Opt: A Domain Specific Language for Non-linear Least Squares Optimization in Graphics and Imaging

Many graphics and vision problems can be expressed as non-linear least squares optimizations of objective functions over visual data, such as images and meshes. The mathematical descriptions of these functions are extremely concise, but their implementation in real code is tedious, especially when optimized for real-time performance on modern GPUs in interactive applications. In this work, we propose a new language, Opt (available under http://optlang.org), for writing these objective functions over image- or graph-structured unknowns concisely and at a high level. Our compiler automatically transforms these specifications into state-of-the-art GPU solvers based on Gauss-Newton or Levenberg-Marquardt methods. Opt can generate different variations of the solver, so users can easily explore tradeoffs in numerical precision, matrix-free methods, and solver approaches. In our results, we implement a variety of real-world graphics and vision applications. Their energy functions are expressible in tens of lines of code, and produce highly-optimized GPU solver implementations. These solver have performance competitive with the best published hand-tuned, application-specific GPU solvers, and orders of magnitude beyond a general-purpose auto-generated solver

Complex dynamics of elementary cellular automata emerging from chaotic rules

We show techniques of analyzing complex dynamics of cellular automata (CA) with chaotic behaviour. CA are well known computational substrates for studying emergent collective behaviour, complexity, randomness and interaction between order and chaotic systems. A number of attempts have been made to classify CA functions on their space-time dynamics and to predict behaviour of any given function. Examples include mechanical computation, \lambda{} and Z-parameters, mean field theory, differential equations and number conserving features. We aim to classify CA based on their behaviour when they act in a historical mode, i.e. as CA with memory. We demonstrate that cell-state transition rules enriched with memory quickly transform a chaotic system converging to a complex global behaviour from almost any initial condition. Thus just in few steps we can select chaotic rules without exhaustive computational experiments or recurring to additional parameters. We provide analysis of well-known chaotic functions in one-dimensional CA, and decompose dynamics of the automata using majority memory exploring glider dynamics and reactions

Computation in generalised probabilistic theories

From the existence of an efficient quantum algorithm for factoring, it is likely that quantum computation is intrinsically more powerful than classical computation. At present, the best upper bound known for the power of quantum computation is that BQP is in AWPP. This work investigates limits on computational power that are imposed by physical principles. To this end, we define a circuit-based model of computation in a class of operationally-defined theories more general than quantum theory, and ask: what is the minimal set of physical assumptions under which the above inclusion still holds? We show that given only an assumption of tomographic locality (roughly, that multipartite states can be characterised by local measurements), efficient computations are contained in AWPP. This inclusion still holds even without assuming a basic notion of causality (where the notion is, roughly, that probabilities for outcomes cannot depend on future measurement choices). Following Aaronson, we extend the computational model by allowing post-selection on measurement outcomes. Aaronson showed that the corresponding quantum complexity class is equal to PP. Given only the assumption of tomographic locality, the inclusion in PP still holds for post-selected computation in general theories. Thus in a world with post-selection, quantum theory is optimal for computation in the space of all general theories. We then consider if relativised complexity results can be obtained for general theories. It is not clear how to define a sensible notion of an oracle in the general framework that reduces to the standard notion in the quantum case. Nevertheless, it is possible to define computation relative to a `classical oracle'. Then, we show there exists a classical oracle relative to which efficient computation in any theory satisfying the causality assumption and tomographic locality does not include NP.Comment: 14+9 pages. Comments welcom

Hard hexagon partition function for complex fugacity

We study the analyticity of the partition function of the hard hexagon model in the complex fugacity plane by computing zeros and transfer matrix eigenvalues for large finite size systems. We find that the partition function per site computed by Baxter in the thermodynamic limit for positive real values of the fugacity is not sufficient to describe the analyticity in the full complex fugacity plane. We also obtain a new algebraic equation for the low density partition function per site.Comment: 49 pages, IoP styles files, lots of figures (png mostly) so using PDFLaTeX. Some minor changes added to version 2 in response to referee report

Two parallel implementations of Ehrlich-Aberth algorithm for root-finding of polynomials on multiple GPUs with OpenMP and MPI

International audienceFinding the roots of polynomials is a very important part of solving real-life problems but the higher the degree of the polynomials is, the less easy it becomes. In this paper, we present two different parallel algorithms of the Ehrlich-Aberth method to find roots of sparse and fully defined polynomials of high degrees. Both algorithms are based on CUDA technology to be implemented on multi-GPU computing platforms but each use different parallel paradigms: OpenMP or MPI. The experiments show a quasi-linear speedup by using up-to 4 GPU devices compared to 1 GPU to find the roots of polynomials of degree up-to 1.4 million. Moreover, other experiments show it is possible to find the roots of polynomials of degree up-to 5 million