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An arbitrarily dense discretization of the Bloch sphere of complex Hilbert states is constructed, where points correspond to bit strings of fixed finite length. Number-theoretic properties of trigonometric functions (not part of the quantum-theoretic canon) are used to show that this constructive discretized representation incorporates many of the defining characteristics of quantum systems: completementarity, uncertainty relationships and (with a simple Cartesian product of discretized spheres) entanglement. Unlike Meyer's earlier discretization of the Bloch Sphere, there are no orthonormal triples, hence the Kocken–Specker theorem is not nullified. A physical interpretation of points on the discretized Bloch sphere is given in terms of ensembles of trajectories on a dynamically invariant fractal set in state space, where states of physical reality correspond to points on the invariant set. This deterministic construction provides a new way to understand the violation of the Bell inequality without violating statistical independence or factorization, where these conditions are defined solely from states on the invariant set. In this finite representation, there is an upper limit to the number of qubits that can be entangled, a property with potential experimental consequences. [ABSTRACT FROM AUTHOR]

The case is made for a much closer synergy between climate science, numerical analysis and computer science. [ABSTRACT FROM AUTHOR]

Given their increasing relevance for society, I suggest that the climate science community itself does not treat the development of error-free ab initio models of the climate system with sufficient urgency. With increasing levels of difficulty, I discuss a number of proposals for speeding up such development. Firstly, I believe that climate science should make better use of the pool of post-PhD talent in mathematics and physics, for developing next-generation climate models. Secondly, I believe there is more scope for the development of modelling systems which link weather and climate prediction more seamlessly. Finally, here in Europe, I call for a new European Programme on Extreme Computing and Climate to advance our ability to simulate climate extremes, and understand the drivers of such extremes. A key goal for such a programme is the development of a 1 km global climate system model to run on the first exascale supercomputers in the early 2020s. [ABSTRACT FROM AUTHOR]

A locally causal hidden-variable theory of quantum physics need not be constrained by the Bell inequalities if this theory also partially violates the measurement independence condition. However, such violation can appear unphysical, implying implausible conspiratorial correlations between the hidden variables of particles being measured and earlier determinants of instrumental settings. A novel physically plausible explanation for such correlations is proposed, based on the hypothesis that states of physical reality lie precisely on a non-computational measure-zero dynamically invariant set in the state space of the universe: the Cosmological Invariant Set Postulate. To illustrate the relevance of the concept of a global invariant set, a simple analogy is considered where a massive object is propelled into a black hole depending on the decay of a radioactive atom. It is claimed that a locally causal hidden-variable theory constrained by the Cosmological Invariant Set Postulate can violate the Clauser-Horne-Shimony-Holt inequality without being conspiratorial, superdeterministic, fine-tuned or retrocausal, and the theory readily accommodates the classical compatibilist notion of (experimenter) free will. [ABSTRACT FROM AUTHOR]

This paper sets out a new methodological approach to solving the equations for simulating and predicting weather and climate. In this approach, the conventionally hard boundary between the dynamical core and the sub-grid parametrizations is blurred. This approach is motivated by the relatively shallow power-law spectrum for atmospheric energy on scales of hundreds of kilometres and less. It is first argued that, because of this, the closure schemes for weather and climate simulators should be based on stochastic-dynamic systems rather than deterministic formulae. Second, as high-wavenumber elements of the dynamical core will necessarily inherit this stochasticity during time integration, it is argued that the dynamical core will be significantly over-engineered if all computations, regardless of scale, are performed completely deterministically and if all variables are represented with maximum numerical precision (in practice using double-precision floating-point numbers). As the era of exascale computing is approached, an energy- and computationally efficient approach to cloud-resolved weather and climate simulation is described where determinism and numerical precision are focused on the largest scales only. [ABSTRACT FROM AUTHOR]

Inexact hardware design, which advocates trading the accuracy of computations in exchange for significant savings in area, power and/or performance of computing hardware, has received increasing prominence in several error-tolerant application domains, particularly those involving perceptual or statistical end-users. In this paper, we evaluate inexact hardware for its applicability in weather and climate modelling. We expand previous studies on inexact techniques, in particular probabilistic pruning, to floating point arithmetic units and derive several simulated set-ups of pruned hardware with reasonable levels of error for applications in atmospheric modelling. The set-up is tested on the Lorenz '96 model, a toy model for atmospheric dynamics, using software emulation for the proposed hardware. The results show that large parts of the computation tolerate the use of pruned hardware blocks without major changes in the quality of short- and long-time diagnostics, such as forecast errors and probability density functions. This could open the door to significant savings in computational cost and to higher resolution simulations with weather and climate models. [ABSTRACT FROM AUTHOR]

Simple chaotic systems are useful tools for testing methods for use in numerical weather simulations owing to their transparency and computational cheapness. The Lorenz system was used here; the full system was defined as 'truth', whereas a truncated version was used as a testbed for parametrization schemes. Several stochastic parametrization schemes were investigated, including additive and multiplicative noise. The forecasts were started from perfect initial conditions, eliminating initial condition uncertainty. The stochastically generated ensembles were compared with perturbed parameter ensembles and deterministic schemes. The stochastic parametrizations showed an improvement in weather and climate forecasting skill over deterministic parametrizations. Including a temporal autocorrelation resulted in a significant improvement over white noise, challenging the standard idea that a parametrization should only represent sub-gridscale variability. The skill of the ensemble at representing model uncertainty was tested; the stochastic ensembles gave better estimates of model uncertainty than the perturbed parameter ensembles. The forecasting skill of the parametrizations was found to be linked to their ability to reproduce the climatology of the full model. This is important in a seamless prediction system, allowing the reliability of short-term forecasts to provide a quantitative constraint on the accuracy of climate predictions from the same system. [ABSTRACT FROM AUTHOR]

Seasonal climate forecasts are being used increasingly across a range of application sectors. A recent UK governmental report asked: how good are seasonal forecasts on a scale of 1-5 (where 5 is very good), and how good can we expect them to be in 30 years time? Seasonal forecasts are made from ensembles of integrations of numerical models of climate. We argue that 'goodness' should be assessed first and foremost in terms of the probabilistic reliability of these ensemble-based forecasts; reliable inputs are essential for any forecast-based decision-making. We propose that a '5' should be reserved for systems that are not only reliable overall, but where, in particular, small ensemble spread is a reliable indicator of low ensemble forecast error. We study the reliability of regional temperature and precipitation forecasts of the current operational seasonal forecast system of the European Centre for Medium-Range Weather Forecasts, universally regarded as one of the world-leading operational institutes producing seasonal climate forecasts. A wide range of 'goodness' rankings, depending on region and variable (with summer forecasts of rainfall over Northern Europe performing exceptionally poorly) is found. Finally, we discuss the prospects of reaching '5' across all regions and variables in 30 years time.

The development of multi-model ensembles for reliable predictions of inter-annual climate fluctuations and climate change, and their application to health, agronomy and water management, are discussed.