Sequences of matrices arise in many applications in science and engineering. In this thesis we consider matrices that are closely related (or closely related in groups), and we take advantage of the small differences between them to efficiently solve sequences of linear systems and eigenproblems. Recycling techniques, such as recycling preconditioners or subspaces, are popular approaches for reducing computational cost. In this thesis, we introduce two novel approaches for recycling previously computed information for a subsequent system or eigenproblem, and demonstrate good results for sequences arising in several applications. Preconditioners are often essential for fast convergence of iterative methods. However, computing a good preconditioner can be very expensive, and when solving a sequence of linear systems, we want to avoid computing a new preconditioner too often. Instead, we can recycle a previously computed preconditioner, for which we have good convergence behavior of the preconditioned system. We propose an update technique we call the sparse approximate map, or SAM update, that approximately maps one matrix to another matrix in our sequence. SAM updates are very cheap to compute and apply, preserve good convergence properties of a previously computed preconditioner, and help to amortize the cost of that preconditioner over many linear solves. When solving a sequence of eigenproblems, we can reduce the computational cost of constructing the Krylov space starting with a single vector by warm-starting the eigensolver with a subspace instead. We propose an algorithm to warm-start the Krylov-Schur method using a previously computed approximate invariant subspace. We first compute the approximate Krylov decomposition for a matrix with minimal residual, and use this space to warm-start the eigensolver. We account for the residual matrix when expanding, truncating, and deflating the decomposition and show that the norm of the residual monotonically decreases. This method is effective in reducing the total number of matrix-vector products, and computes an approximate invariant subspace that is as accurate as the one computed with standard Krylov-Schur. In applications where the matrix-vector products require an implicit linear solve, we incorporate Krylov subspace recycling. Finally, in many applications, sequences of matrices take the special form of the sum of the identity matrix, a very low-rank matrix, and a small-in-norm matrix. We consider convergence rates for GMRES applied to these matrices by identifying the sources of sensitivity. / Doctor of Philosophy / Problems in science and engineering often require the solution to many linear systems, or a sequence of systems, that model the behavior of physical phenomena. In order to construct highly accurate mathematical models to describe this behavior, the resulting matrices can be very large, and therefore the linear system can be very expensive to solve. To efficiently solve a sequence of large linear systems, we often use iterative methods, which can require preconditioning techniques to achieve fast convergence. The preconditioners themselves can be very expensive to compute. So, we propose a cheap update technique that approximately maps one matrix to another in the sequence for which we already have a good preconditioner. We then combine the preconditioner and the map and use the updated preconditioner for the current system. Sequences of eigenvalue problems also arise in many scientific applications, such as those modeling disk brake squeal in a motor vehicle. To accurately represent this physical system, large eigenvalue problems must be solved. The behavior of certain eigenvalues can reveal instability in the physical system but to identify these eigenvalues, we must solve a sequence of very large eigenproblems. The eigensolvers used to solve eigenproblems generally begin with a single vector, and instead, we propose starting the method with several vectors, or a subspace. This allows us to reduce the total number of iterations required by the eigensolver while still producing an accurate solution. We demonstrate good results for both of these approaches using sequences of linear systems and eigenvalue problems arising in several real-world applications. Finally, in many applications, sequences of matrices take the special form of the sum of the identity matrix, a very low-rank matrix, and a small-in-norm matrix. We examine the convergence behavior of the iterative method GMRES when solving such a sequence of matrices.

Material efficiency strategies are of significant importance for the shift towards producing sustainable vehicles and Asynchronous motors (ASM), both for current and new developments. Strategies such as a lightweight design, product lifetime extension, recyclability, and a material selection process based on the three branches of sustainability (ecological, economical, and social). Additionally, this thesis is produced at the request of a case company involved in the production of ASM, with the aim to investigate the sustainability and recyclability of the materials in their current product. This in combination with the importance of circularity, recyclability and resource efficiency, and an increasing market of motors has led to the study’s two research questions (RQ). RQ1: To which degree can carbon dioxide emissions and energy usage be reduced by changing the materials in an ASM rotor? RQ2: How does the changing of the material in the ASM rotor affect the challenges with recycling it? This study contains a theoretical and empirical sub-study. The theoretical sub-study consists of a literature study of published research on the following subjects: Sustainability; sustainable manufacturing; recycled material; recycling processes and circular economy(CE). CE is research in connection to product development. The empirical sub-study is inspired by a policy assessment framework and adapted to fit the thesis. The empirical work contains a life cycle analysis (LCA) with a singular focus on the environmental impact. The work has been conducted during a period of 20 weeks and planned with an agile approach. For the collection of empirical data, six interviews, relevant industrial articles, and the material database Granta EduPack 2022 R2 have been used. Additionally, the thesis contains a comparison between the current material and alternative materials for the ASM rotor. All in all, this has led to a theoretical result with a singular environmental approach to sustainability.     The result of the thesis has partly been the comparison between the current material in the ASM rotor and the proposed materials. The comparison showed that the maximum values for carbon dioxide (CO2) emissions were equal, and an 8.5 per cent reduction occurred for energy usage. The secondary result is that there are factors that enable and prohibit materials and products’ recyclability potential. Moreover, indicated the result of the thesis that the selection of suitable material could reduce these factors. The authors’ conclusion of the thesis is that a material change has the potential to reduce a product’s current environmental impact, secondary is sustainability a complex term that requires a balance between all three sustainability branches. For future research on the subject is it the author’s recommendation that all three branches of sustainability should be studied in more depth, in connection to the ASM. Moreover, factors outside of CO2 and energy should be studied in further depth.   Keywords: Environmental impact; Sustainability; Recycling methods/material/metals; Circular economy/CE; Product development. / Materialeffektivitetsstrategier är av stor betydelse för övergången till att producera hållbara fordon och asynkronmotorer (ASM), både för nuvarande och framtida utveckling. För att uppnå detta är strategier som lättviktsdesign, förlängning av produktens livslängd, återvinningsbarhet och en materialvalsprocess baserad på hållbarhetens tre grenar (ekologiska, ekonomiska och sociala faktorer) avgörande. Detta examensarbete är framtaget på förfrågan från ett fallföretaget involverat i produktionen av ASM. Syftet är att undersöka hållbarheten och återvinningsbarheten av materialen i deras nuvarande produkt. Examensarbetet grundas på cirkuläritet, återvinningsbarhet och resurseffektivitet, samt den växande marknaden för motorer, vilket har lätt till examensarbetets två forskningsfrågor (FF). FF1: I vilken utsträckning kan koldioxidutsläpp och energianvändning minskas genom att byta material i en ASM-rotor? FF2: Hur påverkar bytet av materialet i ASM-rotorn utmaningarna med att återvinna produkten? Detta examensarbete innehåller en teoretisk och empirisk substudie. Den teoretiska studien består av en litteraturstudie av publicerad forskning inom följande ämnen: Hållbarhet; hållbar tillverkning; återvunnet material; återvinningsprocesser och cirkulär ekonomi (CE). CE tas upp i samband med produktutveckling. Den empiriska substudien är inspirerad av ett policyramverk och är anpassad för att passa detta examensarbete. Det empiriska arbetet innehåller en livscykelanalys (LCA) med fokus på den ekologiska aspekten. Arbetet har bedrivits under en period av 20 veckor och planerats med ett agilt förhållningssätt. För insamling av empiri har sex intervjuer, relevanta industriartiklar och materialdatabasen Granta EduPack 2022 R2 använts. Dessutom innehåller examensarbetet en jämförelse mellan nuvarande material och alternativa material för ASM-rotorn. Sammantaget har detta lett till ett teoretiskt resultat med ett fokus på den ekologiska hållbarheten.   Resultatet av examensarbetet har delvis varit jämförelsen mellan det aktuella materialet i ASM-rotorn och de föreslagna materialen. Jämförelsen visade att maxvärdena för koldioxidutsläpp (CO2) var oförändrat medan det förekom en 8.5 procentuell minskning av energianvändningen. Resultatet utav den andra frågeställningen är att det finns faktorer som möjliggör men även försvårar material och produkters återvinningspotential. Dessutom påvisar resultatet av examensarbetet att valet av lämpligt material kan minska dessa faktorer. Författarnas slutsats av examensarbetet är att ett byta av material har potential att minska en produkts nuvarande miljöpåverkan. En annan insikt som författarna tar med sig är att hållbarhet är ett komplext begrepp som kräver en balans mellan alla tre hållbarhetsgrenarna. För framtida forskning i ämnet är det författarens rekommendation att alla tre grenarna av hållbarhet bör studeras mer djupgående, i anslutning till ASM. Dessutom bör faktorer utanför CO2 och energi studeras närmare.   Nyckelord: Miljöpåverkan; Hållbarhet; Återvinningsmetoder/material/metaller; Cirkulär ekonomi/CE; Produktutveckling

Denna studie kartlägger vilka insamlingsmetoder för PET-flaskan som finns för avfalls- och återvinningsfasen på den svenska marknaden samt vilka verksamheter som ansvarar för dessa. Vilken omställning som görs av verksamheterna för att uppnå fossilfrihet, gällande insamlingsmetoder, energi- och drivmedelsanvändning. Tidigare forskning, djupintervjuer och enkäter ligger som grund för de resultat som presenteras. Studien tittar även på miljömässiga påverkansfaktorer men även komplexiteten gällande kommunala och rikstäckande verksamheter, hur de jobbar med hantering av PET på den svenska marknaden samt deras användning av fossila resurser och vad som görs för att minska sin miljömässiga påverkan. Insamlingsmetoderna dit PET-flaskan kan nå är restavfall, fastighetsnära insamling, återvinningsstationer och pantstationer. Verksamheternas ansvarsområde sker både rikstäckande och på kommunal nivå. Nulägesanalysen indikerar att verksamheter behöver göra ytterligare åtgärder för att bli helt fossilfria men att det sker en omställning med åtgärder för energi och transporter för att uppnå detta. Det finns insamlingsmetoder som möjliggör återvinning, men där sorteringsmetoderna kan vara avgörande för att PET ska kunna återgå till rätt materialflöde. Det går att se att verksamheterna använder förnyelsebar energi och drivmedel och att de aktivt arbetar mot att bli helt fossilfria. / This study identifies which collection methods for the PET bottle are available for the waste and recycling phase on the Swedish market and which businesses are responsible for these methods. What adjustment is being made by the operations to achieve fossil-free, current collection methods, energy, and fuel use. Previous research, in-depth interviews and questionnaires form the basis for the results presented. The study also looks at environmental impact factors but also the complexity of municipal and nationwide activities, how they work with handling PET in the Swedish market and their use of fossil resources and what is done to reduce their environmental impact. The collection methods that the PET bottle can reach are residual waste, collection close to the property, recycling stations and pawn shops. The area of ​​responsibility of the operations takes place both nationwide and at the municipal level. The current situation analysis indicates that businesses need to take additional measures to become completely fossil-free, but that there is a change with measures for energy and transport to achieve this. There are collection methods that enable recycling, but the sorting methods can be crucial for PET to be able to return to the correct material flow. You can see that the companies use renewable energy and fuel and that they are actively working towards becoming completely fossil fuel.