A guided Monte Carlo method for optimization problems

We introduce a new Monte Carlo method by incorporating a guided distribution function to the conventional Monte Carlo method. In this way, the efficiency of Monte Carlo methods is drastically improved. To further speed up the algorithm, we include two more ingredients into the algorithm. First, we freeze the sub-patterns that have high probability of appearance during the search for optimal solution, resulting in a reduction of the phase space of the problem. Second, we perform the simulation at a temperature which is within the optimal temperature range of the optimization search in our algorithm. We use this algorithm to search for the optimal path of the traveling salesman problem and the ground state energy of the spin glass model and demonstrate that its performance is comparable with more elaborate and heuristic methods.Comment: 4 pages, ReVTe

Efficient Localization of Discontinuities in Complex Computational Simulations

Surrogate models for computational simulations are input-output approximations that allow computationally intensive analyses, such as uncertainty propagation and inference, to be performed efficiently. When a simulation output does not depend smoothly on its inputs, the error and convergence rate of many approximation methods deteriorate substantially. This paper details a method for efficiently localizing discontinuities in the input parameter domain, so that the model output can be approximated as a piecewise smooth function. The approach comprises an initialization phase, which uses polynomial annihilation to assign function values to different regions and thus seed an automated labeling procedure, followed by a refinement phase that adaptively updates a kernel support vector machine representation of the separating surface via active learning. The overall approach avoids structured grids and exploits any available simplicity in the geometry of the separating surface, thus reducing the number of model evaluations required to localize the discontinuity. The method is illustrated on examples of up to eleven dimensions, including algebraic models and ODE/PDE systems, and demonstrates improved scaling and efficiency over other discontinuity localization approaches