Simplicial Data Analysis: theory, practice, and algorithms

Simplicial complexes store in discrete form key information on a topological space, and have been used in mathematics to introduce combinatorial and discrete tools in geometry and topology. They represent a topological space as a collection of ‘simple elements’ (such as vertices, edges, triangles, tetrahedra, and more general simplices) that are glued to each other in a structured manner. In the last 40 years, they have been a basic tool in computer visualization for storing and classifying different shapes of 3d images, then in the early 2000s these techniques were success- fully applied to more general data, not necessarily sampled from a metric space. The use of techniques borrowed from algebraic topology has been very successfull in analysing data from various fields: genomics, sensor analysis, brain connectomics, fMRI data, trade net- works, and new fields of application are being tested every day. Regrettably, topological data analysis has been used mainly as a qualitative method, the problem being the lack of proper tools to perform effective statistical analysis. Coming from well established techniques in random graph theory, the first models for random simplicial complexes have been introduced in recent years, none of which though can be used effectively in a quantitative analysis of data. We introduce a model that can be successfully used as a null model for simplicial complexes as it fixes the size distribution of facets. Another challenge is to successfully identify a simplicial complex which can correctly encode the topological space from which the initial data set is sampled. The most common solution is to build nesting simplicial complexes, and study the evolution of their features. A recent study uncovered that the problem can reside in making wrong assumption on the space of data. We propose a categorical reasoning which enlightens the cause leading to these misconceptions. We introduce a new category for weighted graphs and study its relation to other common categories when the weights are chosen in a poset. The construction of the appropriate simplicial complex is not the only obstacle one faces when applying topological methods to real data. Available algorithms for homological features extraction have a memory and time complexity which scales exponentially on the number of simplices, making these techniques not suitable for the analysis of ‘big data’. We propose a quantum algorithm which is able to track in logarithmic time the evolution of a quantum version of well known homological features along a filtration of simplicial complexes

Structure-Preserving Model Reduction of Physical Network Systems

This paper considers physical network systems where the energy storage is naturally associated to the nodes of the graph, while the edges of the graph correspond to static couplings. The first sections deal with the linear case, covering examples such as mass-damper and hydraulic systems, which have a structure that is similar to symmetric consensus dynamics. The last section is concerned with a specific class of nonlinear physical network systems; namely detailed-balanced chemical reaction networks governed by mass action kinetics. In both cases, linear and nonlinear, the structure of the dynamics is similar, and is based on a weighted Laplacian matrix, together with an energy function capturing the energy storage at the nodes. We discuss two methods for structure-preserving model reduction. The first one is clustering; aggregating the nodes of the underlying graph to obtain a reduced graph. The second approach is based on neglecting the energy storage at some of the nodes, and subsequently eliminating those nodes (called Kron reduction).</p

Teadusarvutuse algoritmide taandamine hajusarvutuse raamistikele

Teadusarvutuses kasutatakse arvuteid ja algoritme selleks, et lahendada probleeme erinevates reaalteadustes nagu geneetika, bioloogia ja keemia. Tihti on eesmärgiks selliste loodusnähtuste modelleerimine ja simuleerimine, mida päris keskkonnas oleks väga raske uurida. Näiteks on võimalik luua päikesetormi või meteoriiditabamuse mudel ning arvutisimulatsioonide abil hinnata katastroofi mõju keskkonnale. Mida keerulisemad ja täpsemad on sellised simulatsioonid, seda rohkem arvutusvõimsust on vaja. Tihti kasutatakse selleks suurt hulka arvuteid, mis kõik samaaegselt töötavad ühe probleemi kallal. Selliseid arvutusi nimetatakse paralleel- või hajusarvutusteks. Hajusarvutuse programmide loomine on aga keeruline ning nõuab palju rohkem aega ja ressursse, kuna vaja on sünkroniseerida erinevates arvutites samaaegselt tehtavat tööd. On loodud mitmeid tarkvararaamistikke, mis lihtsustavad seda tööd automatiseerides osa hajusprogrammeerimisest. Selle teadustöö eesmärk oli uurida selliste hajusarvutusraamistike sobivust keerulisemate teadusarvutuse algoritmide jaoks. Tulemused näitasid, et olemasolevad raamistikud on üksteisest väga erinevad ning neist ükski ei ole sobiv kõigi erinevat tüüpi algoritmide jaoks. Mõni raamistik on sobiv ainult lihtsamate algoritmide jaoks; mõni ei sobi olukorras, kus andmed ei mahu arvutite mällu. Algoritmi jaoks kõige sobivama hajusarvutisraamistiku valimine võib olla väga keeruline ülesanne, kuna see nõuab olemasolevate raamistike uurimist ja rakendamist. Sellele probleemile lahendust otsides otsustati luua dünaamiline algoritmide modelleerimise rakendus (DAMR), mis oskab simuleerida algoritmi implementatsioone erinevates hajusarvutusraamistikes. DAMR aitab hinnata milline hajusraamistik on kõige sobivam ette antud algoritmi jaoks, ilma algoritmi reaalselt ühegi hajusraamistiku peale implementeerimata. Selle uurimustöö peamine panus on hajusarvutusraamistike kasutuselevõtu lihtsamaks tegemine teadlastele, kes ei ole varem nende kasutamisega kokku puutunud. See peaks märkimisväärselt aega ja ressursse kokku hoidma, kuna ei pea ükshaaval kõiki olemasolevaid hajusraamistikke tundma õppima ja rakendama.Scientific computing uses computers and algorithms to solve problems in various sciences such as genetics, biology and chemistry. Often the goal is to model and simulate different natural phenomena which would otherwise be very difficult to study in real environments. For example, it is possible to create a model of a solar storm or a meteor hit and run computer simulations to assess the impact of the disaster on the environment. The more sophisticated and accurate the simulations are the more computing power is required. It is often necessary to use a large number of computers, all working simultaneously on a single problem. These kind of computations are called parallel or distributed computing. However, creating distributed computing programs is complicated and requires a lot more time and resources, because it is necessary to synchronize different computers working at the same time. A number of software frameworks have been created to simplify this process by automating part of a distributed programming. The goal of this research was to assess the suitability of such distributed computing frameworks for complex scientific computing algorithms. The results showed that existing frameworks are very different from each other and none of them are suitable for all different types of algorithms. Some frameworks are only suitable for simple algorithms; others are not suitable when data does not fit into the computer memory. Choosing the most appropriate distributed computing framework for an algorithm can be a very complex task, because it requires studying and applying the existing frameworks. While searching for a solution to this problem, it was decided to create a Dynamic Algorithms Modelling Application (DAMA), which is able to simulate the implementation of the algorithm in different distributed computing frameworks. DAMA helps to estimate which distributed framework is the most appropriate for a given algorithm, without actually implementing it in any of the available frameworks. This main contribution of this study is simplifying the adoption of distributed computing frameworks for researchers who are not yet familiar with using them. It should save significant time and resources as it is not necessary to study each of the available distributed computing frameworks in detail