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• 51.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Fast multiplication of matrices over a finitely generated semiring2008Ingår i: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, nr 6, s. 230-234Artikel i tidskrift (Refereegranskat)

In this paper we show that n×n matrices with entries from a semiring R which is generated additively by q generators can be multiplied in time O(q2nω), where nω is the complexity for matrix multiplication over a ring (Strassen: ω<2.807, Coppersmith and Winograd: ω<2.376).

We first present a combinatorial matrix multiplication algorithm for the case of semirings with q elements, with complexity O(n3/log2qn), matching the best known methods in this class.

Next we show how the ideas used can be combined with those of the fastest known boolean matrix multiplication algorithms to give an O(q2nω) algorithm for matrices of, not necessarily finite, semirings with q additive generators.

For finite semirings our combinatorial algorithm is simple enough to be a practical algorithm and is expected to be faster than the O(q2nω) algorithm for matrices of practically relevant sizes.

• 52.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
On the complexity of matrix reduction over finite fields2007Ingår i: Advances in Applied Mathematics, ISSN 0196-8858, Vol. 39, nr 4, s. 428-452Artikel i tidskrift (Refereegranskat)

We study matrix elimination over finite fields, and present an algorithm which is asymptotically faster than the traditional Gauss--Jordan elimination. We also bound the average and worst-case complexity for the problem, proving that our algorithm is close to being optimal, and show related concentration results for random matrices.

Next we present the results of a large computational study of the complexities for small matrices and fields. Here we determine the exact distribution of the complexity for matrices from $\mathrm{GL}_{n}(\mathbb{F}_{q})$, with $n$ an $q$ small

Finally we consider an extension of the problems studied for finite fields to finite semifields. We give a conjecture on the behaviour of a natural analogue of $\mathrm{GL}_{n}$ for semifields and prove this for a certain class of semifields.

• 53.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoidability by Latin squares of arrays of even orderManuskript (preprint) (Övrigt vetenskapligt)

We prove that for any k and any 2k × 2k array A such that no cell in A contains more than   k/2550 symbols, and no symbol occurs more than k/2550 times in any row or column, there is a Latin square such that no 2550cell in the Latin square contains a symbol that occurs in the corresponding cell in A. This proves a conjecture of Häggkvist [8] in the special case of arrays with even side.

• 54.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoidability of random arraysManuskript (preprint) (Övrigt vetenskapligt)

An n×n array that in each cell contains a subset of the symbols 1, . . . , n is avoidable if there exists a Latin square of order n such that no cell in the Latin square contains a symbol which belongs to the set of symbols in the corresponding cell of the array. Some results on deterministic conditions for avoidability of arrays have been found, but here we study the problem of having an array with randomly assigned subsets of C in its cells. This is equivalent to the problem of list-edge-coloring $K_{n,n}$ with randomly assigned lists from the set {1, . . . , n}. We show that an array where each symbol appears in each cell with probability p will be avoidable with very high probability even if p is such that the expected number of symbols forbidden in each cell is slightly higher than what deterministic theorems can prove is avoidable.

• 55.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoiding (m, m, m)-arrays of order n = 2kManuskript (preprint) (Övrigt vetenskapligt)

An (m, m, m)-array of order n is an n × n array such that each cell is assigned a set of at most m symbols from {1,...,n} such that no symbol occurs more than m times in any row or column. An (m,m,m)- array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant γ such that if m ≤ γ2k, then any (m,m,m)-array of order 2k is avoidable. Such a constant γ has been conjectured to exist for all n by Häggkvist.

• 56.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoiding (m, m, m)-arrays of order n=2(k)2012Ingår i: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 19, nr 1, s. P63-Artikel i tidskrift (Refereegranskat)

An (m, m, m)-array of order n is an n x n array such that each cell is assigned a set of at most m symbols from f 1,...,n g such that no symbol occurs more than m times in any row or column. An (m, m, m)-array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant gamma such that if m <= gamma 2(k) and k >= 14, then any (m, m, m)-array of order n = 2(k) is avoidable. Such a constant gamma has been conjectured to exist for all n by Haggkvist.

• 57.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On Latin squares and avoidable arrays2010Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

Denna avhandling inehåller de fyra nedan uppräknade artiklarna, samt en översikt av forskningsområdet.

I Lina J. Andrén: Avoiding (m, m, m)-arrays of order n = 2k

II Lina J. Andrén: Avoidability of random arrays

III Lina J. Andrén: Avoidability by Latin squares of arrays with even order

IV Lina J. Andrén, Carl Johan Casselgren and Lars-Daniel Öhman: Avoiding arrays of odd order by Latin squares

Artikel I, III och IV behandlar en förmodan av Häggkvist, som säger att det finns en konstant c sådan att för varje positivt heltal n gäller att om m ≤ cn så finns för varje n × n array A av delmängder till {1, . . . ,n} sådan att ingen cell i A i innehåller fler än m symboler, och ingen symbol förekommer i fler än m celler i någon av raderna eller kolumnerna, så finns en latinsk kvadrat L sådan att ingen cell i L innehåller en symbol som förekommer i motsvarande cell i A. En sådan latinsk kvadrat sägs undvika A. Artikel I innehåller ett bevis av förmodan i specialfallet n = 2k. Artikel III använder och utökar metoderna i Artikel I till ett bevis av förmodan för alla latinska kvadrater av jämn ordning. Förmodan visas slutligen för samtliga ordningar i Artikel IV, där bevismetoden liknar den som finns i i Artikel I och III tillsammans med en omfärgningssats. Artikel II behandlar en annan aspekt av problemet genom att undersöka vad ett deterministiskt resultat om existens av latinska kvadrater som undviker en viss typ av array säger om arrayer där mängderna tilldelas slumpmässigt.

• 58.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Avoiding arrays of odd order by Latin squaresManuskript (preprint) (Övrigt vetenskapligt)

We prove that there exists a constant c such that for each pos- itive integer k every (2k+1)×(2k+1) array A on the symbols 1,...,2k+1 with at most c(2k + 1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1)×(2k+1) Latin square S on the symbols 1,...,2k+1 such that for each cell (i, j) in S the symbol in (i, j) does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist.

• 59.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Convergence of an exponential method for the stochastic Schrödinger equation with power-law nonlinearityManuskript (preprint) (Övrigt vetenskapligt)

A temporal approximation of the stochastic Schrödinger equation with a power-law nonlinearity and driven by multiplicative Ito noise is considered. Observe that the nonlinearity is not globally Lipschitz continuous and, in general, the exact solution cannot be assumed to remain bounded for all times. The first of these issues is handled by considering a truncated version of the equation. The second issue is handled by working with stopping times and random time intervals, on which the solution is almost surely bounded. For the time integration we use a stochastic exponential method. This exponential method has the advantage of being explicit and it does not suffer any CFL-type step size restrictions in general. In this work we prove almost sure convergence and convergence in probability, accompanied by convergence orders of 1/2- and 1/2, respectively. We also find that the regularity assumptions required on the noise and initial value are less restrictive for the convergence of the exponential scheme, compared to the Crank-Nicolson scheme. In addition, we provide numerical experiments that illustrate our theoretical results.

• 60.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Exponential integrators for stochastic partial differential equations2018Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

Stochastic partial differential equations (SPDEs) have during the past decades become an important tool for modeling systems which are influenced by randomness. Because of the complex nature of SPDEs, knowledge of efficient numerical methods with good convergence and geometric properties is of considerable importance. Due to this, numerical analysis of SPDEs has become an important and active research field.

The thesis consists of four papers, all dealing with time integration of different SPDEs using exponential integrators. We analyse exponential integrators for the stochastic wave equation, the stochastic heat equation, and the stochastic Schrödinger equation. Our primary focus is to study strong order of convergence of temporal approximations. However, occasionally, we also analyse space approximations such as finite element and finite difference approximations. In addition to this, for some SPDEs, we consider conservation properties of numerical discretizations.

As seen in this thesis, exponential integrators for SPDEs have many benefits over more traditional integrators such as Euler-Maruyama schemes or the Crank-Nicolson-Maruyama scheme. They are explicit and therefore very easy to implement and use in practice. Also, they are excellent at handling stiff problems, which naturally arise from spatial discretizations of SPDEs. While many explicit integrators suffer step size restrictions due to stability issues, exponential integrators do not in general.

In Paper 1 we consider a full discretization of the stochastic wave equation driven by multiplicative noise. We use a finite element method for the spatial discretization, and for the temporal discretization we use a stochastic trigonometric method. In the first part of the paper, we prove mean-square convergence of the full approximation. In the second part, we study the behavior of the total energy, or Hamiltonian, of the wave equation. It is well known that for deterministic (Hamiltonian) wave equations, the total energy remains constant in time. We prove that for stochastic wave equations with additive noise, the expected energy of the exact solution grows linearly with time. We also prove that the numerical approximation produces a small error in this linear drift.

In the second paper, we study an exponential integrator applied to the time discretization of the stochastic Schrödinger equation with a multiplicative potential. We prove strong convergence order 1 and 1/2 for additive and multiplicative noise, respectively. The deterministic linear Schrödinger equation has several conserved quantities, including the energy, the mass, and the momentum. We first show that for Schrödinger equations driven by additive noise, the expected values of these quantities grow linearly with time. The exponential integrator is shown to preserve these linear drifts for all time in the case of a stochastic Schrödinger equation without potential. For the equation with a multiplicative potential, we obtain a small error in these linear drifts.

The third paper is devoted to studying a full approximation of the one-dimensional stochastic heat equation. For the spatial discretization we use a finite difference method and an exponential integrator is used for the temporal approximation. We prove mean-square convergence and almost sure convergence of the approximation when the coefficients of the problem are assumed to be Lipschitz continuous. For non-Lipschitz coefficients, we prove convergence in probability.

In Paper 4 we revisit the stochastic Schrödinger equation. We consider this SPDE with a power-law nonlinearity. This nonlinearity is not globally Lipschitz continuous and the exact solution is not assumed to remain bounded for all times. These difficulties are handled by considering a truncated version of the equation and by working with stopping times and random time intervals. We prove almost sure convergence and convergence in probability for the exponential integrator as well as convergence orders of ½ − 𝜀, for all 𝜀 > 0, and 1/2, respectively.

• 61.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Department of Mathematics, University of Innsbruck, A–6020 Innsbruck, Austria.
Exponential integrators for stochastic Schrödinger equations driven by Itô noise2018Ingår i: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 36, nr 2, s. 276-309Artikel i tidskrift (Refereegranskat)

We study an explicit exponential scheme for the time discretisation of stochastic Schr¨odinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schr¨odinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

• 62.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Univ Innsbruck, Dept Math, Innsbruck, Austria.
Full discretization of semilinear stochastic wave equations driven by multiplicative noise2016Ingår i: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 54, nr 2, s. 1093-1119Artikel i tidskrift (Refereegranskat)

A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Stormer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.

• 63.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
A fully discrete approximation of the one-dimensional stochastic heat equation2020Ingår i: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 40, nr 1, s. 247-284Artikel i tidskrift (Övrigt vetenskapligt)

A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in Lq(&#x03A9;)" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline-table; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">Lq(Ω)Lq(Ω)⁠, for all q&#x2A7E;2" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline-table; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">q⩾2q⩾2⁠, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.

• 64.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Development and evaluation of stress tests: Utilizing stress tests to complement the current ex-ante analysis at Second Swedish National Pension Fund2017Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

På den finansiella marknaden förekommer termen stresstester med jämna mellanrum, där vissa institutioner har det som krav och andra har det som ett frivilligt sätt att komplettera prediktioner. Stresstester används för att mäta hur robust ett finansiellt instrument eller en portfölj är i olika scenarion, där utmaningen blir att konstruera ett stresstest som är relevant och tillräckligt extremt. Målet med arbetet är att studera olika stresstestmetoder som ska kunna bli tillämpade hos Andra AP-fonden (AP2) i samband med deras prediktion av marknadsrisker. Två olika metoder implementeras med olika scenarion och således utförs unika analyser för respektive metod. Därav jämförs inte metoderna mot varandra utan varje metod analyseras individuellt med för- och nackdelar utifrån valet av metod och typen av scenarion. Resultatet för den första metoden, historiskt stresstest, påvisar att portföljen som stressas skulle minska i värde under det specificerade scenariot. För den andra metoden, koherent stresstest, varierar resultatet för de olika scenarierna.

• 65.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Solvenskrav och riskhantering enligt Solvens II: Implementering av de nya riskbaserade solvensreglerna på ett skadeförsäkringsbolag2014Självständigt arbete på avancerad nivå (yrkesexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

The insurance industryis challengedby major changesthrough internationalizationand thusgrowingcompetitionwithmore optionsand greater availabilityfor customers.Solvency II, a new regulatoryframework and anew standard forinsurance companies,is therefore implemented in a steptowardsa more competitive Europeaninsurancemarket. Solvency II will result in extensive structural changes for many insurance companies. Newstandards andinternal models must also be implemented by the companies. Solvency IIhasa tightening effect on thesolvency capital requirements forinsurance companies. At the same time it also leads tomajor changesandincreased demandon risk management andinternal control, alongside with demands fordisclosure of informationto the market. The purposeof this thesis isthereforeto develop aSolvency IImodel with solvency capital requirements, SCR, and own risk and solvency assessment, ORSA,for a small, fictitious, non-life insurancecompany.The thesis iscovered byfour parts, whereeach of themis a partof developingthe non-life insurance company. The modelis implementedin association withthe consulting firmtheFinancialCompliance Group, FCG, inStockholm and willprovide insight into theSolvency II, its meaning, andhow the frameworkcan be implementedin practice.The modelingis doneprimarilyin Excel. Matlab and VBA, the programming language integrated in Excel, is also used for the implementation. The result shows that market risk is the largest part of the total solvency capital requirement related to the non-life insurance company. The size and allocation of the assets of the insurance company has a direct impact on the size of the capital requirement for market risk. The counterparty credit rating, probability of default, also has a crucial effect on the size of the capital requirement. The outcome in both the base scenario and the stressed scenario in the ORSA show the importance of companies simulating a future solvency situation. They can thereby get an idea of ​​how the business reacts to market changes. The combination of assets in the asset portfolio that generates the minimum SCR is the portfolio with the lowest return and the lowest risk profile. This approach also results in a high solvency ratio. Conversely, the portfolio with the highest yield also implies a maximum level of risk. More weight allocated in assets with high returns and variance generates a higher SCR and thereby a lower solvency ratio. The insurance company can benefit from the strategy with a high return on the asset portfolio in prosperous times with positive market trends. Conversely, more weight allocated to assets with low returns generates a lower SCR and a higher solvency ratio. This approach is preferable when the market suffers from negative trends. The solvency ratio is further affected by changes in the value of both assets and liabilities. A decrease in interest rates results in an increase of the technical provisions which in turn leads to a reduction in own funds and solvency ratio. The same effect arises when the total value of assets is reduced. The thesis also showed the importance of reallocating the assets in the asset portfolio. This is particularly palpable in times when the value of assets suffers from downward trends. Reallocation of the assets will reduce the risk of falling solvency ratio and thereby the risk that the company will become insolvent.

• 66.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Reliable hp finite element computations of scattering resonances in nano optics2019Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.

The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.

In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.

The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.

• 67.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Computation of scattering resonances in absorptive and dispersive media with applications to metal-dielectric nano-structuresManuskript (preprint) (Övrigt vetenskapligt)

In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS.Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

• 68.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On spurious solutions in finite element approximations of resonances in open systems2017Ingår i: Computers and Mathematics with Applications, ISSN 0898-1221, E-ISSN 1873-7668, Vol. 74, nr 10, s. 2385-2402Artikel i tidskrift (Refereegranskat)

In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann–Schwinger equation and we use computations of the pseudospectrum to show that this is a suitable choice. Numerical simulations indicate that the presented test can distinguish between spurious eigenvalues and true eigenvalues also in difficult cases.

• 69.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Removal of spurious solutions encountered in Helmholtz scattering resonance computations in R^dManuskript (preprint) (Övrigt vetenskapligt)

In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs in two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can therefore be applied to graded material properties as well as piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to remove spurious solutions in electromagnetic and acoustic scattering resonances computations at low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

• 70.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Royal Institute of Technology.
Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map2018Ingår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, s. 177-192Artikel i tidskrift (Refereegranskat)

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated eigenvalues of finite multiplicity that only can accumulate at infinity. The proposed method is based on a high order finite element discretization combined with a specialization of the Tensor Infinite Arnoldi method (TIAR). Using Toeplitz matrices, we show how to specialize this method to our specific structure. In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function. The solution scheme can be applied to multiple resonators with a varying refractive index that is not necessarily piecewise constant. We present two test cases to show stability, performance and numerical accuracy of the method. In particular the use of a high order finite element discretization together with TIAR results in an efficient and reliable method to compute resonances.

• 71.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Formativ Bedömning: Återkoppling: Hur lärarens frågor, uppmaningar och undervisning användsformativt i gymnasieskolans matematikundervisning2015Självständigt arbete på avancerad nivå (yrkesexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
• 72.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Allowing Left Truncated and Censored Fertility Data in the Normal Waiting Model1995Rapport (Övrigt vetenskapligt)

Models describing marital fertility are under consideration. In Arnqvist (Research Report 2, Mathematical Statistics, Umeå University), a normal approximation of the waiting model was introduced. In this report a modification of the normal approximation is suggested. This specification allows the data to be left truncated and censored, which gives the possibility to apply the normally approximated waiting model in datasets as from the United Nations World Fertility Services. The model is appropriate except for extremely high fertility intensities, when it gives rise to bias in the parameter estimations. In this case, therefore, a bootstrap method is suggested to estimate and correct the bias. This means that the normal approximated waiting model is a good competitor to the well known Poisson or Coale-Trussell model. It also uses an understandable fertility specification.

• 73.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Approximation of the waiting model1995Rapport (Övrigt vetenskapligt)

In an attempt to estimate the level of family planning in a population, Coale-Trussell suggested an intensity model based on five year summarized data as given in the reports of the United Nations. Their model is denoted Coale-Trussell model. To make inference in the model, it was assumed that the pregnancy data in the model followed a Poisson process. In Arnqvist (research report 1, 1995, Mathematical Statistics, Umeå University), a modification of the Poisson assumption in the intensity model was suggested, introducing waiting time after the pregnancies. The resulting model was named the waiting model. The aim of this paper is to compare the Coale-Trussell model with the waiting model when data of the form given in UN World Fertility Surveys are used (Table 1). By using a normal approximation of the first two moments of the number of pregnancies, the asymptotic variance of the estimators of the interesting parameters is given. Simulation studies show that the Coale-Trussell model and the normal approximation model both approximate the lower intensities in the waiting model well. However, the Coale-Trussell model gives essentially biased estimates of the intensities for high birth intensities.

• 74.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Aspects of the Coale-Trussell Model1995Rapport (Övrigt vetenskapligt)

Coale-Trussell model for marital fertility is investigated. The assumption that the data follow a Poisson model is invalidated by empirical evidence. The data are less spread than assumed, which indicates a point process which is undersdispersed relative to the Poisson model. By generalizing the Poisson model, by using a more realistic assumption about spacing between births, eg. allowing for a constant delay after each birth, we produce a better and more natural descripton of human reproduction.

• 75.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Functional clustering methods and marital fertility modelling2017Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

This thesis consists of two parts.The first part considers further development of a model used for marital fertility, the Coale-Trussell's fertility model, which is based on age-specific fertility rates. A new model is suggested using individual fertility data and a waiting time after pregnancies. The model is named the waiting model and can be understood as an alternating renewal process with age-specific intensities. Due to the complicated form of the waiting model and the way data is presented, as given in the United Nation Demographic Year Book 1965, a normal approximation is suggested together with a normal approximation of the mean and variance of the number of births per summarized interval. A further refinement of the model was then introduced to allow for left truncated and censored individual data, summarized as table data. The waiting model suggested gives better understanding of marital fertility and by a simulation study it is shown that the waiting model outperforms the Coale-Trussell model when it comes to estimating the fertility intensity and to predict the mean and variance of the number of births for a population.

The second part of the thesis focus on developing functional clustering methods.The methods are motivated by and applied to varved (annually laminated) sediment data from lake Kassj\"on in northern Sweden. The rich but complex information (with respect to climate) in the varves, including the shapes of the seasonal patterns, the varying varve thickness, and the non-linear sediment accumulation rates makes it non-trivial to cluster the varves. Functional representations, smoothing and alignment are functional data tools used to make the seasonal patterns comparable.Functional clustering is used to group the seasonal patterns into different types, which can be associated with different weather conditions.

A new non-parametric functional clustering method is suggested, the Bagging Voronoi K-mediod Alignment algorithm, (BVKMA), which simultaneously clusters and aligns spatially dependent curves. BVKMA is used on the varved lake sediment, to infer on climate, defined as frequencies of different weather types, over longer time periods.

Furthermore, a functional model-based clustering method is proposed that clusters subjects for which both functional data and covariates are observed, allowing different covariance structures in the different clusters. The model extends a model-based functional clustering method proposed by James and Suger (2003). An EM algorithm is derived to estimate the parameters of the model.

• 76.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för ekologi, miljö och geovetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för ekologi, miljö och geovetenskap. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Functional clustering of varved lake sediment to reconstruct past seasonal climate2016Ingår i: Environmental and Ecological Statistics, ISSN 1352-8505, E-ISSN 1573-3009, Vol. 23, nr 4, s. 513-529Artikel i tidskrift (Refereegranskat)

Annually laminated (varved) lake sediments constitutes excellent environmental archives, and have the potential to play an important role for understanding past seasonal climate with their inherent annual time resolution and within-year seasonal patterns. We propose to use functional data analysis methods to extract the relevant information with respect to climate reconstruction from the rich but complex information in the varves, including the shapes of the seasonal patterns, the varying varve thickness, and the non-linear sediment accumulation rates. In particular we analyze varved sediment from lake Kassjon in northern Sweden, covering the past 6400 years. The properties of each varve reflect to a large extent weather conditions and internal biological processes in the lake the year that the varve was deposited. Functional clustering is used to group the seasonal patterns into different types, that can be associated with different weather conditions. The seasonal patterns were described by penalized splines and clustered by the k-means algorithm, after alignment. The observed (within-year) variability in the data was used to determine the degree of smoothing for the penalized spline approximations. The resulting clusters and their time dynamics show great potential for seasonal climate interpretation, in particular for winter climate changes.

• 77.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Model based functional clustering of varved lake sedimentsManuskript (preprint) (Övrigt vetenskapligt)

Climate and environmental changes are today widely discussed, and in particular the impact of human activity. To understand variations in past climate over longer time periods, historical documents, year rings from trees, ice cores from glaciers as well as lake and sea sediments are being used.In this paper we introduce a model based functional cluster analysis, giving us possibility to use both the functional form and covariates in our analysis. It also allow us to model the dependency of the chosen basis coefficients and the covariates. We also allow for different covariance structure within each cluster and give suggestions on how to determine how many clusters to use.In particular we analyze varved sediment from lake Kassjön (N Sweden) which cover more than 6400 years.

• 78.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On interval edge colorings of (a,b)-biregular bipartitie graphs2006Ingår i: Discrete Mathematics, ISSN 0012-365X, E-ISSN 1872-681X, Vol. 307, nr 15, s. 1951-1956Artikel i tidskrift (Refereegranskat)
• 79.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
On Path Factors of (3,4)-Biregular Bigraphs2008Ingår i: Graphs and Combinatorics, ISSN 0911-0119, E-ISSN 1435-5914, Vol. 24, nr 5, s. 405-411Artikel i tidskrift (Refereegranskat)

A (3, 4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a simple (3,4)-biregular bigraph always has a path factor such that the endpoints of each path have degree three. Moreover we suggest a polynomial algorithm for the construction of such a path factor.

• 80.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Southern Polytechnic State University, Marietta, Georgia. University of Illinois, Urbana, Illinois.
Proper path-factors and interval edge-coloring of (3,4)-biregular bigraphs2009Ingår i: Journal of Graph Theory, ISSN 0364-9024, E-ISSN 1097-0118, Vol. 61, nr 2, s. 88-97Artikel i tidskrift (Refereegranskat)

An interval coloring of a graph G is a proper coloring of E(G) by positive integers such that the colors on the edges incident to any vertex are consecutive. A (3,4)-biregular bigraph is a bipartite graph in which each vertex of one part has degree 3 and each vertex of the other has degree 4; it is unknown whether these all have interval colorings. We prove that G has an interval coloring using 6 colors when G is a (3,4)-biregular bigraph having a spanning subgraph whose components are paths with endpoints at 3-valent vertices and lengths in {2, 4, 6, 8}. We provide several sufficient conditions for the existence of such a subgraph.

• 81. Avelin, Benny
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
A Note on the Hyperconvexity of Pseudoconvex Domains Beyond Lipschitz Regularity2015Ingår i: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, Vol. 43, nr 3, s. 531-545Artikel i tidskrift (Refereegranskat)

We show that bounded pseudoconvex domains that are Holder continuous for all alpha < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 194(4) 519-564, 1987) beyond Lipschitz regularity.

• 82. Avelin, Benny
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Approximation of plurisubharmonic functions2016Ingår i: Complex Variables and Elliptic Equations, ISSN 1747-6933, E-ISSN 1747-6941, Vol. 61, nr 1, s. 23-28Artikel i tidskrift (Refereegranskat)

We extend a result by Fornaaess and Wiegerinck [Ark. Mat. 1989;27:257-272] on plurisubharmonic Mergelyan type approximation to domains with boundaries locally given by graphs of continuous functions.

• 83.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Boundary estimates for solutions to operators of p-Laplace type with lower order terms2011Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 250, nr 1, s. 264-291Artikel i tidskrift (Refereegranskat)

In this paper we study the boundary behavior of solutions to equations of the form∇⋅A(x,∇u)+B(x,∇u)=0, in a domain ΩRn, assuming that Ω is a δ-Reifenberg flat domain for δ sufficiently small. The function A is assumed to be of p-Laplace character. Concerning B, we assume that |∇ηB(x,η)|⩽c|η|p−2, |B(x,η)|⩽c|η|p−1, for some constant c, and that B(x,η)=|η|p−1B(x,η/|η|), whenever xRn, ηRn∖{0}. In particular, we generalize the results proved in J. Lewis et al. (2008) [12] concerning the equation ∇⋅A(x,∇u)=0, to equations including lower order terms.

• 84.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Optimal doubling, reifenberg flatness and operators of p-laplace type2011Ingår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 74, nr 17, s. 5943-5955Artikel i tidskrift (Refereegranskat)

In this paper, we consider equations of p-Laplace type of the form A(x,u)=0. Concerning A we assume, for p∈(1,) fixed, an appropriate ellipticity type condition, Hölder continuity in x and that A(x,η)=|η|p−1A(x,η/|η|) whenever xRn and ηRn∖{0}. Let ΩRn be a bounded domain, let D be a compact subset of Ω. We say that is the A-capacitary function for D in Ω if on D, on Ω in the sense of and in ΩD in the weak sense. We extend to RnΩ by putting on RnΩ. Then there exists a unique finite positive Borel measure on Rn, with support in Ω, such that In this paper, we prove that if Ω is Reifenberg flat with vanishing constant, then for every τ, 0<τ≤1. In particular, we prove that is an asymptotically optimal doubling measure on Ω.

• 85.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Kalkylprogram som hjälpmedel i matematikundervisningen2009Självständigt arbete på grundnivå (yrkesexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

• 86.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
X och Y bland ettor och nollor: En inblick i några gymnasielärares resonemang kring datorstöd i matematikundervisningen2009Självständigt arbete på grundnivå (yrkesexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

Det blir allt vanligare att elever som börjar gymnasieskolan erbjuds varsin laptop, men få matematiklärare verkar använda dem i sin undervisning. Syftet med detta examensarbete är därför att undersöka hur lärare resonerar kring datorstöd i matematikundervisningen nu när eleverna har tillgång till varsin dator. De forskningsfrågor som besvaras i undersökningen berör gymnasielärares användning och syn på datorstöd samt vad som krävs för att lärare ska använda datorstöd. I undersökningen har fyra matematiklärare på gymnasiet intervjuats. Resultatet är att när datorstöd används i matematikundervisningen så är det främst läraren som använder datorn, och då till att demonstrera, åskådliggöra svårritade figurer, visa simuleringar samt till att variera koefficienter i olika funktioner och visa på de förändringar som sker. För att lärare ska använda datorstöd behöver de få motiv, inspiration, konkreta tips, teknisk utrustning, relevant utbildning samt slutligen mer tid till att planera datorstöd.

• 87.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Envelopes of holomorphy for bounded holomorphic functions1992Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

Some problems concerning holomorphic continuation of the class of bounded holo­morphic functions from bounded domains in Cn that are domains of holomorphy are solved. A bounded domain of holomorphy Ω in C2 with nonschlicht H°°-envelope of holomorphy is constructed and it is shown that there is a point in D for which Glea­son’s Problem for H°°(Ω) cannot be solved. Furthermore a proof of the existence of a bounded domain of holomorphy in C2 for which the volume of the H°°-envelope of holomorphy is infinite is given. The idea of the proof is to put a family of so-called ”Sibony domains” into the unit bidisk by a packing procedure and patch them together by thin neighbourhoods of suitably chosen curves.

If H°°(Ω) is the Banach algebra of bounded holomorphic functions on a bounded domain Ω in Cn and if p is a point in Ω, then the following problem is known as Gleason’s Problem for Hoo(Ω) :

Is the maximal ideal in H°°(Ω) consisting of functions vanishing at p generated

by (z1 -p1) , ... ,   (zn - pn) ?

A sufficient condition for solving Gleason’s Problem for 77°° (Ω) for all points in Ω is given. In particular, this condition is fulfilled by a convex domain Ω with Lip1+e boundary (0 < e < 1) and thus generalizes a theorem of S.L.Leibenson. It is also proved that Gleason’s Problem can be solved for all points in certain unions of two polydisks in C2. One of the ideas in the methods of proof is integration along specific polygonal lines.

Certain properties of some open sets defined by global plurisubharmonic func­tions in Cn are studied. More precisely, the sets Du = {z e Cn : u(z) < 0} and Eh = {{z,w) e Cn X C : h(z,w) < 1} are considered where ti is a plurisubharmonic function of minimal growth and h≠0 is a non-negative homogeneous plurisubharmonic function. (That is, the functions u and h belong to the classes L(Cn) and H+(Cn x C) respectively.) It is examined how the fact that Eh and the connected components of Du are H°°-domains of holomorphy is related to the structure of the set of disconti­nuity points of the global defining functions and to polynomial convexity. Moreover it is studied whether these notions are preserved under a certain bijective mapping from L(Cn) to H+(Cn x C). Two counterexamples are given which show that polynomial convexity is not preserved under this bijection. It is also proved, for example, that if Du is bounded and if the set of discontinuity points of u is pluripolar then Du is of type H°°.

A survey paper on general properties of envelopes of holomorphy is included. In particular, the paper treats aspects of the theory for the bounded holomorphic functions. The results for the bounded holomorphic functions are compared with the corresponding ones for the holomorphic functions.

• 88. Backlund, Ulf
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Semi-Bloch Functions in Several Complex Variables2016Ingår i: Journal of Geometric Analysis, ISSN 1050-6926, E-ISSN 1559-002X, Vol. 26, nr 1, s. 463-473Artikel i tidskrift (Refereegranskat)

Let M be an n-dimensional complex manifold. A holomorphic function f : M -> C is said to be semi-Bloch if for every lambda is an element of C the function g(lambda) = exp(lambda f(z)) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.

• 89. Backlund, Ulf
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
On the algebra of Ak-functions2006Ingår i: Mathematica Bohemica, ISSN 0862-7959, Vol. 131Artikel i tidskrift (Refereegranskat)
• 90. Baculikova, Blanka
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Oscillation of third order trinomial delay differential equations2012Ingår i: Applied Mathematics and Computation, ISSN 0096-3003, E-ISSN 1873-5649, Vol. 218, nr 13, s. 7023-7033Artikel i tidskrift (Refereegranskat)

The purpose of this paper is to study oscillation and asymptotic behavior of solutions to a third order linear delay differential equation y'''(t) + p(t)y'(t) + q(t)y(tau(t)) = 0. New comparison theorems deduce oscillation of the given third order delay differential equation via application of known oscillation criteria to associated first order delay differential equations or inequalities. Contrary to most known results, we establish oscillation of all solutions to a given delay differential equation. Conditions in all theorems are not difficult to verify; computer algebra systems can be efficiently used to assist the computations. Main results in the paper are illustrated with a number of examples.

• 91.
Christian-Albrechts Universität Kiel.
Christian-Albrechts Universität Kiel. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Christian-Albrechts Universität Kiel. Christian-Albrechts Universität Kiel.
Exact and heuristic algorithms for the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection2015Ingår i: Journal of the Operational Research Society, ISSN 0160-5682, E-ISSN 1476-9360, Vol. 66, nr 4, s. 615-626Artikel i tidskrift (Refereegranskat)

We introduce and study the Travelling Salesman Problem with Multiple Time Windows and Hotel Selection (TSP-MTWHS), which generalises the well-known Travelling Salesman Problem with Time Windows and the recently introduced Travelling Salesman Problem with Hotel Selection. The TSP-MTWHS consists in determining a route for a salesman (eg, an employee of a services company) who visits various customers at different locations and different time windows. The salesman may require a several-day tour during which he may need to stay in hotels. The goal is to minimise the tour costs consisting of wage, hotel costs, travelling expenses and penalty fees for possibly omitted customers. We present a mixed integer linear programming (MILP) model for this practical problem and a heuristic combining cheapest insert, 2-OPT and randomised restarting. We show on random instances and on real world instances from industry that the MILP model can be solved to optimality in reasonable time with a standard MILP solver for several small instances. We also show that the heuristic gives the same solutions for most of the small instances, and is also fast, efficient and practical for large instances.

• 92. Banerjee, M.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Stabilizing Role of Nonlocal Interaction on Spatio-temporal Pattern Formation2016Ingår i: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 11, nr 5, s. 103-118Artikel i tidskrift (Refereegranskat)

Here we study a spatio-temporal prey-predator model with ratio-dependent functional response and nonlocal interaction term in the prey growth. For a clear understanding of the effect of nonlocal interaction on the resulting stationary and non-stationary patterns, we consider the nonlocal interaction term in prey growth only to describe the nonlocal intra-specific competition due to limited resources for the prey. First we obtain the patterns exhibited by the basic model in the absence of nonlocal interaction and then explore the effect of nonlocal interaction on the resulting patterns. We demonstrate the stabilizing role of nonlocal interaction as it induces stationary pattern from periodic and chaotic regimes with an increase in the range of nonlocal interaction. The existence of multiple branches of stationary solutions, bifurcating from homogeneous steady-state as well as non-stationary patterns, is illustrated with the help of numerical continuation technique.

• 93. Banerjee, Malay
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Influence of discrete delay on pattern formation in a ratio-dependent prey-predator model2014Ingår i: Chaos, Solitons & Fractals, ISSN 0960-0779, E-ISSN 1873-2887, Vol. 67, s. 73-81Artikel i tidskrift (Refereegranskat)

In this paper we explore how the two mechanisms, Turing instability and Hopf bifurcation, interact to determine the formation of spatial patterns in a ratio-dependent prey predator model with discrete time delay. We conduct both rigorous analysis and extensive numerical simulations. Results show that four types of patterns, cold spot, labyrinthine, chaotic as well as mixture of spots and labyrinthine can be observed with and without time delay. However, in the absence of time delay, the two aforementioned mechanisms have a significant impact on the emergence of spatial patterns, whereas only Hopf bifurcation threshold is derived by considering the discrete time delay as the bifurcation parameter. Moreover, time delay promotes the emergence of spatial patterns via spatio-temporal Hopf bifurcation compared to the non-delayed counterpart, implying the destabilizing role of time delay. In addition, the destabilizing role is prominent when the magnitude of time delay and the ratio of diffusivity are comparatively large.

• 94. Banerjee, Malay
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Time delay can enhance spatio-temporal chaos in a prey-predator model2016Ingår i: Ecological Complexity: An International Journal on Biocomplexity in the Environment and Theoretical Ecology, ISSN 1476-945X, E-ISSN 1476-9840, Vol. 27, s. 17-28Artikel i tidskrift (Refereegranskat)

In this paper we explore how the time delay induced Hopf-bifurcation interacts with Turing instability to determine the resulting spatial patterns. For this study, we consider a delayed prey-predator model with Holling type-II functional response and intra-specific competition among the predators. Analytical criteria for the delay induced Hopf-bifurcation and for the delayed spatio-temporal model are provided with numerical example to validate the analytical results. Exhaustive numerical simulation reveals the appearance of three types of stationary patterns, cold spot, labyrinthine, mixture of stripe-spot and two non-stationary patterns, quasi-periodic and spatio-temporal chaotic patterns. The qualitative features of the patterns for the non-delayed and the delayed spatio-temporal model are the same but their occurrence is solely controlled by the temporal parameters, rate of diffusivity and magnitude of the time delay. It is evident that the magnitude of time delay parameter beyond the Hopf-bifurcation threshold mostly produces spatio-temporal chaotic patterns.

• 95.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Institute of Mathematics and Statistics, University of Tartu, Tartu, Estonia. Umeå universitet, Medicinska fakulteten, Institutionen för strålningsvetenskaper. Department of Mathematical Sciences, Chalmers and University of Gothenburg, Gothenburg, Sweden. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Prediction of CT images from MR images with hidden Markov and random field models2016Ingår i: Proceedings of the 8th International Workshop on Spatio-Temporal Modelling / [ed] A. Iftimi, J. Mateu and F. Montes, 2016, s. 163-163Konferensbidrag (Övrigt vetenskapligt)
• 96.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Medicinska fakulteten, Institutionen för strålningsvetenskaper. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Computed Tomography Image Estimation by Statistical Learning MethodsManuskript (preprint) (Övrigt vetenskapligt)

There is increasing interest in computed tomography (CT) image estimations from magnetic resonance (MR) images. The estimated CT images canbe utilised for attenuation correction, patient positioning, and dose planningin diagnostic and radiotherapy workflows. This study presents a statisticallearning method for CT image estimation. We have used predefined tissuetype information in a Gaussian mixture model to explore the estimation.The performance of our method was evaluated using cross-validation on realdata. In comparison with the existing model-based CT image estimationmethods, the proposed method has improved the estimation, particularly inbone tissues. Evaluation of our method shows that it is a promising methodto generate CT image substitutes for the implementation of fully MR-basedradiotherapy and PET/MRI applications.

• 97.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Statistical methods in medical image estimation and sparse signal recovery2018Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

Denna avhandling presenterar arbete kring metoder för skattning av datortomografibilder (CT) från magnetiska resonanstomografibilder (MR) för ett antal diagnostiska och terapeutiska arbetsflöden. Studien demonstrerar även en metod för gles signalrekonstruktion, vilket är en mellanliggande metod för rekonstruktion av MR-bilder. Avhandlingen består av fyra artiklar. De tre första artiklarna handlar om att utveckla statistiska metoder för uppskattning av CT-bilder från MR-bilder. Här formuleras rumsliga och icke-rumsliga modeller för skattning av CT-bilder från MR-bilder, där de rumsliga modellerna inkluderar dolda Markov-modeller (HMM) och dolda Markov-slumpfältmodeller (HMRF), medan de icke-rumsliga modellerna består av Gaussiska mix-modeller (GMM) och skeva Gaussiska mixmodeller (SGMM). De statistiska modellerna skattas via en maximum-likelihoodansats, där EM-algoritmen används för GMM och SGMM, EM-gradientalgoritmen för HMRF samt Baum-Welch-algoritmen för HMM. Vi har även undersökt CTbildskattning med hjälp av GMM och övervakade statistiska inlärningsmetoder. Modellernas prestanda har utvärderats med hjälp av korsvalidering på faktiska data. Genom att jämföra prestandan hos modellernas CT-bildskattningar har vi observerat att GMM kombinerat med övervakad statistisk inlärning har den bästa prestandan, i synnerhet ifråga om benvävnad. Den fjärde artikeln behandlar en gles modellering inom signalrekonstruktion. Med hjälp av så kallade ”spike and slab priors” för signalen formulerade vi ett glest signalrekonstruktionsproblem och utvecklade en adaptiv algoritm för gles signalrekonstruktion. Den utvecklade algoritmen har bättre prestanda än den nyligen föreslagna iterativ konvex förfining (ICR)-algoritmen. De metoder som introducerats i detta arbete är bidrag till litteraturen inom så kallade ”lattice-processer” och signalbehandling. Resultaten levererar ett bidrag till forskningen kring ersättandet av CT-bilder med syntetiska eller pseudo-CTbilder, samt till effektiv gles signalrekonstruktion.

• 98.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik. Umeå universitet, Medicinska fakulteten, Institutionen för strålningsvetenskaper. Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Statistical learning in computed tomography image estimation2018Ingår i: Medical physics (Lancaster), ISSN 0094-2405, Vol. 45, nr 12, s. 5450-5460Artikel i tidskrift (Refereegranskat)

Purpose: There is increasing interest in computed tomography (CT) image estimations from magneticresonance (MR) images. The estimated CT images can be utilized for attenuation correction, patientpositioning, and dose planning in diagnostic and radiotherapy workflows. This study aims to introducea novel statistical learning approach for improving CT estimation from MR images and to compare theperformance of our method with the existing model-based CT image estimation methods.

Methods: The statistical learning approach proposed here consists of two stages. At the trainingstage, prior knowledge about tissue types from CT images was used together with a Gaussian mixturemodel (GMM) to explore CT image estimations from MR images. Since the prior knowledge is notavailable at the prediction stage, a classifier based on RUSBoost algorithm was trained to estimatethe tissue types from MR images. For a new patient, the trained classifier and GMMs were used topredict CT image from MR images. The classifier and GMMs were validated by using voxel-leveltenfold cross-validation and patient-level leave-one-out cross-validation, respectively.

Results: The proposed approach has outperformance in CT estimation quality in comparison withthe existing model-based methods, especially on bone tissues. Our method improved CT image estimationby 5% and 23% on the whole brain and bone tissues, respectively.

Conclusions: Evaluation of our method shows that it is a promising method to generate CTimage substitutes for the implementation of fully MR-based radiotherapy and PET/MRI applications

• 99.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Model-based computed tomography image estimation: partitioning approach2019Ingår i: Journal of Applied Statistics, ISSN 0266-4763, E-ISSN 1360-0532, Vol. 46, nr 14, s. 2627-2648Artikel i tidskrift (Refereegranskat)

There is a growing interest to get a fully MR based radiotherapy. The most important development needed is to obtain improved bone tissue estimation. The existing model-based methods perform poorly on bone tissues. This paper was aimed at obtaining improved bone tissue estimation. Skew-Gaussian mixture model and Gaussian mixture model were proposed to investigate CT image estimation from MR images by partitioning the data into two major tissue types. The performance of the proposed models was evaluated using the leaveone-out cross-validation method on real data. In comparison with the existing model-based approaches, the model-based partitioning approach outperformed in bone tissue estimation, especially in dense bone tissue estimation.

• 100.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för matematik och matematisk statistik.
Model-based Estimation of Computed Tomography Images2017Konferensbidrag (Övrigt vetenskapligt)
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