Assessing the robustness and scalability of the accelerated pseudo-transient method towards exascale computing

The development of highly efficient, robust and scalable numerical algorithms lags behind the rapid increase in massive parallelism of modern hardware. We address this challenge with the accelerated pseudo-transient iterative method and present here a physically motivated derivation. We analytically determine optimal iteration parameters for a variety of basic physical processes and confirm the validity of theoretical predictions with numerical experiments. We provide an efficient numerical implementation of pseudo-transient solvers on graphical processing units (GPUs) using the Julia language. We achieve a parallel efficiency over 96 % on 2197 GPUs in distributed memory parallelisation weak scaling benchmarks. 2197 GPUs allow for unprecedented terascale solutions of 3D variable viscosity Stokes flow on 49953 grid cells involving over 1.2 trillion degrees of freedom. We verify the robustness of the method by handling contrasts up to 9 orders of magnitude in material parameters such as viscosity, and arbitrary distribution of viscous inclusions for different flow configurations. Moreover, we show that this method is well suited to tackle strongly nonlinear problems such as shear-banding in a visco-elasto-plastic medium. A GPU-based implementation can outperform CPU-based direct-iterative solvers in terms of wall-time even at relatively low resolution. We additionally motivate the accessibility of the method by its conciseness, flexibility, physically motivated derivation and ease of implementation. This solution strategy has thus a great potential for future high-performance computing applications, and for paving the road to exascale in the geosciences and beyond.

A Method for the Reduction of Out-of-Band Measurement Errors in Multi-band Instruments using Synthetic Source Distributions

A method to reduce multi-band sensor measurement biases due to finite out-of-band response is described. The method takes advantage of the fact that out-of-band measurement errors cancel if the calibration source and the measured source have the same spectral distributions—independent of their spectral distributions or the magnitude of a sensor band’s out-of-band response. Using a known spectral responsivity, a synthetic, arbitrary source spectral distribution can replace a realized spectral distribution in the measurement equation and the signal can be calculated rather than measured. Given the freedom to select any arbitrary distribution for the synthetic source, the efficacy of the approach depends on the fidelity of the replication of the measured spectrum by the synthetic source spectrum. To illustrate the method, an example application is given of top-of-the-atmosphere measurements of water-leaving radiance by multi-band filter radiometers on celestial Earth-viewing sensors.

Mathematical Model of the Delay in Communication Networks Based on QS with a Time Lag

In the mathematical modeling of modern computer networks, telecommunication networks, traffic flows, logistics and many others, the methods of queuing theory are widely used. In turn, in studies of queuing systems (QS) G/G/1 with arbitrary distribution laws of intervals between adjacent requirements of the incoming flow and their service time, the spectral decomposition method (MSD) of solving the Lindley integral equation is often used. This method is based on the search for zeros and poles of the constructed spectral decomposition in the form of some fractional-rational function using numerical methods to determine the roots of polynomials. In this case, the coefficients of the polynomial in the numerator of the expansion are expressed through the unknown parameters of the distribution laws used to describe the QS. In the case of teletraffic research, usually these unknown parameters of the distribution laws can be determined through the numerical characteristics of the intervals between traffic packets by the method of moments. The purpose of this article is to present a fundamentally new mathematical model of a system formed by two flows with distribution laws shifted to the right. This is possible only for those probability distribution laws whose density functions are Laplace transformable. The main advantages of such systems, let us call them time lag systems, are that they provide less queue latency compared to conventional systems, and that they extend the range of traffic parameters. The article presents the results obtained on the average delay of requests in the queue for a system with exponential and hyper-Erlang distributions, an algorithm for calculating the average delay and the results of computational experiments in the Mathcad package.

Mathematical Model of Call Center in the Form of Multi-Server Queueing System

The paper considers the model of a call center in the form of a multi-server queueing system with Poisson arrivals and an unlimited waiting area. In the model under consideration, incoming calls do not differ in terms of service conditions, requested service, and interarrival periods. It is assumed that an incoming call can use any free server and they are all identical in terms of capabilities and quality. The goal problem is to find the stationary distribution of the number of calls in the system for an arbitrary recurrent service. This will allow us to evaluate the performance measures of such systems and solve various optimization problems for them. Considering models with non-exponential service times provides solutions for a wide class of mathematical models, making the results more adequate for real call centers. The solution is based on the approximation of the given distribution function of the service time by the hyperexponential distribution function. Therefore, first, the problem of studying a system with hyperexponential service is solved using the matrix-geometric method. Further, on the basis of this result, an approximation of the stationary distribution of the number of calls in a multi-server system with an arbitrary distribution function of the service time is constructed. Various issues in the application of this approximation are considered, and its accuracy is analyzed based on comparison with the known analytical result for a particular case, as well as with the results of the simulation.

Determination of the Probability Density Function of the Mixture Signal and Additive Noise under Influence of Multiplicative Noise

Issues related to effect of multiplicative (modulating) (otherwise amplitude distortions) and additive noises on processed signal are considered. Analysis of statistical characteristics of probability density function (PDF) of instantaneous signal values against background of modulating interference is carried out. Expressions of the joint PDF envelope, phase and instantaneous signal values are obtained, for the case of only phase distortions, as well as the case when fluctuations in amplitude and phase are independent of each other. It is noted that in the second case, with the independence of phase and amplitude distortions, the PDF of the signal, first of all, is determined by the PDF of its instantaneous values and practically does not depend on the PDF of the signal phase. Expressions are obtained for the PDF useful signal and the most common PDF envelope when affecting the signal with modulating noise for deep phase distortions and for its uniform distribution over the interval (0, 2π). In addition, for the case of some functional relationship of phase and amplitude fluctuations, the expression for PDF instantaneous signal values is defined. The presence of such a functional connection allows calculating the PDF of the signal through the statistical characteristics of its envelope. It is obtained that PDF of mixture of signal and additive noise contains arbitrary distribution of phase and amplitude, as well as arbitrary law of distribution of envelope of processed signal. It is obtained that weight coefficients are determined by derivative of characteristic function of amplitude distortion in case when PDFs of said mixture are distributed according to normal law and functional connection exists between amplitude and phase distortions.

Amplification and attenuation of noisy expression by export processes

Inside mammalian cells, single genes are known to be transcribed in stochastic bursts leading to the synthesis of nuclear RNAs that are subsequently exported to the cytoplasm to create mRNAs. We systematically characterize the role of export processes in shaping the extent of random fluctuations (i.e. noise) in the mRNA level of a given gene. Using the method of Partitioning of Poisson arrivals, we derive an exact analytical expression for the noise in mRNA level assuming that the nuclear retention time of each RNA is an independent and identically distributed random variable following an arbitrary distribution. These results confirm recent experimental/theoretical findings that decreasing the nuclear export rate buffers the noise in mRNA level, and counterintuitively, decreasing the noise in the nuclear retention time enhances the noise in the mRNA level. Next, we further generalize the model to consider a dynamic extrinsic disturbance that affects the nuclear-to-cytoplasm export. Our results show that noise in the mRNA level varies non-monotonically with the disturbance timescale. More specifically, high- and low-frequency external disturbances have little impact on the mRNA noise level, while noise is amplified at intermediate frequencies. In summary, our results systematically uncover how the coupling of bursty transcription with nuclear export can both attenuate or amplify noise in mRNA levels depending on the nuclear retention time distribution and the presence of extrinsic fluctuations.

Excitation of the Round Infinite Impedance Cylinder by Internal Extrinsic Current: Far-Field Radiation

The paper considers the method of solving the problem of excitation a circular infinite impedance cylinder by means of extrinsic current with an arbitrary spatial orientation. The formal solution of the problem is based on the method of expanding the fields in a complete system of eigenfunctions. Arbitrary distribution of extrinsic currents is taken into account factors excitation. An approximate solution of the problem in the far field is presented. The effect of a dielectric cylinder on the directional diagram of an axial electric dipole is studied.

Mathematical model of delay based on a system with gamma distribution

This article is devoted to the analysis of a queuing system formed by two flows with density functions of the gamma distribution law in order to derive a solution for the average delay of requests in the queue, which is the main characteristic for any queuing system. According to this characteristic, for example, packet delays in packet-switched networks are estimated when they are modeled using the queuing system. In queuing theory, studies of G/G/1 systems are especially relevant because there is no solution in the final form for the general case. Therefore, in the study of such systems, various particular distribution laws are used as an arbitrary distribution law for G. In the study of G/G/1 systems, an important role is played by the method of spectral decomposition of the solution of the Lindley integral equation, and most of the results in the theory of queuing were obtained using this method. The article presents the derivation of the calculation formula for the average delay of requests in the queue in the system under consideration, also based on the spectral decomposition method.

Rarefied gas pipe network analysis via kinetic theory

Το θέμα της παρούσας διδακτορικής διατριβής εστιάζει στην ανάπτυξη ενός αλγορίθμου προσομοίωσης δικτύων αερίων υπό αραιοποιημένες συνθήκες μέσω κινητικής θεωρίας. Η περιοχή αυτή είναι γνωστή σαν αραιοποιημένη αεριοδυναμική (rarefied gas dynamics) με πολλές τεχνολογικές εφαρμογές εκ των οποίων τα μικρο-ηλεκτρομηχανολογικά συστήματα και η τεχνολογία κενού να είναι αυτές που εξελίσσονται με πολύ γρήγορους ρυθμούς. Σημειώνεται ότι ενώ στο συνεχές όριο, υπολογιστικά πακέτα (αλγόριθμοι) σχεδιασμού δικτύων σωληνώσεων (όπως δίκτυα συμπιεσμένου αέρα ή φυσικού αερίου, κτλ.) είναι ευρέως διαδεδομένα, αντίστοιχα υπολογιστικά εργαλεία για το σχεδιασμό δικτύων σε συνθήκες χαμηλής πίεσης (υψηλού, μέτριου ή χαμηλού κενού) είναι αρκετά περιορισμένα. Αξιοποιώντας την πολύχρονη εμπειρία του Eργαστηρίου Φυσικών & Χημικών Διεργασιών σε αριθμητικές μεθόδους μεσοκλίμακας, στο πλαίσιο της διατριβής αναπτύσσεται και εφαρμόζεται σύνθετο λογισμικό προσομοίωσης κυκλοφορίας αραιοποιημένων αερίων σε δίκτυα σωληνώσεων σε όλο το εύρος του αριθμού Knudsen. Για την υλοποίηση αυτού του εγχειρήματος δημιουργείται εκτενής βάση δεδομένων για πλήρως ανεπτυγμένες και αναπτυσσόμενες ροές μέσα από αγωγούς μεγάλου, μεσαίου και μικρού μήκους, συμπεριλαμβανομένων των φαινομένων εισόδου/εξόδου στα άκρα των αγωγών. Αυτό επιτυγχάνεται επιλύοντας κινητικές εξισώσεις με αναβαθμισμένους και παράλληλους αλγορίθμους διακριτών μοριακών ταχυτήτων, ενώ η χρήση του λογισμικού υποβοηθείται από ένα γραφικό περιβάλλον το οποίο αναπτύχθηκε για το σκοπό αυτό. Η τελική μορφή του αλγορίθμου αποτελείται από α) την εισαγωγή των γεωμετρικών χαρακτηριστικών και λειτουργικών χαρακτηριστικών του δικτύου προς επίλυση, β) τον ορισμό των βρόχων και των ψευδο-βρόχων, γ) τη διαμόρφωση και επίλυση των εξισώσεων διατήρησης μάζας και ενέργειας, δ) τη βάση δεδομένων που έχει προκύψει από αποτελέσματα κινητικών εξισώσεων και χρησιμοποιείται για την επίλυση των εξισώσεων διατήρησης και ε) τα αποτελέσματα του αλγορίθμου τα οποία αναφέρονται σε τιμές πίεσης στους κόμβους του δικτύου καθώς και σε τιμές μαζικής και ογκομετρικής παροχής μέσα από τις σωληνώσεις του δικτύου. Πιο συγκεκριμένα, το σύστημα των εξισώσεων το οποίο περιγράφει το δίκτυο αποτελείται από τις εξισώσεις πτώσης πίεσης σε κάθε έναν από τους αγωγούς του δικτύου σε συνδυασμό με τις εξισώσεις διατήρησης μάζας σε κάθε έναν από τους κόμβους του δικτύου. Στην περίπτωση που το δίκτυο είναι πλήρως ορισμένο, οι εξισώσεις πτώσης πίεσης ανάγονται σε ισοζύγια ενέργειας ανάμεσα στους κλειστούς βρόχους του δικτύου και τους ψευδο-βρόχους που ενώνουν κόμβους γνωστών ιδιοτήτων (π.χ. γνωστές τιμές πίεσης). Το σύστημα των εξισώσεων λύνεται επαναληπτικά έχοντας ως άγνωστο τις τιμές των παροχών υποθέτοντας αρχικά τις τιμές των πιέσεων στους κόμβους του δικτύου, όπου αυτές δεν είναι από την αρχή γνωστές. Σε κάθε βήμα, μετά την επίλυση του συστήματος των εξισώσεων, οι τιμές των πιέσεων ανανεώνονται σύμφωνα με τις τιμές της αδιάστατης παροχής κάνοντας χρήση των σχέσεων πτώσης πίεσης για κάθε έναν από τους αγωγούς του δικτύου. Τα βήματα αυτά αποτελούν τον πυρήνα μιας επαναληπτικής διαδικασίας η οποία τερματίζεται με την ικανοποίηση του κριτηρίου σύγκλισης το οποίο εφαρμόζεται στις τιμές των πιέσεων στους κόμβους. Το υπολογιστικό πακέτο φέρει το όνομα ARIADNE (Algorithm for Rarefied gas flow in Arbitrary Distribution Networks). Η αποτελεσματικότητα και η ακρίβεια του αλγορίθμου ελέγχθηκε με την προσομοίωση διαφόρων δικτύων σωληνώσεων στο υδροδυναμικό όριο, συγκρίνοντας τα αποτελέσματα με αντίστοιχα που προκύπτουν από τυπικό αλγόριθμο που βασίζεται στις υδροδυναμικές εξισώσεις επιστρέφοντας πολύ καλή συμφωνία τόσο στις παροχές των αγωγών, όσο και στις πιέσεις των κόμβων του δικτύου. Επιπλέον, στην ελεύθερη μοριακή περιοχή και στην μεταβατική πραγματοποιήθηκαν συγκρίσεις με το λογισμικό ITERVAC, το οποίο είναι ένας ημι-εμπειρικός αλγόριθμος, επιστρέφοντας πολύ καλή συμφωνία. Τέλος, το λογισμικό εφαρμόστηκε στην επίλυση δικτύων που προσεγγίζουν το σύστημα άντλησης αερίων του αντιδραστήρα σύντηξης ITER. Με βάση τα παραπάνω η παρούσα διδακτορική διατριβή αναμένεται να αποτελέσει τον πυρήνα ενός γενικότερου αλγορίθμου για τη μελέτη ροών αερίων σε δίκτυα σωληνώσεων σε όλο το εύρος αραιοποίησης.