Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked, and in the handling of singular solutions, have made probability-one homotopy methods even more practical. This thesis describes the theory behind and performance of the new code POLSYS_PLP, which consists of Fortran 90 modules for finding all isolated solutions of a complex coefficient polynomial system of equations by a probability-one homotopy method. The package is intended to be used in conjunction with HOMPACK90, and makes extensive use of Fortran 90 derived data types to support a partitioned linear product (PLP) polynomial system structure. PLP structure is a generalization of m-homogeneous structure, whereby each component of the system can have a different m-homogeneous structure. POLSYS_PLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different PLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding. / Master of Science

Abstract The aim of this work is to improve the overall robustness in equation-oriented chemical engineering simulation work. Because the performance of locally convergent solving methods is strongly dependent on a favourable initial guess, bounded homotopy methods were investigated as a way to enlarge the domain of convergence. Bounded homotopies make it possible to keep the homotopy path inside a feasible problem domain. Thus the fatal errors possibly caused by unfeasible variable values in thermodynamic subroutines can be avoided. To enable the utilization of a narrow bounding zone, modifications were proposed for bounded homotopies. The performance of the modifications was studied with simple test problems and several types of distillation systems in the MATLAB environment. The findings illustrate that modified bounded homotopies with variables mapping make it possible to bound the homotopy path strictly to run inside a feasible problem domain. The homotopy path can be tracked accurately and flexibly also inside a narrow bounding zone. It was also noticed that by utilizing the concept of bounding the homotopy path with respect to the homotopy parameter, the possibility of approaching starting point and solution multiplicities is increased in cases where the traditional problem-independent homotopy method fails. The concept aims to connect separate homotopy path branches thus offering a trackable path with real space arithmetic. Even though the modified bounded homotopies were found to overcome several challenges often encountered with traditional problem-independent homotopy continuation methods, alone they are not enough to guarantee that the solution is approached from an arbitrary starting point. Therefore, problem-tailored solving procedures were implemented in the consideration of complex column configurations. Problem-tailored solving procedures aim to offer feasible consecutive sub-problems and thus direct the solving towards the state distribution that fulfils exact product purity specifications. As a whole, the modified bounded homotopies and problem-tailored solving procedures were found to improve the overall robustness of an equation-oriented solving approach. Thus the threshold for designing and implementing complex process systems such as complex distillation configurations for practical use could be lowered.

This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-Teichmüller group. In Paper I, we present an algorithm for the computation of degrees of Segre classes of closed subschemes of complex projective space. The algorithm is based on the residual intersection theorem and can be implemented both symbolically and numerically. In Paper II, we describe an algorithm for the computation of the degrees of Chern-Schwartz-MacPherson classes and the topological Euler characteristic of closed subschemes of complex projective space, provided an algorithm for the computation of degrees of Segre classes. We also explain in detail how the algorithm in Paper I can be implemented numerically. Together this yields a symbolical and a numerical version of the algorithm. Paper III describes the Macaulay2 package CharacteristicClasses. It implements the algorithms from papers I and II, as well as an algorithm for the computation of degrees of Chern classes. In Paper IV, we show that L-infinity-automorphisms of the Schouten algebra T_poly(R^d) of polyvector fields on affine space R^d which satisfy certain conditions can be globalized. This means that from a given L-infinity-automorphism of T_poly(R^d) an L-infinity-automorphism of T_poly(M) can be constructed, for a general smooth manifold M. It follows that Willwacher's action of the Grothendieck-Teichmüller group on T_poly(R^d) can be globalized, i.e., the Grothendieck-Teichmüller group acts on the Schouten algebra T_poly(M) of polyvector fields on a general manifold M. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Accepted.</p>

Abstract The aim of this work was to formulate separation process models with both predictive capabilities and robust solution strategies. Although all separation process models should have predictive capabilities, the current literature still has multiple applications in which predictive models having the combination of a clear phenomenon base and robust solving strategy are not available. The separation process models investigated in this work were liquid-liquid phase separation and membrane separation models. The robust solving of a liquid-liquid phase separation model typically demands the solution of a phase stability analysis problem. In addition, predicting the liquid-liquid phase compositions reliably depends on robust phase stability analysis. A phase stability analysis problem has multiple feasible solutions, all of which have to be sought to ensure both the robust solving of the model and predictive process model. Finding all the solutions with a local solving method is difficult and generally inexact. Therefore, the modified bounded homotopy methods, a global solving method, were further developed to solve the problem robustly. Robust solving demanded the application of both variables and homotopy parameter bounding features and the usage of the trivial solution in the solving strategy. This was shown in multiple liquid-liquid equilibrium cases. In the context of membrane separation models, predictive capabilities are achieved with the application of a Maxwell-Stefan based model. With the Maxwell-Stefan approach, multicomponent separation can be predicted based on pure component permeation data alone. On the other hand, the solving of the model demands a robust solving strategy with application-dependent knowledge. These issues were illustrated in the separation of a H2/CO2 mixture with a high-silica MFI zeolite membrane at high pressure and low temperature. Similarly, the prediction of mixture adsorption based on pure component adsorption data alone was successfully demonstrated. In the context of membrane separation models, predictive capabilities are achieved with the application of a Maxwell-Stefan based model. With the Maxwell-Stefan approach, multicomponent separation can be predicted based on pure component permeation data alone. On the other hand, the solving of the model demands a robust solving strategy with application-dependent knowledge. These issues were illustrated in the separation of a H2/CO2 mixture with a high-silica MFI zeolite membrane at high pressure and low temperature. Similarly, the prediction of mixture adsorption based on pure component adsorption data alone was successfully demonstrated. / Tiivistelmä Työn tavoitteena oli muotoilla prosessin käyttäytymisen ennustamiseen kykeneviä erotusprosessimalleja ja niiden ratkaisuun käytettäviä luotettavia strategioita. Vaikka kaikkien erotusprosessimallien tulisi olla ennustavia, on tällä hetkellä useita kohteita, joissa prosessin käyttäytymistä ei voida ennustaa siten, että käytettävissä olisi sekä ilmiöpohjainen malli että ratkaisuun soveltuva luotettava strategia. Tässä työssä erotusprosessimalleista kohteina tarkasteltiin neste-neste-erotuksen ja membraanierotuksen kuvaukseen käytettäviä malleja. Neste-neste-erotusmallien luotettava ratkaisu vaatii yleensä faasistabiilisuusongelman ratkaisua. Lisäksi faasien koostumusten luotettava ennustaminen pohjautuu faasistabiilisuusanalyysiin. Faasistabiilisuusongelmalla on useita mahdollisia ratkaisuja, jotka kaikki tulee löytää, jotta voitaisiin varmistaa luotettava mallin ratkaisu sekä prosessimallin ennustuskyvyn säilyminen. Kaikkien ratkaisujen löytäminen on sekä vaikeaa että epätarkkaa paikallisesti konvergoituvilla ratkaisumenetelmillä. Tämän vuoksi globaaleihin ratkaisumenetelmiin kuuluvia modifioituja rajoitettuja homotopiamenetelmiä kehitettiin edelleen, jotta faasistabiilisuusongelma saataisiin ratkaistua luotettavasti. Ratkaisun luotettavuus vaati sekä muuttujien että homotopiaparametrin rajoittamista ja ongelman triviaalin ratkaisun käyttöä ratkaisustrategiassa. Tämä käyttäytyminen todennettiin useissa neste-nestetasa-painoa kuvaavissa esimerkeissä. Membraanierotusta tarkasteltaessa ennustava malli voidaan muotoilla käyttämällä Maxwell-Stefan pohjaista mallia. Maxwell-Stefan lähestymistavalla voidaan ennustaa monikomponenttiseosten erotusta perustuen puhtaiden komponenttien membraanin läpäisystä saatuun mittausaineistoon. Toisaalta mallin ratkaisu vaatii luotettavan ratkaisustrategian, jossa hyötykäytetään kohteesta riippuvaa tietoa. Näitä kysymyksiä havainnollistettiin H2/CO2 seoksen erotuksessa MFI-zeoliitti-membraanilla korkeassa paineessa. Samoin seosten adsorboitumiskäyttäytymistä ennustettiin onnistuneesti pelkästään puhtaiden komponenttien adsorptiodatan pohjalta. Kokonaisuutena voidaan todeta, että tarkasteltujen erotusprosessimallien ennustavuutta voidaan parantaa yhdistämällä malli, jolla on selkeä ilmiöpohja ja luotettava ratkaisustrategia. Lisäksi mallien käytettävyys erotusprosessien suunnittelussa on parantunut työn tulosten pohjalta.