"Algorithmic Differentiation Through Automatic Graph Elimination Ordering (ADTAGEO) is based on the principle of Instant Elimination: At runtime we dynamically maintain a DAG representing only active variables that are alive at any time. Whenever an active variable is deallocated or its value is overwritten the corresponding vertex in the Live-DAG will be eliminated immediately by the well known vertex elimination rule [1]. Consequently, the total memory requirement is equal to that of the sparse forward mode."

"...Common misconceptions regarding the embedding approach are addressed including whether or not it results in an average value control model (no), is necessary to "tweak" the algorithm to get bang-bang solutions (no), requires infinite switching (no), has real-time capability (yes), or reduction to a classical nonlinear optimization problem (a desirable yes)."

"We present a new algorithm to solve polynomial equations, and publish its code, which is 1.6-3 times faster than the ZROOTS subroutine that is commercially available from Numerical Recipes, depending on application. The largest improvement, when compared to naive solvers, comes from a fail-safe procedure that permits us to skip the majority of the calculations in the great majority of cases, without risking catastrophic failure in the few cases that these are actually required. Second, we identify a discriminant that enables a rational choice between Laguerre's Method and Newton's Method (or a new intermediate method) on a case-by-case basis. We briefly review the history of root solving and demonstrate that "Newton's Method" was discovered neither by Newton (1671) nor by Raphson (1690), but only by Simpson (1740). Some of the arguments leading to this conclusion were first given by the British historian of science Nick Kollerstrom in 1992, but these do not appear to have penetrated the astronomical community. Finally, we argue that Numerical Recipes should voluntarily surrender its copyright protection for non-profit applications, despite the fact that, in this particular case, such protection was the major stimulant for developing our improved algorithm."

REBOUND is a new multi-purpose N-body code which is freely available under an open-source license. It was designed for collisional dynamics such as planetary rings but can also solve the classical N-body problem. It is highly modular and can be customized easily to work on a wide variety of different problems in astrophysics and beyond.

"A simplified model of natural convection, similar to the Lorenz (1963) system, is compared to computational fluid dynamics simulations in order to test data assimilation methods and better understand the dynamics of convection. The thermosyphon is represented by a long time flow simulation, which serves as a reference "truth". Forecasts are then made using the Lorenz-like model and synchronized to noisy and limited observations of the truth using data assimilation. The resulting analysis is observed to infer dynamics absent from the model when using short assimilation windows.

Furthermore, chaotic flow reversal occurrence and residency times in each rotational state are forecast using analysis data. Flow reversals have been successfully forecast in the related Lorenz system, as part of a perfect model experiment, but never in the presence of significant model error or unobserved variables. Finally, we provide new details concerning the fluid dynamical processes present in the thermosyphon during these flow reversals."

"…While several practical uses of the method have been proposed in Monte Carlo applications, these require an implementation framework that is flexible, general and efficient. We present such a framework for functions that are either strictly linear, concave, or convex on the unit interval using a series of envelope functions defined through a cascade, and show that this method not only greatly reduces the number of input bits needed in practice compared to other currently proposed solutions for more specific problems, but can easily be coupled to more asymptotically efficient methods to allow for theoretically strong results."

"This paper considers the fundamental problem of recovering a general signal, an image for example, from the magnitude of its Fourier transform. This problem, also known as phase retrieval, arises in many applications and has challenged engineers, physicists, and mathematicians for decades. Its origin comes from the fact that detectors can often times only record the squared modulus of the Fresnel or Fraunhofer diffraction pattern of the radiation that is scattered from an object. In such settings, one cannot measure the phase of the optical wave reaching the detector and, therefore, much information about the scattered object or the optical field is lost since, as is well known, the phase encodes a lot of the structural content of the image we wish to form."

How to sum a large array of floats while keeping errors in check

Runge Kutta method, suitable for adaptive time-stepping.

"We present a very fast algorithm for general matrix factorization of a data matrix for use in the statistical analysis of high-dimensional data via latent factors. Such data are prevalent across many application areas and generate an ever-increasing demand for methods of dimension reduction in order to undertake the statistical analysis of interest. Our algorithm uses a gradient-based approach which can be used with an arbitrary loss function provided the latter is differentiable. The speed and effectiveness of our algorithm for dimension reduction is demonstrated in the context of supervised classification of some real high-dimensional data sets from the bioinformatics literature."

"We consider the problem of parameter estimation for a system of ordinary differential equations from noisy observations on a solution of the system. In case the system is nonlinear, as it typically is in practical applications, an analytic solution to it usually does not exist. Consequently, straightforward estimation methods like the ordinary least squares method depend on repetitive use of numerical integration in order to determine the solution of the system for each of the parameter values considered, and to find subsequently the parameter estimate that minimises the objective function. This induces a huge computational load to such estimation methods. We propose an estimator that is defined as a minimiser of an appropriate distance between a nonparametrically estimated derivative of the solution and the right-hand side of the system applied to a nonparametrically estimated solution.…"

"In this paper, a parametric level set method for reconstruction of obstacles in general inverse problems is considered. General evolution equations for the reconstruction of unknown obstacles are derived in terms of the underlying level set parameters. We show that using the appropriate form of parameterizing the level set function results a significantly lower dimensional problem, which bypasses many difficulties with traditional level set methods, such as regularization, re-initialization and use of signed distance function.…"

"We propose a degree-based coarse graining approach that not just accelerates the evaluation of dynamics on complex networks, but also satisfies the consistency conditions for both equilibrium statistical distributions and nonequilibrium dynamical flows. For the Ising model and susceptible-infected-susceptible epidemic model, we introduce these required conditions explicitly and further prove that they are satisfied by our coarse-grained network construction within the annealed network approximation. Finally, we numerically show that the phase transitions and fluctuations on the coarse-grained network are all in good agreements with those on the original one."

"We introduced an effective algorithm for k-nearest neighbor problem which works on multiple GPUs. By an experiment, we have shown that it runs more than 330 times faster than an implementation on a single core of an up-to-date CPU. We have also shown that the algorithm is effective from the viewpoint of parallelism of GPUs. That is because 1) there is no synchronization between GPUs until the very end of the process and 2) the workload is well balanced."

"Many computerized methods for RNA-RNA interaction structure prediction have been developed. Recently, $O(N^6)$ time and $O(N^4)$ space dynamic programming algorithms have become available that compute the partition function of RNA-RNA interaction complexes. However, few of these methods incorporate the knowledge concerning related sequences, thus relevant evolutionary information is often neglected from the structure determination. Therefore, it is of considerable practical interest to introduce a method taking into consideration both thermodynamic stability and sequence covariation. We present the \emph{a priori} folding algorithm \texttt{ripalign}, whose input consists of two (given) multiple sequence alignments (MSA). \texttt{ripalign} outputs (1) the partition function, (2) base-pairing probabilities, (3) hybrid probabilities and (4) a set of Boltzmann-sampled suboptimal structures consisting of canonical joint structures that are compatible to the alignments.…"

"A new floating-point arithmetic called precision arithmetic is developed to track precision for arithmetic calculations. It uses a novel rounding scheme to avoid excessive rounding error propagation of conventional floating-point arithmetic. Unlike interval arithmetic, its uncertainty tracking is based on statistics and its bounding range is much tighter. Generic standards and systematic methods for validating uncertainty-bearing arithmetics are discussed. The precision arithmetic is found to be better than interval arithmetic in uncertainty-tracking for linear algorithms. "

"The well known algorithm of VOLKER STRASSEN for matrix multiplication can only be used for $(m2^k \times m2^k)$ matrices. For arbitrary $(n \times n)$ matrices one has to add zero rows and columns to the given matrices to use STRASSEN's algorithm. … The aim of this work is to give a detailed analysis of the number of additional zero rows and columns and the additional work caused by STRASSEN's bad parameters. STRASSEN used the parameters $m$ and $k$ to show that his matrix multiplication algorithm needs less than $4.7n^{\log_2 7}$ flops. We can show in this paper, that these parameters cause an additional work of approx. 20 % in the worst case in comparison to the optimal strategy for the worst case. This is the main reason for the search for better parameters."

"We present a novel method for the calculation of free energy landscapes. Our approach involves a history dependent bias potential which is evaluated on a grid. The corresponding free energy landscape is constructed via a histogram reweighting procedure a posteriori. Due to the presence of the bias potential, our method can also be used to accelerate rare events. In addition, the calculated free energy landscape is not restricted to the actual choice of collective variables and can in principle be extended to all variables of interest without further numerical effort. We present numerical results for the alanine dipeptide and the Met-Enkephalin in explicit solution to illustrate our approach."