Efficient implementation of the Gutzwiller variational method

We present a self-consistent numerical approach to solve the Gutzwiller variational problem for general multi-band models with arbitrary on-site interaction. The proposed method generalizes and improves the procedure derived by Deng et al., Phys. Rev. B. 79 075114 (2009), overcoming the restriction to density-density interaction without increasing the complexity of the computational algorithm. Our approach drastically reduces the problem of the high-dimensional Gutzwiller minimization by mapping it to a minimization only in the variational density matrix, in the spirit of the Levy and Lieb formulation of DFT. For fixed density the Gutzwiller renormalization matrix is determined as a fixpoint of a proper functional, whose evaluation only requires ground-state calculations of matrices defined in the Gutzwiller variational space. Furthermore, the proposed method is able to account for the symmetries of the variational function in a controlled way, reducing the number of variational parameters. After a detailed description of the method we present calculations for multi-band Hubbard models with full (rotationally invariant) Hund's rule on-site interaction. Our analysis shows that the numerical algorithm is very efficient, stable and easy to implement. For these reasons this method is particularly suitable for first principle studies -- e.g., in combination with DFT -- of many complex real materials, where the full intra-atomic interaction is important to obtain correct results.Comment: 19 pages, 7 figure

Symmetric Strategy Improvement

Symmetry is inherent in the definition of most of the two-player zero-sum games, including parity, mean-payoff, and discounted-payoff games. It is therefore quite surprising that no symmetric analysis techniques for these games exist. We develop a novel symmetric strategy improvement algorithm where, in each iteration, the strategies of both players are improved simultaneously. We show that symmetric strategy improvement defies Friedmann's traps, which shook the belief in the potential of classic strategy improvement to be polynomial

Elixir: synthesis of parallel irregular algorithms

Algorithms in new application areas like machine learning and data analytics usually operate on unstructured sparse graphs. Writing efficient parallel code to implement these algorithms is very challenging for a number of reasons. First, there may be many algorithms to solve a problem and each algorithm may have many implementations. Second, synchronization, which is necessary for correct parallel execution, introduces potential problems such as data-races and deadlocks. These issues interact in subtle ways, making the best solution dependent both on the parallel platform and on properties of the input graph. Consequently, implementing and selecting the best parallel solution can be a daunting task for non-experts, since we have few performance models for predicting the performance of parallel sparse graph programs on parallel hardware. This dissertation presents a synthesis methodology and a system, Elixir, that addresses these problems by (i) allowing programmers to specify solutions at a high level of abstraction, and (ii) generating many parallel implementations automatically and using search to find the best one. An Elixir specification consists of a set of operators capturing the main algorithm logic and a schedule specifying how to efficiently apply the operators. Elixir employs sophisticated automated reasoning to merge these two components, and uses techniques based on automated planning to insert synchronization and synthesize efficient parallel code. Experimental evaluation of our approach demonstrates that the performance of the Elixir generated code is competitive to, and can even outperform, hand-optimized code written by expert programmers for many interesting graph benchmarks.Computer Science

Cyclic proof systems for modal fixpoint logics

This thesis is about cyclic and ill-founded proof systems for modal fixpoint logics, with and without explicit fixpoint quantifiers.Cyclic and ill-founded proof-theory allow proofs with infinite branches or paths, as long as they satisfy some correctness conditions ensuring the validity of the conclusion. In this dissertation we design a few cyclic and ill-founded systems: a cyclic one for the weak Grzegorczyk modal logic K4Grz, based on our explanation of the phenomenon of cyclic companionship; and ill-founded and cyclic ones for the full computation tree logic CTL* and the intuitionistic linear-time temporal logic iLTL. All systems are cut-free, and the cyclic ones for K4Grz and iLTL have fully finitary correctness conditions.Lastly, we use a cyclic system for the modal mu-calculus to obtain a proof of the uniform interpolation property for the logic which differs from the original, automata-based one