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  • 1. Aalto, Daniel
    et al.
    Berkovits, Lauri
    Kansanen, Outi Elina
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Yue, Hong
    John-Nirenberg lemmas for a doubling measure2011In: Studia Mathematica, ISSN 0039-3223, E-ISSN 1730-6337, Vol. 204, no 1, p. 21-37Article in journal (Refereed)
    Abstract [en]

    We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderon-Zygmund decomposition in metric spaces and use it to prove the corresponding John Nirenberg inequality.

  • 2. Aas, E.
    et al.
    Ayyer, A.
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Potka, Samu
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    The exact phase diagram for a semipermeable TASEP with nonlocal boundary jumps2019In: Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8113, E-ISSN 1751-8121, Vol. 52, no 35, article id 355001Article in journal (Refereed)
    Abstract [en]

    We consider a finite one-dimensional totally asymmetric simple exclusion process with four types of particles, {1, 0, 1, }, in contact with reservoirs. Particles of species 0 can neither enter nor exit the lattice, and those of species are constrained to lie at the first and last site. Particles of species 1 enter from the left reservoir into either the first or second site, move rightwards, and leave from either the last or penultimate site. Conversely, particles of species 1 enter from the right reservoir into either the last or penultimate site, move leftwards, and leave from either the first or last site. This dynamics is motivated by a natural random walk on the Weyl group of type D. We compute the exact nonequilibrium steady state distribution using a matrix ansatz building on earlier work of Arita. We then give explicit formulas for the nonequilibrium partition function as well as densities and currents of all species in the steady state, and derive the phase diagram.

  • 3.
    Aas, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A Markov Process on Cyclic Words2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    The TASEP (totally asymmetric simple exclusion process) studied here is a Markov chain on cyclic words over the alphabet{1,2,...,n} given by at each time step sorting an adjacent pair of letters chosen uniformly at random. For example, from the word 3124 one may go to 1324, 3124, 3124, 4123 by sorting the pair 31, 12, 24, or 43.

    Two words have the sametype if they are permutations of each other. If we restrict TASEP to words of some particular type m we get an ergodic Markov chain whose stationary distribution we denote by ζm. Soζm (u) is the asymptotic proportion of time spent in the state u if the chain started in some word of type m. The distribution ζ is the main object of study in this thesis. This distribution turns out to have several remarkable properties, and alternative characterizations. It has previously been studied both from physical, combinatorial, and probabilitistic viewpoints.

    In the first chapter we give an extended summary of known results and results in this thesis concerning ζ. The new results are described (and proved) in detail in Papers I - IV.

    The new results in Papers I and II include an explicit formula for the value ofζat sorted words and a product formula for decomposable words. We also compute some correlation functions for ζ. In Paper III we study of a generalization of TASEP to Weyl groups. In Paper IV we study a certain scaling limit of ζ, finding several interesting patterns of which we prove some. We also study an inhomogenous version of TASEP, in which different particles get sorted at different rates, which generalizes the homogenous version in several aspects. In the first chapter we compute some correlation functions for ζ

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  • 4.
    Aas, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Limit points of the iterative scaling procedure2014In: Annals of Operations Research, ISSN 0254-5330, E-ISSN 1572-9338, Vol. 215, no 1, p. 15-23Article in journal (Refereed)
    Abstract [en]

    The iterative scaling procedure (ISP) is an algorithm which computes a sequence of matrices, starting from some given matrix. The objective is to find a matrix 'proportional' to the given matrix, having given row and column sums. In many cases, for example if the initial matrix is strictly positive, the sequence is convergent. It is known that the sequence has at most two limit points. When these are distinct, convergence to these two points can be slow. We give an efficient algorithm which finds the limit points, invoking the ISP only on subproblems for which the procedure is convergent.

  • 5.
    Aas, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Stationary probability of the identity for the TASEP on a Ring2012Other (Other academic)
    Abstract [en]

    Consider the following Markov chain on permutations of length n. At each time step we choose a random position. If the letter at that position is smaller than the letter immediately to the left (cyclically) then these letters swap positions. Otherwise nothing happens, corresponding to a loop in the Markov chain. This is the circular TASEP. We compute the average proportion of time the chain spends at the identity permutation (and, in greater generality, at sorted words). This answers a conjecture by Thomas Lam.

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  • 6.
    Aas, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    TASEP in any Weyl groupManuscript (preprint) (Other academic)
    Abstract [en]

    We investigate a Markov chain dened by Thomas Lam [6], whichgeneralizes the multi-type TASEP on a ring to any Weyl group. For groups of typeC we dene an analogue of the multiline queues of Ferrari and Martin (which com-pute the stationary distribution for the classical TASEP). While our constructiondoes not suce for nding the stationary distribution, the construction gives thestationary distribution of a certain projection of Lam's chain. Also, our approach isincremental, in the sense that the construction appears to t into a pattern of 'con-jugation matrices', which remains to be fully worked out. We conjecture an explicitformula for the partition function of the model. Finally, we prove a theorem for theclassical TASEP which ts into the picture of viewing TASEP in a permutation-freeway.

  • 7. Aas, Erik
    et al.
    Ayyer, Arvind
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Potka, Samu
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Limiting directions for random walks in classical affine Weyl groupsManuscript (preprint) (Other academic)
    Abstract [en]

    Let be a finite Weyl group and the corresponding affine Weyl group. A random element of can be obtained as a reduced random walk on the alcoves of . By a theorem of Lam (Ann. Probab. 2015), such a walk almost surely approaches one of many directions. We compute these directions when is , and and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types and in the affirmative and for type in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process called the -TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.

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  • 8.
    Aas, Erik
    et al.
    Penn State Univ, Dept Math, University Pk, PA 16802 USA..
    Ayyer, Arvind
    Indian Inst Sci, Dept Math, Bangalore 560012, Karnataka, India..
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Potka, Samu
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Limiting Directions for Random Walks in Classical Affine Weyl Groups2021In: International mathematics research notices, ISSN 1073-7928, E-ISSN 1687-0247, Vol. 2023, no 4, p. 3092-3137Article in journal (Refereed)
    Abstract [en]

    Let W be a finite Weyl group and (W) over tilde the corresponding affine Weyl group. A random element of (W) over tilde can be obtained as a reduced random walk on the alcoves of (W) over tilde. By a theorem of Lam (Ann. Prob. 2015), such a walk almost surely approaches one of vertical bar W vertical bar many directions. We compute these directions when W is B-n, C-n, and D-n, and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types B and C in the affirmative and for type D in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process (TASEP) called the D*-TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.

  • 9.
    Aas, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Continuous multiline queues and TASEPManuscript (preprint) (Other academic)
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  • 10.
    Aas, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Continuous multi-line queues and TASEP2018In: ANNALES DE L INSTITUT HENRI POINCARE D, ISSN 2308-5827, Vol. 5, no 1, p. 127-152Article in journal (Refereed)
    Abstract [en]

    In this paper, we study a distribution Xi of labeled particles on a continuous ring. It arises in three different ways, all related to the multi-type TASEP on a ring. We prove formulas for the probability density function for some permutations and give conjectures for a larger class. We give a complete conjecture for the probability of two particles i, j being next to each other on the cycle, for which we prove some cases. We also find that two natural events associated to the process have exactly the same probability expressed as a Vandermonde determinant. It is unclear whether this is just a coincidence or a consequence of a deeper connection.

  • 11.
    Aas, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Sjöstrand, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A product formula for the TASEP on a ring2016In: Random structures & algorithms (Print), ISSN 1042-9832, E-ISSN 1098-2418, Vol. 48, no 2, p. 247-259Article in journal (Refereed)
    Abstract [en]

    For a random permutation sampled from the stationary distributionof the TASEP on a ring, we show that, conditioned on the event that the rstentries are strictly larger than the last entries, the order of the rst entries isindependent of the order of the last entries. The proof uses multi-line queues asdened by Ferrari and Martin, and the theorem has an enumerative combinatorialinterpretation in that setting.As an application we prove a conjecture of Lam and Williams concerningSchubert factors of the stationary probability of certain states.Finally, we present a conjecture for the case where the small and large entriesare not separated.

  • 12.
    Aas, Erik
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Sjöstrand, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    A product formula for the TASEP on a ring - Extended Abstract2014In: Discrete Mathematics and Theoretical Computer Science, Discrete Mathematics and Theoretical Computer Science , 2014, p. 633-641Conference paper (Refereed)
    Abstract [en]

    For a random permutation sampled from the stationary distribution of the TASEP on a ring, we show that, conditioned on the event that the first entries are strictly larger than the last entries, the order of the first entries is independent of the order of the last entries. The proof uses multi-line queues as defined by Ferrari and Martin, and the theorem has an enumerative combinatorial interpretation in that setting. Finally, we present a conjecture for the case where the small and large entries are not separated. 

  • 13.
    Abazari, Tina
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Baghchesara, Sherwin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Predicting Stock Price Direction for Asian Small Cap Stocks with Machine Learning Methods2021Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Portfolio managers have a great interest in detecting high-performing stocks early on. Detecting outperforming stocks has for long been of interest from a research as well as financial point of view. Quantitative methods to predict stock movements have been widely studied in diverse contexts, where some present promising results. The quantitative algorithms for such prediction models can be, to name a few, support vector machines, tree-based methods, and regression models, where each one can carry different predictive power. Most previous research focuses on indices such as S&P 500 or large-cap stocks, while small- and micro-cap stocks have been examined to a lesser extent. These types of stocks also commonly share the characteristic of high volatility, with prospects that can be difficult to assess. This study examines to which extent widely studied quantitative methods such as random forest, support vector machine, and logistic regression can produce accurate predictions of stock price directions on a quarterly and yearly basis. The problem is modeled as a binary classification task, where the aim is to predict whether a stock achieves a return above or below a benchmark index. The focus lies on Asian small- and micro-cap stocks. The study concludes that the random forest method for a binary yearly prediction produces the highest accuracy of 69.64%, where all three models produced higher accuracy than a binary quarterly prediction. Although the statistical power of the models can be ruled adequate, more extensive studies are desirable to examine whether other models or variables can increase the prediction accuracy for small- and micro-cap stocks.

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  • 14.
    Abdallah, Nancy
    et al.
    Univ Borås, Borås, Sweden..
    Altafi, Nasrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH Royal Inst Technol, Dept Math, S-10044 Stockholm, Sweden.;Queens Univ, Dept Math & Stat, Kingston, ON K7L 3N6, Canada..
    Iarrobino, Anthony
    Northeastern Univ, Dept Math, Boston, MA 02115 USA..
    Seceleanu, Alexandra
    Univ Nebraska Lincoln, Dept Math, Lincoln, NE 68588 USA..
    Yameogo, Joachim
    Univ Cote dAzur, CNRS, LJAD, Nice, France..
    Lefschetz properties of some codimension three Artinian Gorenstein algebras2023In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 625, p. 28-45Article in journal (Refereed)
    Abstract [en]

    Codimension two Artinian algebras have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three Artinian Gorenstein algebras. Despite much work, the strong Lefschetz property for codimension three Artinian Gorenstein algebra has remained largely mysterious; our results build on and strengthen some of the previous results. We here show that every standard-graded codimension three Artinian Gorenstein algebra A having maximum value of the Hilbert function at most six has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of A, we show that A is almost strong Lefschetz, they are strong Lefschetz except in the extremal pair of degrees.

  • 15.
    Abdullah Al Ahad, Muhammed
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Non-linearstates in parallel Blasius boundary layer2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    There is large theoretical, experimental and numerical interest in studying boundary layers, which develop around any body moving through a fluid. The simplest of these boundary layers lead to the theoretical abstraction of a so-called Blasius boundary layer, which can be derived under the assumption of a flat plate and zero external pressure gradient. The Blasius solution is characterised by a slow growth of the boundary layer in the streamwise direction. For practical purposes, in particular related to studying transition scenarios, non-linear finite-amplitude states (exact coherent states, edge states), but also for turbulence, a major simplification of the problem could be attained by removing this slow streamwise growth, and instead consider a parallel boundary layer. Parallel boundary layers are found in reality, e.g. when applying suction (asymptotic suction boundary layer) or rotation (Ekman boundary layer), but not in the Blasius case. As this is only a model which is not an exact solution to the Navier-Stokes (or boundary-layer) equations, some modifications have to be introduced into the governing equations in order for such an approach to be feasible. Spalart and Yang introduced a modification term to the governing Navier-Stokes equations in 1987. In this thesis work, we adapted the amplitude of the modification term introduced by Spalart and Yang to identify the nonlinear states in the parallel Blasius boundary layer. A final application of this modification was in determining the so-called edge states for boundary layers, previously found in the asymptotic suction boundary layer

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  • 16.
    Abdullah Mohamad, Ormia
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Westin, Anna
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Pricing and Modeling Heavy Tailed Reinsurance Treaties - A Pricing Application to Risk XL Contracts2023Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    To estimate the risk of a loss occurring for insurance takers is a difficult task in the insurance industry. It is an even more difficult task to price the risk for reinsurance companies which insures the primary insurers. Insurance that is bought by an insurance company, the cedent, from another insurance company, the reinsurer, is called treaty reinsurance. This type of reinsurance is the main focus in this thesis. A very common risk to insure, is the risk of fire in municipal and commercial properties which is the risk that is priced in this thesis. This thesis evaluates Länsförsäkringar AB's current pricing model which calculates the risk premium for Risk XL contracts. The goal of this thesis is to find areas of improvement for tail risk pricing. The risk premium can be calculated commonly by using one of three different types of pricing models, experience rating, exposure rating and frequency-severity rating. This thesis focuses on frequency-severity pricing, which is a model that assumes independence between the frequency and the severity of losses, and therefore splits the two into separate models. This is a very common model used when pricing Risk XL contracts. The risk premium is calculated with the help of loss data from two insurance companies, from a Norwegian and a Finnish insurance company. The main focus of this thesis is to price the risk with the help of extreme value theory, mainly with the method of moments method to model the frequency of losses, and peaks over threshold model to model the severity of the losses. In order to model the estimated frequency of losses by using the method of moments method, two distributions are compared, the Poisson and the negative binomial distribution. There are different distributions that can be used to model the severity of losses. In order to evaluate which distribution is optimal to use, two different Goodness of Fit tests are applied, the Kolmogorov-Smirnov and the Anderson-Darling test. The Peaks over threshold model is a model that can be used with the Pareto distribution. With the help of the Hill estimator we are able to calculate a threshold $u$, which regulates the tail of the Pareto curve. To estimate the rest of the ingoing parameters in the generalized Pareto distribution, the maximum likelihood and the least squares method are used. Lastly, the bootstrap method is used to estimate the uncertainty in the price which was calculated with the help of the estimated parameters. From this, empirical percentiles are calculated and set as guidelines to where the risk premium should lie between, in order for both the data sets to be considered fairly priced.

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  • 17. Abdulle, A.
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Multiscale Methods for Wave Problems in Heterogeneous Media2017In: Handbook of Numerical Analysis, Elsevier B.V. , 2017, p. 545-576Conference paper (Refereed)
    Abstract [en]

    In this chapter we give a survey on various multiscale methods for the numerical solution of second-order hyperbolic equations in highly heterogeneous media. We concentrate on the wave equation and distinguish between two classes of applications. First we discuss numerical methods for the wave equation in heterogeneous media without scale separation. Such a setting is for instance encountered in the geosciences, where natural structures often exhibit a continuum of different scales, that all need to be resolved numerically to get meaningful approximations. Approaches tailored for these settings typically involve the construction of generalized finite element spaces, where the basis functions incorporate information about the data variations. In the second part of the chapter, we discuss numerical methods for the case of structured media with scale separation. This setting is for instance encountered in engineering sciences, where materials are often artificially designed. If this is the case, the structure and the scale separation can be explicitly exploited to compute appropriate homogenized/upscaled wave models that only exhibit a single coarse scale and that can be hence solved at significantly reduced computational costs. 

  • 18. Abdulle, Assyr
    et al.
    Henning, Patrick
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Localized orthogonal decomposition method for the wave equation with a continuum of scales2017In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 86, no 304, p. 549-587Article in journal (Refereed)
    Abstract [en]

    This paper is devoted to numerical approximations for the wave equation with a multiscale character. Our approach is formulated in the framework of the Localized Orthogonal Decomposition (LOD) interpreted as a numerical homogenization with an L2-projection. We derive explicit convergence rates of the method in the L∞(L2)-, W1,∞(L2)-and L∞(H1)-norms without any assumptions on higher order space regularity or scale-separation. The order of the convergence rates depends on further graded assumptions on the initial data. We also prove the convergence of the method in the framework of G-convergence without any structural assumptions on the initial data, i.e. without assuming that it is well-prepared. This rigorously justifies the method. Finally, the performance of the method is demonstrated in numerical experiments.

  • 19.
    Abedin, Arian
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Ligai, Wolmir
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Automatingand optimizing pile group design using a Genetic Algorithm2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In bridge design, a set of piles is referred to as a pile group. The design process of pile groups employed by many firms is currently manual, time consuming, and produces pile groups that are not robust against placement errors.

    This thesis applies the metaheuristic method Genetic Algorithm to automate and improve the design of pile groups for bridge column foundations. A software is developed and improved by implementing modifications to the Genetic Algorithm. The algorithm is evaluated by the pile groups it produces, using the Monte Carlo method to simulate errors for the purpose of testing the robustness. The results are compared with designs provided by the consulting firm Tyrens AB.

    The software is terminated manually, and generally takes less than half an hour to produce acceptable pile groups. The developed Genetic Algorithm Software produces pile groups that are more robust than the manually designed pile groups to which they are compared, using the Monte Carlo method. However, due to the visually disorganized designs, the pile groups produced by the algorithm may be di cult to get approved by Trafikverket. The software might require further modifications addressing this problem before it can be of practical use.

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  • 20.
    Aboud, Stephanie
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Predicting Customer Conversion using Supervised Machine Learning2021Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The growth of e-commerce has been evident over the past years and for companies like Klarna that provides payment solutions, focusing on the purchase experience is more important than ever. With that goal in mind, more companies are using machine learning methods and tools to make predictions and forecast future outcomes, giving them a competitive advantage on the market. This thesis aims to apply supervised machine learning techniques to predict customer conversion, i.e. predict if a customer with a started shopping session will complete the purchase. The purpose of the project is to also determine which supervised learning algorithm performs the best when predicting customer conversion, with regards to a set of model evaluation metrics. The classical classification method Logistic Regression was tested, as well as the machine learning methods Support vector Machine, Random forest and XGBoost. The metrics used to evaluate the model performances were Precision, Recall, F1- and AUC-scores. Furthermore, the SHapley Additive exPlanations approach was implemented for feature importance and for interpreting tree-based models. The results showed that it is in fact possible to predict customer conversion using machine learning. All models yielded good performance and the difference in performance was relatively small. XGBoost performed slightly better than the rest of the models.

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  • 21.
    Abrahamsson, Peter
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Ahlqvist, Niklas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Evaluation of Machine Learning Methods for Time Series Forecasting on E-commerce Data2022Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Within demand forecasting, and specifically within the field of e-commerce, the provided data often contains erratic behaviours which are difficult to explain. This induces contradictions to the common assumptions within classical approaches for time series analysis. Yet, classical and naive approaches are still commonly used. Machine learning could be used to alleviate such problems. This thesis evaluates four models together with Swedish fin-tech company QLIRO AB. More specifically, a MLR (Multiple Linear Regression) model, a classic Box-Jenkins model (SARIMAX), an XGBoost model, and a LSTM-network (Long Short-Term Memory). The provided data consists of aggregated total daily reservations by e-merchants within the Nordic market from 2014. Some data pre processing was required and a smoothed version of the data set was created for comparison. Each model was constructed according to their specific requirements but with similar feature engineering. Evaluation was then made on a monthly level with a forecast horizon of 30 days during 2021. The results shows that both the MLR and the XGBoost provides the most consistent results together with perks for being easy to use. After these two, the LSTM-network showed the best results for November and December on the original data set but worst overall. Yet it had good performance on the smoothed data set and was then comparable to the first two. The SARIMAX was the worst performing of all the models considered in this thesis and was not as easy to implement.

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  • 22.
    Abrahamsson, Ville
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Ekblom, Julia
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Sustainable Investments - The impact of the EU green taxonomy2021Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The increasing environmental issues and the measures taken to tackle them, is a topic of high significance in today's society. In light of this, the EU is underway with developing a taxonomy classifying sustainable economic activities in hopes to raise awareness, increase transparency regarding environmental impact, and motivate investors to invest sustainable. This paper aims to examine if the taxonomy is relevant to its cause, as well as if sustainability factors can be identified with linear regression connected to growth in a company's value, which may motivate sustainable investments. Several interviews were conducted, along with the creation of a mathematical model. The conclusions drawn was that it is not viable to determine a company's growth in value using solely sustainability factors. However, the results were promising regarding the implementation of sustainability factors in more comprehensive models. Furthermore, the impact of the taxonomy was hard to predict at this time, however, the consensus of the majority of the interviews conducted with experts on the subject, is that it has potential to impact sustainable investments in the future. Future research on the taxonomy may yield results of higher interest since more comprehensive data will be available, and the impact of the taxonomy will be more concrete.

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  • 23. Abuzyarova, Natalia
    et al.
    Hedenmalm, Håkan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Branch point area methods in conformal mapping2006In: Journal d'Analyse Mathematique, ISSN 0021-7670, E-ISSN 1565-8538, Vol. 99, p. 177-198Article in journal (Refereed)
    Abstract [en]

    The classical estimate of Bieberbach that vertical bar a(2)vertical bar <= 2 for a given univalent function phi(z) = z + a(2)z(2) +... in the class S leads to the best possible pointwise estimates of the ratio phi''(z)/phi'(z) for phi is an element of S, first obtained by K oe be and Bieberbach. For the corresponding class E of univalent functions in the exterior disk, Goluzin found in 1943 by variational methods the corresponding best possible pointwise estimates of psi(z)/psi'(z) for psi is an element of Sigma. It was perhaps surprising that this time, the expressions involve elliptic integrals. Here, we obtain an area-type theorem which has Goluzin's pointwise estimate as a corollary. This shows that Goluzin's estimate, like the K oe be-Bieberbach estimate, is firmly rooted in area-based methods. The appearance of elliptic integrals finds a natural explanation: they arise because a certain associated covering surface of the Riemann sphere is a torus.

  • 24. Achdou, Yves
    et al.
    Dao, Manh Khang
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Ley, Olivier
    Tchou, Nicoletta
    Finite horizon mean field games on networks2020In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 59, no 5, article id 157Article in journal (Refereed)
    Abstract [en]

    We consider finite horizon stochastic mean field games in which the state space is a network. They are described by a system coupling a backward in time Hamilton-Jacobi-Bellman equation and a forward in time Fokker-Planck equation. The value functionuis continuous and satisfies general Kirchhoff conditions at the vertices. The densitymof the distribution of states satisfies dual transmission conditions: in particular,mis generally discontinuous across the vertices, and the values ofmon each side of the vertices satisfy some compatibility conditions. The stress is put on the case when the Hamiltonian is Lipschitz continuous. Existence, uniqueness and regularity results are proven.

  • 25. Acker, Andrew
    et al.
    Poghosyan, Michael
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Convex configurations for solutions to semilinear elliptic problems in convex rings2006In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 31, no 9, p. 1273-1287Article in journal (Refereed)
    Abstract [en]

    For a given convex ring Omega = Omega(2)\(Omega) over bar (1) and an L-1 function f : Omega x R -> R+ we show, under suitable assumptions on f, that there exists a solution (in the weak sense) to Delta(p)u = f(x, u) in Omega u = 0 on partial derivative Omega(2) u = M on partial derivative Omega(1) with {x is an element of Omega : u(x) > s} boolean OR Omega(1) convex, for all s is an element of (0, M).

  • 26.
    Ackland, Marcus
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Wargentin, Robin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    How prices of condominiums vary with respect to distance from the city center in 20 major cities in Sweden2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    This report will examine how condominium prices vary with respect to the distance from the city center in 20 major cities in Sweden. With regression analysis three models are construct-ed for each city to predict the price of a condominium in the city with a known set of variables such as area, monthly fee and distance from city center. The three models each depend on the distance parameter in different ways; linearly, exponentially, and exponentially with a higher degree polynomial as an exponent. The models are then examined statistically between cities to determine if there is any correlation between price function with regards to distance and population size. Results show that prices do decline substantially when distance to city center increases in all observed cities. There is a significant correlation between price function of distance and population size, but the relation is not enough to, by itself, explain the differ-ences between cities.

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    Marcus Ackland & Robin Wargentin kandidatexamensarbete inom teknisk fysik vid inst för matematik, matematisk statistik
  • 27. Adamaszek, Michal
    et al.
    Barmak, Jonathan Ariel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On a lower bound for the connectivity of the independence complex of a graph2011In: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 311, no 21, p. 2566-2569Article in journal (Refereed)
    Abstract [en]

    Aharoni, Berger and Ziv proposed a function which is a lower bound for the connectivity of the independence complex of a graph. They conjectured that this bound is optimal for every graph. We give two different arguments which show that the conjecture is false.

  • 28.
    Adelstrand, Carl
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Gavefalk, Sofia
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    In times of regional geopolitical turmoil – Why do some equity funds performbetter than others?2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    In times of regional geopolitical turmoil – why do some investment portfolios, equity funds, perform better than others? Is it simply luck, the effects of systematic risk or do factors such as investment styles and managerial skills play a significant part in the performance of a fund?

    As financial markets often reflect the macro environment, much of the previous year’s fluctuations of Eastern European stocks can be seen to derive from a number of geopolitical events; from the 2013 summer clashes between the Turkish police and opposing parties to the current issue concerning Russia and Ukraine. Needless to say, these events have affected return on equity in their regions and created a distressed environment for investors and equity fund managers investing in Eastern Europe.

    This thesis aims to explore how the aforementioned macroeconomic events impact the market and thus the portfolios of asset managers. The thesis also intends to provide aspects of eventual investment strategies that are more preferable than others under such circumstances, in order to mitigate the subsequent risks.

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  • 29.
    Adiprasito, Karim
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    FAQ on the g-theorem and the hard Lefschetz theorem for face rings2019In: Rendiconti di Matematica e delle sue Applicazioni. Serie VII, ISSN 1120-7183, Vol. 40, no 2, p. 97-111Article in journal (Refereed)
    Abstract [en]

    We discuss the hard Lefschetz theorem for simplicial spheres, as well as the theory at its core: Perturbations of maps, biased Poincaré pairings and a cobordism argument that relates the Lefschetz property of a manifold to the Lefschetz property on its boundary, and sketch an alternative argument based on edge-contractions.

  • 30. Adiprasito, Karim
    et al.
    Björner, Anders
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Goodarzi, Afshin
    Freie Universität, Germany.
    Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals2017In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, no 12, p. 3851-3865Article in journal (Refereed)
    Abstract [en]

    A numerical characterization is given of the h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree <= d and shifted pure. (d - 1)-dimensional simplicial complexes.

  • 31. Adiprasito, Karim
    et al.
    Goodarzi, Afshin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Varbaro, Matteo
    Connectivity of pseudomanifold graphs from an algebraic point of view2015In: Comptes Rendus Mathematiques de l'Academie des Sciences = Mathematical reports of the academy of science, ISSN 0706-1994, Vol. 353, no 12, p. 1061-1065Article in journal (Refereed)
    Abstract [en]

    The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis, and Bjorner. In this note, we provide a unifying approach that allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley-Reisner rings.

  • 32.
    Adler, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Data-driven Methods in Inverse Problems2019Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    In this thesis on data-driven methods in inverse problems we introduce several new methods to solve inverse problems using recent advancements in machine learning and specifically deep learning. The main goal has been to develop practically applicable methods, scalable to medical applications and with the ability to handle all the complexities associated with them.

    In total, the thesis contains six papers. Some of them are focused on more theoretical questions such as characterizing the optimal solutions of reconstruction schemes or extending current methods to new domains, while others have focused on practical applicability. A significant portion of the papers also aim to bringing knowledge from the machine learning community into the imaging community, with considerable effort spent on translating many of the concepts. The papers have been published in a range of venues: machine learning, medical imaging and inverse problems.

    The first two papers contribute to a class of methods now called learned iterative reconstruction where we introduce two ways of combining classical model driven reconstruction methods with deep neural networks. The next two papers look forward, aiming to address the question of "what do we want?" by proposing two very different but novel loss functions for training neural networks in inverse problems. The final papers dwelve into the statistical side, one gives a generalization of a class of deep generative models to Banach spaces while the next introduces two ways in which such methods can be used to perform Bayesian inversion at scale.

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  • 33.
    Adler, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    GPU Monte Carlo scatter calculations for Cone Beam Computed Tomography2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    A GPU Monte Carlo code for x-ray photon transport has been implemented and extensively tested. The code is intended for scatter compensation of cone beam computed tomography images.

    The code was tested to agree with other well known codes within 5% for a set of simple scenarios. The scatter compensation was also tested using an artificial head phantom. The errors in the reconstructed Hounsfield values were reduced by approximately 70%.

    Several variance reduction methods have been tested, although most were found infeasible on GPUs. The code is nonetheless fast, and can simulate approximately 3 ·109 photons per minute on a NVIDIA Quadro 4000 graphics card. With the use of appropriate filtering methods, the code can be used to calculate patient specific scatter distributions for a full CBCT scan in approximately one minute, allowing scatter reduction in clinical applications.

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  • 34.
    Adler, Jonas
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Learned Iterative Reconstruction2023In: Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging: Mathematical Imaging and Vision, Springer Nature , 2023, p. 751-771Chapter in book (Other academic)
    Abstract [en]

    Learned iterative reconstruction methods have recently emerged as a powerful tool to solve inverse problems. These deep learning techniques for image reconstruction achieve remarkable speed and accuracy by combining hard knowledge about the physics of the image formation process, represented by the forward operator, with soft knowledge about how the reconstructions should look like, represented by deep neural networks. A diverse set of such methods have been proposed, and this chapter seeks to give an overview of their similarities and differences, as well as discussing some of the commonly used methods to improve their performance.

  • 35.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lunz, Sebastian
    Univ Cambridge, Dept Appl Math & Theoret Phys, Cambridge, England..
    Banach Wasserstein GAN2018In: Advances in Neural Information Processing Systems 31 (NIPS 2018) / [ed] Bengio, S Wallach, H Larochelle, H Grauman, K CesaBianchi, N Garnett, R, Neural Information Processing Systems (NIPS) , 2018Conference paper (Refereed)
    Abstract [en]

    Wasserstein Generative Adversarial Networks (WGANs) can be used to generate realistic samples from complicated image distributions. The Wasserstein metric used in WGANs is based on a notion of distance between individual images, which induces a notion of distance between probability distributions of images. So far the community has considered l(2) as the underlying distance. We generalize the theory of WGAN with gradient penalty to Banach spaces, allowing practitioners to select the features to emphasize in the generator. We further discuss the effect of some particular choices of underlying norms, focusing on Sobolev norms. Finally, we demonstrate a boost in performance for an appropriate choice of norm on CIFAR-10 and CelebA.

  • 36.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). DeepMind, 6 Pancras Square, London, N1C 4AG, United Kingdom.
    Lunz, Sebastian
    Univ Cambridge, Ctr Math Sci, Cambridge CB3 0WA, England..
    Verdier, Olivier
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences, Bergen, Norway.
    Schonlieb, Carola-Bibiane
    Univ Cambridge, Ctr Math Sci, Cambridge CB3 0WA, England..
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Division of Scientific Computing, Department of Information Technology, Uppsala University.
    Task adapted reconstruction for inverse problems2022In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 38, no 7, article id 075006Article in journal (Refereed)
    Abstract [en]

    The paper considers the problem of performing a post-processing task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and post-processing as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the post-processing task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any post-processing that can be encoded as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation.

  • 37.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Elekta.
    Lunz, Sebastian
    Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom.
    Verdier, Olivier
    Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden ; Department of Computing, Mathematics and Physics, Western Norway University of Applied Sciences, Bergen, Norway.
    Schönlieb, Carola-Bibiane
    Centre for Mathematical Sciences, University of Cambridge, Cambridge CB3 0WA, United Kingdom.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Task adapted reconstruction for inverse problemsManuscript (preprint) (Other academic)
    Abstract [en]

    The paper considers the problem of performing a task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and task as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any task that is encodable as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation.

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  • 38.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Elekta, Box 7593, 103 93 Stockholm, Sweden.
    Ringh, Axel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Karlsson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Learning to solve inverse problems using Wasserstein lossManuscript (preprint) (Other academic)
    Abstract [en]

    We propose using the Wasserstein loss for training in inverse problems. In particular, we consider a learned primal-dual reconstruction scheme for ill-posed inverse problems using the Wasserstein distance as loss function in the learning. This is motivated by miss-alignments in training data, which when using standard mean squared error loss could severely degrade reconstruction quality. We prove that training with the Wasserstein loss gives a reconstruction operator that correctly compensates for miss-alignments in certain cases, whereas training with the mean squared error gives a smeared reconstruction. Moreover, we demonstrate these effects by training a reconstruction algorithm using both mean squared error and optimal transport loss for a problem in computerized tomography.

  • 39.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Elekta.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Deep Bayesian InversionManuscript (preprint) (Other academic)
    Abstract [en]

    Characterizing statistical properties of solutions of inverse problems is essential for decision making. Bayesian inversion offers a tractable framework for this purpose, but current approaches are computationally unfeasible for most realistic imaging applications in the clinic. We introduce two novel deep learning based methods for solving large-scale inverse problems using Bayesian inversion: a sampling based method using a WGAN with a novel mini-discriminator and a direct approach that trains a neural network using a novel loss function. The performance of both methods is demonstrated on image reconstruction in ultra low dose 3D helical CT. We compute the posterior mean and standard deviation of the 3D images followed by a hypothesis test to assess whether a "dark spot" in the liver of a cancer stricken patient is present. Both methods are computationally efficient and our evaluation shows very promising performance that clearly supports the claim that Bayesian inversion is usable for 3D imaging in time critical applications.

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  • 40.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Elekta Instrument AB, Stockholm, Sweden.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Learned Primal-Dual Reconstruction2018In: IEEE Transactions on Medical Imaging, ISSN 0278-0062, E-ISSN 1558-254X, Vol. 37, no 6, p. 1322-1332Article in journal (Refereed)
    Abstract [en]

    We propose the Learned Primal-Dual algorithm for tomographic reconstruction. The algorithm accounts for a (possibly non-linear) forward operator in a deep neural network by unrolling a proximal primal-dual optimization method, but where the proximal operators have been replaced with convolutional neural networks. The algorithm is trained end-to-end, working directly from raw measured data and it does not depend on any initial reconstruction such as filtered back-projection (FBP). We compare performance of the proposed method on low dose computed tomography reconstruction against FBP, total variation (TV), and deep learning based post-processing of FBP. For the Shepp-Logan phantom we obtain >6 dB peak signal to noise ratio improvement against all compared methods. For human phantoms the corresponding improvement is 6.6 dB over TV and 2.2 dB over learned post-processing along with a substantial improvement in the structural similarity index. Finally, our algorithm involves only ten forward-back-projection computations, making the method feasible for time critical clinical applications.

  • 41.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Solving ill-posed inverse problems using iterative deep neural networks2017In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 33, no 12, article id 124007Article in journal (Refereed)
    Abstract [en]

    We propose a partially learned approach for the solution of ill-posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularisation theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularising functional. The method results in a gradient-like iterative scheme, where the 'gradient' component is learned using a convolutional network that includes the gradients of the data discrepancy and regulariser as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against filtered backprojection and total variation reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the total variation reconstruction while being significantly faster, giving reconstructions of 512 x 512 pixel images in about 0.4 s using a single graphics processing unit (GPU).

  • 42. Adler, M.
    et al.
    Chhita, S.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    van Moerbeke, P.
    Tacnode GUE-minor processes and double Aztec diamonds2015In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 162, no 1-2, p. 275-325Article in journal (Refereed)
    Abstract [en]

    We study determinantal point processes arising in random domino tilings of a double Aztec diamond, a region consisting of two overlapping Aztec diamonds. At a turning point in a single Aztec diamond where the disordered region touches the boundary, the natural limiting process is the GUE-minor process. Increasing the size of a double Aztec diamond while keeping the overlap between the two Aztec diamonds finite, we obtain a new determinantal point process which we call the tacnode GUE-minor process. This process can be thought of as two colliding GUE-minor processes. As part of the derivation of the particle kernel whose scaling limit naturally gives the tacnode GUE-minor process, we find the inverse Kasteleyn matrix for the dimer model version of the Double Aztec diamond.

  • 43.
    Adler, Mark
    et al.
    Brandeis Univ, Dept Math, Waltham, MA 02254 USA..
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    van Moerbeke, Pierre
    UCLouvain, Dept Math, Louvain, Belgium.;Brandeis Univ, Waltham, MA 02254 USA..
    A SINGULAR TOEPLITZ DETERMINANT AND THE DISCRETE TACNODE KERNEL FOR SKEW-AZTEC RECTANGLES2022In: The Annals of Applied Probability, ISSN 1050-5164, E-ISSN 2168-8737, Vol. 32, no 2, p. 1234-1294Article in journal (Refereed)
    Abstract [en]

    Random tilings of geometrical shapes with dominos or lozenges have been a rich source of universal statistical distributions. This paper deals with domino tilings of checker board rectangular shapes such that the top two and bottom two adjacent squares have the same orientation and the two most left and two most right ones as well. It forces these so-called "skew-Aztec rectangles" to have cuts on either side. For large sizes of the domain and upon an appropriate scaling of the location of the cuts, one finds split tacnodes between liquid regions with two distinct adjacent frozen phases descending into the tacnode. Zooming about such split tacnodes, filaments appear between the liquid patches evolving in a bricklike sea of dimers of another type. This work shows that the random fluctuations in a neighborhood of the split tacnode are governed asymptotically by the discrete tacnode kernel, providing strong evidence that this kernel is a universal discrete-continuous limiting kernel occurring naturally whenever we have doubly interlacing patterns. The analysis involves the inversion of a singular Toeplitz matrix which leads to considerable difficulties.

  • 44. Adler, Mark
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    van Moerbeke, Pierre
    Double Aztec diamonds and the tacnode process2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 252, p. 518-571Article in journal (Refereed)
    Abstract [en]

    Discrete and continuous non-intersecting random processes have given rise to critical "infinite-dimensional diffusions", like the Airy process, the Pearcey process and variations thereof. It has been known that domino tilings of very large Aztec diamonds lead macroscopically to a disordered region within an inscribed ellipse (arctic circle in the homogeneous case), and a regular brick-like region outside the ellipse. The fluctuations near the ellipse, appropriately magnified and away from the boundary of the Aztec diamond, form an Airy process, run with time tangential to the boundary. This paper investigates the domino tiling of two overlapping Aztec diamonds; this situation also leads to non-intersecting random walks and an induced point process; this process is shown to be determinantal. In the large size limit, when the overlap is such that the two arctic ellipses for the single Aztec diamonds merely touch, a new critical process will appear near the point of osculation (tacnode), which is run with a time in the direction of the common tangent to the ellipses: this is the tacnode process. It is also-shown here that this tacnode process is universal: it coincides with the one found in the context of two groups of non-intersecting random walks or also Brownian motions, meeting momentarily.

  • 45. Adler, Mark
    et al.
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    van Moerbeke, Pierre
    Lozenge Tilings of Hexagons with Cuts and Asymptotic Fluctuations: a New Universality Class2018In: Mathematical physics, analysis and geometry, ISSN 1385-0172, E-ISSN 1572-9656, Vol. 21, no 1, article id 9Article in journal (Refereed)
    Abstract [en]

    This paper investigates lozenge tilings of non-convex hexagonal regions and more specifically the asymptotic fluctuations of the tilings within and near the strip formed by opposite cuts in the regions, when the size of the regions tend to infinity, together with the cuts. It leads to a new kernel, which is expected to have universality properties.

  • 46.
    Adler, Mark
    et al.
    Brandeis Univ, Dept Math, Waltham, MA 02453 USA..
    Johansson, Kurt
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    van Moerbeke, Pierre
    Univ Louvain, Dept Math, B-1348 Louvain, Belgium.;Brandeis Univ, Waltham, MA 02453 USA..
    Tilings of Non-convex Polygons, Skew-Young Tableaux and Determinantal Processes2018In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 364, no 1, p. 287-342Article in journal (Refereed)
    Abstract [en]

    This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The precise geometrical figure here consists of a hexagon with cuts along opposite edges. For this model, we take limits when the size of the hexagon and the cuts tend to infinity, while keeping certain geometric data fixed in order to guarantee sufficient interaction between the cuts in the limit. We show in this paper that the kernel for the finite tiling model can be expressed as a multiple integral, where the number of integrations is related to the fixed geometric data above. The limiting kernel is believed to be a universal master kernel.

  • 47.
    af Klinteberg, Ludvig
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Computational methods for microfluidics2013Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

    The first paper of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an explicit interface tracking method, wherein the interface between two fluids is decomposed into segments, which are represented locally on an Eulerian grid. The framework of this method provides a natural setting for solving the advection-diffusion equation governing the surfactant concentration on the interface. Open interfaces and moving contact lines are also incorporated into the method in a natural way, though we show that care must be taken when regularizing interface forces to the grid near the boundary of the computational domain.

    In the second paper we present a boundary integral formulation for sedimenting particles in periodic Stokes flow, using the completed double layer boundary integral formulation. The long-range nature of the particle-particle interactions lead to the formulation containing sums which are not absolutely convergent if computed directly. This is solved by applying the method of Ewald summation, which in turn is computed in a fast manner by using the FFT-based spectral Ewald method. The complexity of the resulting method is O(N log N), as the system size is scaled up with the number of discretization points N. We apply the method to systems of sedimenting spheroids, which are discretized using the Nyström method and a basic quadrature rule.

    The Ewald summation method used in the boundary integral method of the second paper requires a decomposition of the potential being summed. In the introductory chapters of the thesis we present an overview of the available methods for creating Ewald decompositions, and show how the methods and decompositions can be related to each other.

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  • 48.
    af Klinteberg, Ludvig
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Ewald summation for the rotlet singularity of Stokes flow2016Report (Other academic)
    Abstract [en]

    Ewald summation is an efficient method for computing the periodic sums that appear when considering the Green's functions of Stokes flow together with periodic boundary conditions. We show how Ewald summation, and accompanying truncation error estimates, can be easily derived for the rotlet, by considering it as a superposition of electrostatic force calculations.

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  • 49.
    af Klinteberg, Ludvig
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
    Fast and accurate integral equation methods with applications in microfluidics2016Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this.

    The first, and smaller, part of the thesis presents a numerical method for simulating multiphase flows involving insoluble surfactants and moving contact lines. The method is based on an interface decomposition resulting in local, Eulerian grid representations. This provides a natural setting for solving the PDE governing the surfactant concentration on the interface.

    The second, and larger, part of the thesis is concerned with a framework for simulating large systems of rigid particles in three-dimensional, periodic viscous flow using a boundary integral formulation. This framework can solve the underlying flow equations to high accuracy, due to the accurate nature of surface quadrature. It is also fast, due to the natural coupling between boundary integral methods and fast summation methods.

    The development of the boundary integral framework spans several different fields of numerical analysis. For fast computations of large systems, a fast Ewald summation method known as Spectral Ewald is adapted to work with the Stokes double layer potential. For accurate numerical integration, a method known as Quadrature by Expansion is developed for this same potential, and also accelerated through a scheme based on geometrical symmetries. To better understand the errors accompanying this quadrature method, an error analysis based on contour integration and calculus of residues is carried out, resulting in highly accurate error estimates.

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  • 50.
    af Klinteberg, Ludvig
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Lindbo, Dag
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    Tornberg, Anna-Karin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA. KTH, School of Engineering Sciences (SCI), Centres, Linné Flow Center, FLOW.
    An explicit Eulerian method for multiphase flow with contact line dynamics and insoluble surfactant2014In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 101, p. 50-63Article in journal (Refereed)
    Abstract [en]

    The flow behavior of many multiphase flow applications is greatly influenced by wetting properties and the presence of surfactants. We present a numerical method for two-phase flow with insoluble surfactants and contact line dynamics in two dimensions. The method is based on decomposing the interface between two fluids into segments, which are explicitly represented on a local Eulerian grid. It provides a natural framework for treating the surfactant concentration equation, which is solved locally on each segment. An accurate numerical method for the coupled interface/surfactant system is given. The system is coupled to the Navier-Stokes equations through the immersed boundary method, and we discuss the issue of force regularization in wetting problems, when the interface touches the boundary of the domain. We use the method to illustrate how the presence of surfactants influences the behavior of free and wetting drops.

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