Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems

We present a new variational method, based on the matrix product operator (MPO) ansatz, for finding the steady state of dissipative quantum chains governed by master equations of the Lindblad form. Instead of requiring an accurate representation of the system evolution until the stationary state is attained, the algorithm directly targets the final state, thus allowing for a faster convergence when the steady state is a MPO with small bond dimension. Our numerical simulations for several dissipative spin models over a wide range of parameters illustrate the performance of the method and show that indeed the stationary state is often well described by a MPO of very moderate dimensions.Comment: Accepted versio

A fast recursive coordinate bisection tree for neighbour search and gravity

We introduce our new binary tree code for neighbour search and gravitational force calculations in an N-particle system. The tree is built in a "top-down" fashion by "recursive coordinate bisection" where on each tree level we split the longest side of a cell through its centre of mass. This procedure continues until the average number of particles in the lowest tree level has dropped below a prescribed value. To calculate the forces on the particles in each lowest-level cell we split the gravitational interaction into a near- and a far-field. Since our main intended applications are SPH simulations, we calculate the near-field by a direct, kernel-smoothed summation, while the far field is evaluated via a Cartesian Taylor expansion up to quadrupole order. Instead of applying the far-field approach for each particle separately, we use another Taylor expansion around the centre of mass of each lowest-level cell to determine the forces at the particle positions. Due to this "cell-cell interaction" the code performance is close to O(N) where N is the number of used particles. We describe in detail various technicalities that ensure a low memory footprint and an efficient cache use. In a set of benchmark tests we scrutinize our new tree and compare it to the "Press tree" that we have previously made ample use of. At a slightly higher force accuracy than the Press tree, our tree turns out to be substantially faster and increasingly more so for larger particle numbers. For four million particles our tree build is faster by a factor of 25 and the time for neighbour search and gravity is reduced by more than a factor of 6. In single processor tests with up to 10^8 particles we confirm experimentally that the scaling behaviour is close to O(N). The current Fortran 90 code version is OpenMP-parallel and scales excellently with the processor number (=24) of our test machine.Comment: 12 pages, 16 figures, 1 table, accepted for publication in MNRAS on July 28, 201

Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement

The typical goal of surface remeshing consists in finding a mesh that is (1) geometrically faithful to the original geometry, (2) as coarse as possible to obtain a low-complexity representation and (3) free of bad elements that would hamper the desired application. In this paper, we design an algorithm to address all three optimization goals simultaneously. The user specifies desired bounds on approximation error {\delta}, minimal interior angle {\theta} and maximum mesh complexity N (number of vertices). Since such a desired mesh might not even exist, our optimization framework treats only the approximation error bound {\delta} as a hard constraint and the other two criteria as optimization goals. More specifically, we iteratively perform carefully prioritized local operators, whenever they do not violate the approximation error bound and improve the mesh otherwise. In this way our optimization framework greedily searches for the coarsest mesh with minimal interior angle above {\theta} and approximation error bounded by {\delta}. Fast runtime is enabled by a local approximation error estimation, while implicit feature preservation is obtained by specifically designed vertex relocation operators. Experiments show that our approach delivers high-quality meshes with implicitly preserved features and better balances between geometric fidelity, mesh complexity and element quality than the state-of-the-art.Comment: 14 pages, 20 figures. Submitted to IEEE Transactions on Visualization and Computer Graphic