Self-Specifying Machines

We study the computational power of machines that specify their own acceptance types, and show that they accept exactly the languages that \manyonesharp-reduce to NP sets. A natural variant accepts exactly the languages that \manyonesharp-reduce to P sets. We show that these two classes coincide if and only if \psone = \psnnoplusbigohone, where the latter class denotes the sets acceptable via at most one question to \sharpp followed by at most a constant number of questions to \np.Comment: 15 pages, to appear in IJFC

On the Power of Probabilistic Polynomial Time: PNP[log] ⊆ PP

We show that every set in the ΘP2 level of the polynomial hierarchy -- that is, every set polynomial-time truth-table reducible to SAT -- is accepted by a probabilistic polynomialtime Turing machine: PNP[log] ⊆ PP

Three perceptions of the evapotranspiration landscape: comparing spatial patterns from a distributed hydrological model, remotely sensed surface temperatures, and sub-basin water balances

A problem encountered by many distributed hydrological modelling studies is high simulation errors at interior gauges when the model is only globally calibrated at the outlet. We simulated river runoff in the Elbe River basin in central Europe (148 268 km2) with the semi-distributed eco-hydrological model SWIM (Soil and Water Integrated Model). While global parameter optimisation led to Nash–Sutcliffe efficiencies of 0.9 at the main outlet gauge, comparisons with measured runoff series at interior points revealed large deviations. Therefore, we compared three different strategies for deriving sub-basin evapotranspiration: (1) modelled by SWIM without any spatial calibration, (2) derived from remotely sensed surface temperatures, and (3) calculated from long-term precipitation and discharge data. The results show certain consistencies between the modelled and the remote sensing based evapotranspiration rates, but there seems to be no correlation between remote sensing and water balance based estimations. Subsequent analyses for single sub-basins identify amongst others input weather data and systematic error amplification in inter-gauge discharge calculations as sources of uncertainty. The results encourage careful utilisation of different data sources for enhancements in distributed hydrological modelling

Fireshape: a shape optimization toolbox for Firedrake

Shape optimization studies how to design a domain such that a shape function is minimized. Ubiquitous in industrial applications, shape optimization is often constrained to partial differential equations (PDEs). One of the main challenges in PDE-constrained shape optimization is the coupling of domain updates and PDE-solvers. Fireshape addresses this challenge by elegantly coupling the finite element library Firedrake and the Rapid Optimization Library (ROL). The main features of Fireshape are: accessibility to users with minimal shape optimization knowledge; decoupled discretization of control and state variables; full access to Firedrake's PDE-solvers; automated derivation of adjoint equations and shape derivatives; different metrics to define shape gradients; access to ROL's optimization algorithms via PyROL. Fireshape is available at https://github.com/fireshape/fireshape. Fireshape's documentation comprises several tutorials and is available at https://fireshape.readthedocs.io/en/latest/

Reducing the Number of Solutions of NP Functions

AbstractWe study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines, we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses

Permanent magnet optimization for stellarators as sparse regression

A common scientific inverse problem is the placement of magnets that produce a desired magnetic field inside a prescribed volume. This is a key component of stellarator design, and recently permanent magnets have been proposed as a potentially useful tool for magnetic field shaping. Here, we take a closer look at possible objective functions for permanent magnet optimization, reformulate the problem as sparse regression, and propose an algorithm that can efficiently solve many convex and nonconvex variants. The algorithm generates sparse solutions that are independent of the initial guess, explicitly enforces maximum strengths for the permanent magnets, and accurately produces the desired magnetic field. The algorithm is flexible, and our implementation is open-source and computationally fast. We conclude with two new permanent magnet configurations for the NCSX and MUSE stellarators. Our methodology can be additionally applied for effectively solving permanent magnet optimizations in other scientific fields, as well as for solving quite general high-dimensional, constrained, sparse regression problems, even if a binary solution is required