAK—a fast and effective performance tuning system via compiler optimization orchestration

Compile-time optimizations generally improve program performance. Nevertheless, degradations caused by individual compiler optimization techniques are to be expected. Feedback-directed optimization orchestration systems generate optimized code versions under a series of optimization combinations, evaluate their performance, and search for the best version. One challenge to such systems is to tune program performance quickly in an exponential search space. Another challenge is to achieve high program performance, considering that optimizations interact. Aiming at these two goals, this article presents an automated performance tuning system, PEAK, which searches for the best compiler optimization combinations for the important code sections in a program. The major contributions made in this work are as follows: (1) An algorithm called Combined Elimination (CE) is developed to explore the optimization space quickly and effectively; (2) Three fast and accurate rating methods are designed to evaluate the performance of an optimized code section based on a partial execution of the program; (3) An algorithm is developed to identify important code sections as candidates for performance tuning by trading off tuning speed and tuned program performance; and (4) A set of compiler tools are implemented to automate optimization orchestration. Orchestrating optimization options in SUN Forte compilers at the whole-program level, our CE algorithm improves performance by 10.8 percent; over the SPEC CPU2000 FP baseline setting, compared to 5.6 percent; improved by manual tuning. Orchestrating GCC O3 optimizations, CE improves performance by 12 percent; over O3, the highest optimization level. Applying the rating methods, PEAK reduces tuning time from 2.19 hours to 5.85 minutes on average, while achieving equal or better program performance