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• 1.
Linköping University, Department of Mathematics.
A note on the (n, p)-asymptotics of ATS for high dimensional multivariate data2010Report (Other academic)
• 2.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A two-sample test statistic for high-dimensional multivariate data under non-normality2011Report (Other academic)

Ahmad, Ohlson, and von Rosen (2011a) present asymptotic distribution of a one-sample test statistic under non-normality, when the data are high dimensional, i.e., when the dimension of the vector, p, may exceed the sample size, n. This paper extends the case to a two-sample statistic to test the difference of mean vectors of two independent multivariate distributions, again under high-dimensional set up. Using the asymptotic theory of U-statistics, and under mild assumptions on the traces of the unknown covariance matrices, the statistic is shown to follow an approximate normal distribution when n and p are large. However, no relationship between n and p is assumed. An extension to the paired case is given, which, being essentially a one-sample statistic, supplements the asymptotic results obtained in Ahmad, Ohlson, and von Rosen (2011a).

• 3.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Robustness of a one-sample statistic for mean testing of high dimensional longitudinal data2010Report (Other academic)
• 4.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A U-statistics Based Approach to Mean Testing for High Dimensional Multivariate Data Under Non-normality2011Report (Other academic)

A test statistic is considered for testing a hypothesis for the mean vector for multivariate data, when the dimension of the vector, p, may exceed the number of vectors, n, and the underlying distribution need not necessarily be normal. With n, p large, and under mild assumptions, the statistic is shown to asymptotically follow a normal distribution. A by product of the paper is the approximate distribution of a quadratic form, based on the reformulation of well-known Box's approximation, under high-dimensional set up.

• 5.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology. Department of Energy and Technology, Swedish Univerity of Agricultural Sciences, SE-750 07 Uppsala, Sweden.
Some Tests of Covariance Matrices for High Dimensional Multivariate Data2011Report (Other academic)

Test statistics for sphericity and identity of the covariance matrix are presented, when the data are multivariate normal and the dimension, p, can exceed the sample size, n. Using the asymptotic theory of U-statistics, the test statistics are shown to follow an approximate normal distribution for large p, also when p >> n. The statistics are derived under very general conditions, particularly avoiding any strict assumptions on the traces of the unknown covariance matrix. Neither any relationship between n and p is assumed. The accuracy of the statistics is shown through simulation results, particularly emphasizing the case when p can be much larger than n. The validity of the commonly used assumptions for high-dimensional set up is also briefly discussed.

• 6.
Department of Aeronautics and Astronautics, Stanford University, Stanford, USA.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
High-order accurate difference schemes for the Hodgkin-Huxley equations2012Report (Other academic)

A novel approach for simulating potential propagation in neuronal branches with high accuracy is developed. The method relies on high-order accurate dierence schemes using the Summation-By-Parts operators with weak boundary and interface conditions applied to the Hodgkin-Huxley equations. This work is the rst demonstrating high accuracy for that equation. Several boundary conditions are considered including the non-standard one accounting for the soma presence, which is characterized by its own partial dierential equation. Well-posedness for the continuous problem as well as stability of the discrete approximation is proved for all the boundary conditions. Gains in terms of CPU times are observed when high-order operators are used, demonstrating the advantage of the high-order schemes for simulating potential propagation in large neuronal trees.

• 7.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Quasistatic frictional contact problems with finitely many degrees of freedom.1999Report (Other academic)

In the present paper results on existence and uniqueness of solutions to discrete frictional quasi-static unilateral contact problems are given under a condition that the coefficients of friction are smaller than a certain upper bound. This upper bound is defined in terms of an influence matrix for the contact nodes. The results of existence and uniqueness may be ordered into two classes depending on whether regularity conditions for the applied forces are imposed or not. For general loading which has a time derivative almost everywhere it is shown that a solution exists which satisfies governing equations for almost all times. Uniqueness of the solution has been shown only when the problem is restricted to two degrees of freedom. For a loading which is right piecewise analytic, additional results can be obtained. For instance, if each contact node has only two degrees of freedom a unique solution which satisfies governing equeations for all times exists. For the constructed solutions a priori estimates of the displacement field and its time derivate in terms of the applied forces are also given.

• 8.
Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Global Search Strategies for Solving Multilinear Least-squares Problems2011Report (Other academic)

The multilinear least-squares (MLLS) problem is an extension of the linear least-squares problem. The difference is that a multilinearoperator is used in place of a matrix-vector product. The MLLS istypically a large-scale problem characterized by a large number of local minimizers. It originates, for instance, from the design of filter networks. We present a global search strategy that allows formoving from one local minimizer to a better one. The efficiencyof this strategy isillustrated by results of numerical experiments performed forsome problems related to the design of filter networks.

• 9.
Linköping University, Department of Biomedical Engineering. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Sparsity Optimization in Design of Multidimensional Filter Networks2013Report (Other academic)

Filter networks is a powerful tool used for reducing the image processing time, while maintaining its reasonably high quality.They are composed of sparse sub-filters whose low sparsity ensures fast image processing.The filter network design is related to solvinga sparse optimization problem where a cardinality constraint bounds above the sparsity level.In the case of sequentially connected sub-filters, which is the simplest network structure of those considered in this paper, a cardinality-constrained multilinear least-squares (MLLS) problem is to be solved. If to disregard the cardinality constraint, the MLLS is typically a large-scale problem characterized by a large number of local minimizers. Each of the local minimizers is singular and non-isolated.The cardinality constraint makes the problem even more difficult to solve.An approach for approximately solving the cardinality-constrained MLLS problem is presented.It is then applied to solving a bi-criteria optimization problem in which both thetime and quality of image processing are optimized. The developed approach is extended to designing filter networks of a more general structure. Its efficiency is demonstrated by designing certain 2D and 3D filter networks. It is also compared with the existing approaches.

• 10.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics .
A conditional perspective of weighted variance estimation of regression estimator2003Report (Other academic)
• 11.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics .
Alternative confidence intervals for the total of a skewed biological population2003Report (Other academic)
• 12.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematical Statistics .
An optimal calibration distance leading to optimal regression estimator2003Report (Other academic)
• 13.
Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Asymptotic distribution of the estimators of a harmonic component in a multivariate time series under m-dependence2011Report (Other academic)

Multivariate time series with definite harmonic structure is considered, in the special case when the marginal univariate time series are independent and asymptotically stationary to second order. The asymptotic distribution of the estimators of a harmonic component under \$m\$-dependence is found

• 14.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
Examples of infinity harmonic functions having singular lines2006Report (Other academic)
• 15.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Matematisk modellering i svenska gymnasieskolans kursplaner i matematik 1965-20002009Report (Other academic)

Denna rapport studerar hur begreppen matematisk modell och matematisk modellering behandlats i kursplanerna i matematik för motsvarande svenska gymnasieskolans naturvetenskapliga program mellan åren 1965 och 2000. En översikt över sex kursplaners olika form och innehåll ges och tre olika analyser på tre olika nivåer genomförs; en innehållsanalys, en kvalitativ analys, och en analys som beaktar relationen mellan begreppen modellering, tillämpningar och problemlösning. Analysen visar att matematiska modeller och modellering förekommit implicit sedan kursplanen från 1965, men att begreppen främst i de två senaste kursplanerna lyfts fram explicit och fått en central position i kursplanerna. I den nuvarande kursplanen Gy2000 framgår att matematiska modeller kan tolkas både som ett undervisningsmål i sig och som ett didaktiskt hjälpmedel att lära matematik.

• 16.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Symbolic computing and mathematics education1994Report (Other academic)

A lab sequence with the computer algebra system Maple was integrated in one of the analysis courses for beginning engineering students. The students appreciated the labs, but questionned their value for the learning of analysis. There was a significant shift over time toward a higher appreciation of the labs, and most students wanted Maple in the course. No significant differences were found in mathematical achievement or attitudes between the lab and control groups. The plotting function of Maple was valued for learning and understanding analysis. Data indicate, however, that the symbolic computational feature of a computer algebra system will not be fully appreciated until students are familiar with the computations. It thus seems necessary, when planning to use a computer algebra system as a learning tool, to explicate the intended learning paradigm, how the computational facilities of the system are to be used, and what kind of mathematical competence students develop and get out of the course. A conclusion from this study is that the use in education of thus type of technology is appreciated by the students, but that early in an analysis course it should be used mainly as a support to imagery, and that only later in the course automated algebraic computations should be used for example as a problem solving tool.

• 17.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Boundary identification for an elliptic equation2001Report (Other academic)

We consider an inverse problem for the two dimensional steady state heat equation. More precisely, the heat equation is valid in a domain Ω, that is a subset of the unit square, temperature and heat-flux measurements are available on the line y = 0, and the sides x = 0 and x = 1 are assumed to be insulated. From these we wish to determine the temperature in the domain Ω. Furthermore, a part of the boundry ∂Ω is considered to be unknown, and must also be determined.

The problem is ill-posed in the sense that the solution does not depend continuously on the data. We stabilize the computations by replacing the x-derivative in the heat equation by an operator, representing differentiation of least squares cubic splines. We discretize in the x-coordinate, and obtain an initial value problem for a system of ordinary differential equation, which can be solved using standard numerical methods.

The inverse problem, that we consider in this paper, arises in iron production, where the walls of a melting furnace are subject to physical and chemical wear. In order to avoid a situation where molten metal breaks out the remaining thickness of the walls should constantly be monitored. This is done by recording the temperature at several locations inside the walls. The shape of the interface boundary between the molten iron and the walls of the furnace can then be determined by solving an invers heat conduction problem.

• 18.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation2001Report (Other academic)

Numerical procedures for solving a non-Characteristic Cauchy problem for the heat equation are discussed. More precisely we consider a problem, where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. Mathematically it is formulated as a Cauchy problem for the heat equation in a quarter plane, with data given along the line x = 1, where the solution is wanted 0 ≤ x <1. The problem is often referred to as the sideways heat equation.

The problem is analyzed, using both Fourier analysis and the singular value decomposition, and is found to be severely ill-posed. The literature is vast, and many authors have proposed numerical methods that regularize the IHCP. In this paper we attempt to give an overview that covers the most popular methods that have been considered.

Numerical examples that illustrate the numerical algorithms are given.

• 19.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Management and Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, Faculty of Science & Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. St Petersburg State University, Russia; St Petersburg State Polytech University, Russia.
A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients2018Report (Other academic)

One dimensional models for fluid flow in tubes are frequently used tomodel complex systems, such as the arterial tree where a large numberof vessels are linked together at bifurcations. At the junctions transmission conditions are needed. One popular option is the classic Kirchhoffconditions which means conservation of mass at the bifurcation andprescribes a continuous pressure at the joint.

In reality the boundary layer phenomena predicts fast local changesto both velocity and pressure inside the bifurcation. Thus it is not appropriate for a one dimensional model to assume a continuous pressure. In this work we present a modification to the classic Kirchhoff condi-tions, with a symmetric pressure drop matrix, that is more suitable forone dimensional flow models. An asymptotic analysis, that has beencarried out previously shows that the new transmission conditions hasen exponentially small error.

The modified transmission conditions take the geometry of the bifurcation into account and can treat two outlets differently. The conditions can also be written in a form that is suitable for implementationin a finite difference solver. Also, by appropriate choice of the pressuredrop matrix we show that the new transmission conditions can producehead loss coefficients similar to experimentally obtained ones.

• 20.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology. University of Rwanda. Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
Numerical Solution of the Cauchy Problem for the Helmholtz Equation2014Report (Other academic)

The Cauchy problem for the Helmholtz equation appears in applications related to acoustic or electromagnetic wave phenomena. The problem is ill–posed in the sense that the solution does not depend on the data in a stable way. In this paper we give a detailed study of the problem. Specifically we investigate how the ill–posedness depends on the shape of the computational domain and also on the wave number. Furthermore, we give an overview over standard techniques for dealing with ill–posed problems and apply them to the problem.

• 21.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
National University of Rwanda, Box 117, Butare, Rwanda.
A Data Assimilation Approach to Coefficient Identification2011Report (Other academic)

The thermal conductivity properties of a material can be determined experimentally by using temperature measurements taken at specified locations inside the material. We examine a situation where the properties of a (previously known) material changed locally. Mathematically we aim to find the coefficient k(x) in the stationary heat equation (kTx)x = 0;under the assumption that the function k(x) can be parametrized using only a few degrees of freedom.

The coefficient identification problem is solved using a least squares approach; where the (non-linear) control functional is weighted according to the distribution of the measurement locations. Though we only discuss the 1D case the ideas extend naturally to 2D or 3D. Experimentsdemonstrate that the proposed method works well.

• 22.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
More on Estimation of Banded and Banded Toeplitz Covariance Matrices2017Report (Other academic)

In this paper we consider two different linear covariance structures, e.g., banded and bended Toeplitz, and how to estimate them using different methods, e.g., by minimizing different norms.

One way to estimate the parameters in a linear covariance structure is to use tapering, which has been shown to be the solution to a universal least squares problem. We know that tapering not always guarantee the positive definite constraints on the estimated covariance matrix and may not be a suitable method. We propose some new methods which preserves the positive definiteness and still give the correct structure.

More specific we consider the problem of estimating parameters of a multivariate normal p–dimensional random vector for (i) a banded covariance structure reflecting m–dependence, and (ii) a banded Toeplitz covariance structure.

• 23.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
A regularity classification of boundary points for p-harmonic funktions and quasiminimizers2006Report (Other academic)
• 24.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
p-harmonic functions with boundary data having jump discontinuities and Baernstein´s problem2009Report (Other academic)

For p-harmonic functions on unweighted R-2, with 1 andlt; p andlt; infinity, we show that if the boundary values f has a jump at an (asymptotic) corner point zo, then the Perron solution Pf is asymptotically a + b arg(z - z(0)) + o(vertical bar z z(0)vertical bar). We use this to obtain a positive answer to Baernsteins problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the p. harmonic measure of (G) over bar. We also obtain various invariance results for functions with jumps and perturbations on small sets. For p andgt; 2 these results are new also for continuous functions. Finally we look at generalizations to R-n and metric spaces.

• 25.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology. Helsinki University of Technology.
Lebesgue points and the fundamental convergence theorem for superharmonic functions on metric spaces2008Report (Other academic)
• 26.
Saab AB, Sweden.
Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering. Saab AB, Sweden. Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
An Optimisation Approach for Pre-Runtime Scheduling of Tasks and Communication in an Integrated Modular Avionic System2017Report (Other academic)

In modern integrated modular avionic systems, applications share hardware resources on a common avionic platform. Such an architecture necessitates strict requirements on the spatial and temporal partitioning of the system to prevent fault propagation between different aircraft functions. One way to establish a temporal partitioning is through pre-runtime scheduling of the system, which involves creating a schedule for both tasks and a communication network.

While the avionic systems are growing more and more complex, so is the challenge of scheduling them. Scheduling of the system has an important role in the development of new avionic systems since functionality typically is added to the system over a period of several years and a scheduling tool is used both to detect if the platform can host the new functionality and, in case this is possible, to create a new schedule. For this reason an exact solution strategy for avionics scheduling is preferred over a heuristic one.

In this paper we present a mathematical model for an industrially relevant avionic system and present a constraint generation  procedure for scheduling of such systems. We apply our optimisation approach to instances provided by our industrial partner. These instances are of relevance for the development of future avionic systems and contain up to 20 000 tasks to be scheduled. The computational results show that our optimisation approach can be used to create schedules for such instances within reasonable time.

• 27.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
A genetic algorithm for a pulp distribution problem2003Report (Other academic)

In this paper we present a genetic algorithm for the pulp distribution problem at a large pulp producer in Scandinavia. The distribution is a major part of the company's supply chain and includes transports with cargo vessels, by train and trucks and storages at terminals in port, at pulp mills and in customer locations. The problem we focus on is to find ship schedules and pulp deliveries in order to minimize the total cost of distribution.

The genetic algorithm utilizes two linear programming models. The first model optimizes all transport flows given a schedule and the second model approximates the performance of a schedule, measured in the total distribution cost. In the computational experiments we use instances from the real world and compare the results with an exact mixed integer programming approach.

• 28.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Determining the Non-Existence of a Compatible OSPF Metric2004Report (Other academic)

Many telecommunication networks use Internet Protocol for deciding the routing of traffic. The specifications OSPF (Open Shortest Path First) and ECM (Equal Cost Multipath) are very common, and state that each router sends the traffic on the shortest path to the destination. If there are several shortest path, the router splits the traffic evenly. In order to have some control over the traffic distribution, the operator can assign weights to the links in the network, and these weights are used by the routers when calculating the shortest paths. It has been shown that by optimizing over the values of the weights, the performance of a network can be much improved. A difficult question is whether or not for a set of desired traffic patterns there exists a compatible metric, i.e. weights making the routers give the specified traffic patterns. There is one known necessary condition for the existence of such a metric, but up to now no sufficient conditions. We investigate this problem, and find more general necessary conditions for the existence of compatible weights for a set of given desired "shortest path"-graphs. A polynomial algorithm that for most cases verifies the non-existence of a compatible metric is presented. The algorithm also indicates which parts of the traffic patterns that are in conflict. A few numer;cal examples are used to illustrate the procedure, and some computational tests are reported.

• 29.
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Optimization .
On the Extremal Structure of an OSPF Related Cone2006Report (Other academic)
• 30.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Stronger Necessary Conditions for the Existence of a Compatible OSPF Metric2004Report (Other academic)

A dominating standard for routing in telecommunication networks is Internet Protocol with OSPF (Open Shortest Path First) and ECM (Equal Cost Multipath). Each router sends the traffic on the shortest path to the destination. If there are several shortest path, the router splits the traffic evenly. The operator can assign weights to the links in the network, which are used by the routers when calculating the shortest paths. An important question is whether or not for a set of desired traffic patterns there exists a compatible metric, i.e. weights making the routers give the specified traffic patterns. We describe necessary conditions, stronger than those previously discovered, for the existence of compatible weights for a set of given desired shortest path-graphs.

• 31.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Computer and Information Science, Artificial Intelligence and Integrated Computer Systems. Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, Artificial Intelligence and Integrated Computer Systems. Linköping University, The Institute of Technology.
Local Search for Hop-constrained Directed Steiner Tree Problem with Application to UAV-based Multi-target Surveillance2014Report (Other academic)

We consider the directed Steiner tree problem (DSTP) with a constraint on the total number of arcs (hops) in the tree. This problem is known to be NP-hard, and therefore, only heuristics can be applied in the case of its large-scale instances.   For the hop-constrained DSTP, we propose local search strategies aimed at improving any heuristically produced initial Steiner tree. They are based on solving a sequence of hop-constrained shortest path problems for which we have recently developed ecient label correcting algorithms.   The presented approach is applied to nding suitable 3D locations where unmanned aerial vehicles (UAVs) can be placed to relay information gathered in multi-target monitoring and surveillance. The eciency of our algorithms is illustrated by results of numerical experiments involving problem instances with up to 40 000 nodes and up to 20 million arcs.

• 32.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Computer and Information Science, Artificial Intelligence and Integrated Computer Systems. Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, Artificial Intelligence and Integrated Computer Systems. Linköping University, The Institute of Technology.
Optimal Scheduling for Replacing Perimeter Guarding Unmanned Aerial Vehicles2014Report (Other academic)

Guarding the perimeter of an area in order to detect potential intruders is an important task in a variety of security-related applications. This task can in many circumstances be performed by a set of camera-equipped unmanned aerial vehicles (UAVs). Such UAVs will occasionally require refueling or recharging, in which case they must temporarily be replaced by other UAVs in order to maintain complete surveillance of the perimeter. In this paper we consider the problem of scheduling such replacements. We present optimal replacement strategies and justify their optimality.

• 33.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Samsung Advanced Institute of Technology, China Lab, Beijing, China. State Key Laboratory of Scientic and Engineering Computing, Institute of Computational. Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
On Efficiently Combining Limited Memory and Trust-Region Techniques2013Report (Other academic)

Limited memory quasi-Newton methods and trust-region methods represent two efficient approaches used for solving unconstrained optimization problems. A straightforward combination of them deteriorates the efficiency of the former approach, especially in the case of large-scale problems. For this reason, the limited memory methods are usually combined with a line search. We show how to efficiently combine limited memory and trust-region techniques. One of our approaches is based on the eigenvalue decomposition of the limited memory quasi-Newton approximation of the Hessian matrix. The decomposition allows for finding a nearly-exact solution to the trust-region subproblem defined by the Euclidean norm with an insignificant computational overhead compared with the cost of computing the quasi-Newton direction in line-search limited memory methods. The other approach is based on two new eigenvalue-based norms. The advantage of the new norms is that the trust-region subproblem is separable and each of the smaller subproblems is easy to solve. We show that our eigenvalue-based limited-memory trust-region methods are globally convergent. Moreover, we propose improved versions of the existing limited-memory trust-region algorithms. The presented results of numerical experiments demonstrate the efficiency of our approach which is competitive with line-search versions of the L-BFGS method.

• 34.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, KPLAB - Knowledge Processing Lab. Linköping University, The Institute of Technology. Linköping University, Department of Computer and Information Science, KPLAB - Knowledge Processing Lab. Linköping University, The Institute of Technology.
Optimal placement of communications relay nodes2009Report (Other (popular science, discussion, etc.))

We consider a constrained optimization problem with mixed integer and real variables. It models optimal placement of communications relay nodes in the presence of obstacles. This problem is widely encountered, for instance, in robotics, where it is required to survey some target located in one point and convey the gathered information back to a base station located in another point. One or more unmanned aerial or ground vehicles (UAVs or UGVs) can be used for this purpose as communications relays. The decision variables are the number of unmanned vehicles (UVs) and the UV positions. The objective function is assumed to access the placement quality. We suggest one instance of such a function which is more suitable for accessing UAV placement. The constraints are determined by, firstly, a free line of sight requirement for every consecutive pair in the chain and, secondly, a limited communication range. Because of these requirements, our constrained optimization problem is a difficult multi-extremal problem for any fixed number of UVs. Moreover, the feasible set of real variables is typically disjoint. We present an approach that allows us to efficiently find a practically acceptable approximation to a global minimum in the problem of optimal placement of communications relay nodes. It is based on a spatial discretization with a subsequent reduction to a shortest path problem. The case of a restricted number of available UVs is also considered here. We introduce two label correcting algorithms which are able to take advantage of using some peculiarities of the resulting restricted shortest path problem. The algorithms produce a Pareto solution to the two-objective problem of minimizing the path cost and the number of hops. We justify their correctness. The presented results of numerical 3D experiments show that our algorithms are superior to the conventional Bellman-Ford algorithm tailored to solving this problem.

• 35.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia. Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia.
Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions2011Report (Other academic)

We suggest here a least-change correction to available finite element (FE) solution.This postprocessing procedure is aimed at recoveringthe monotonicity and some other important properties that may not beexhibited by the FE solution. It is based on solvinga monotonic regression problem with some extra constraints.One of them is a linear equality-type constraint which models the conservativityrequirement. The other ones are box-type constraints, andthey originate from the discrete maximum principle.The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessedFE solution preserves the accuracy of the discrete FE approximation.We introduce an algorithm for solving the postprocessingproblem. It can be viewed as a dual ascent method basedon the Lagrangian relaxation of the equality constraint.We justify theoretically its correctness.Its efficiency is demonstrated by the presented results of numerical experiments.

• 36.
Linköping University, Department of Mathematics, Optimization . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Computer and Information Science, Statistics. Linköping University, Faculty of Arts and Sciences.
Regularized monotonic regression2016Report (Other academic)

Monotonic (isotonic) Regression (MR) is a powerful tool used for solving a wide range of important applied problems. One of its features, which poses a limitation on its use in some areas, is that it produces a piecewise constant fitted response. For smoothing the fitted response, we introduce a regularization term in the MR formulated as a least distance problem with monotonicity constraints. The resulting Smoothed Monotonic Regrassion (SMR) is a convex quadratic optimization problem. We focus on the SMR, where the set of observations is completely (linearly) ordered. Our Smoothed Pool-Adjacent-Violators (SPAV) algorithm is designed for solving the SMR. It belongs to the class of dual activeset algorithms. We proved its finite convergence to the optimal solution in, at most, n iterations, where n is the problem size. One of its advantages is that the active set is progressively enlarging by including one or, typically, more constraints per iteration. This resulted in solving large-scale SMR test problems in a few iterations, whereas the size of that problems was prohibitively too large for the conventional quadratic optimization solvers. Although the complexity of the SPAV algorithm is O(n2), its running time was growing in our computational experiments in proportion to n1:16.

• 37.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Optimization . Linköping University, The Institute of Technology.
A Conjugate Direction Frank-Wolfe Method for Nonconvex Problems2003Report (Refereed)

In this paper we propose an algorithm for solving problems with nonconvex objective function and linear constraints. We extend the previously suggested Conjugate direction Frank–Wolfe algorithm to nonconvex problems. We apply our method to multi-class user equilibria under social marginal cost pricing. Results of numerical experiments on Sioux Falls and Winnipeg are reported.

• 38.
Department of Geological Science, San Diego State University, 5500 Campanile Drive, San Diego, California, 92182-1020..
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
Stable, High Order Accurate Adaptive Schemes for Long Time, Highly Intermittent Geophysics Problems2013Report (Other academic)

Many geophysical phenomena are characterized by properties that evolve over a wide range of scales which introduce difficulties when attempting to model these features in one computational method. We have developed a high-order finite difference method for the elastic wave equation that is able to efficiently handle varying temporal scales in a single, stand-alone framework. We apply this method to earthquake cycle models characterized by extremely long interseismic periods interspersed with abrupt, short periods of dynamic rupture. Through the use of summation-by-parts operators and weak enforcement of boundary conditions we derive a provably stable discretization. Time stepping is achieved through the implicit θ-method which allows us to take large time steps during the intermittent period between earthquakes and adapts appropriately to fully resolve rupture.

• 39.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Finite difference schemes with transferable interfaces for parabolic problems2018Report (Other academic)

We derive a method to locally change the order of accuracy of finite difference schemes that approximate the second derivative. The derivation is based on summation-by-parts operators, which are connected at interfaces using penalty terms. At such interfaces, the numerical solution has a double representation, with one representation in each domain. We merge this double representation into a single one, yielding a new scheme with unique solution values in all grid points. The resulting scheme is proven to be stable, accurate and dual consistent.

• 40.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Mathematics, University of Dar es Salaam, Tanzania.
Regime Switching models on Temperature Dynamics2016Report (Other academic)

Two regime switching models for predicting temperature dynamics are presented in this study for the purpose to be used for weather derivatives pricing. One is an existing model in the literature (Elias model) and the other is presented in this paper. The new model we propose in this study has a mean reverting heteroskedastic process in the base regime and a Brownian motion in the shifted regime. The parameter estimation of the two models is done by the use expectation-maximization (EM) method using historical temperature data. The performance of the two models on prediction of temperature dynamics is compared using historical daily average temperature data from five weather stations across Sweden. The comparison is based on the heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. The expected HDDs, CDDs and CAT of the models are compared to the true indices from the real data. Results from the expected HDDs, CDDs and CAT together with their corresponding daily average plots demonstrate that, our model captures temperature dynamics relatively better than Elias model.

• 41.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
Weather derivatives pricing using regim switching models2017Report (Other academic)

In this study we discuss the pricing of weather derivatives whose underlying weather variable is temperature. The dynamics of temperature in this study follows a two state regime switching model with a heteroskedastic mean reverting process as the base regime and a shifted regime defined by Brownian motion with mean different from zero. We develop the mathematical formulas for pricing futures contract on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. We also present the mathematical expressions for pricing the corresponding options on futures contracts for the same temperature indices. The local volatility nature of the model in the base regime captures very well the dynamics of the underlying process, thus leading to a better pricing processes for temperature derivatives contracts written on various index variables. We provide the description of Montecarlo simulation method for pricing weather derivatives under this model and use it to price a few weather derivatives call option contracts.

• 42.
Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
Teachers’ conceptions of mathematical modelling at Swedish Upper Secondary school2011Report (Other academic)

Researchers argue that teachers’ conceptions (beliefs) about mathematical modelling have an impact on the low integration of modelling activities into the mathematics classroom. This paper presents a case study of 18 teachers, investigating their conceptions about mathematical modelling and their experiences of working with modelling activities in Swedish upper secondary school. The results, based on an analysis with a grounded theory inspired coding strategy, indicate that the teachers’ conceptions of the notion mathematical modelling relates to designing a mathematical model based on a situation. It is also concluded that the teachers have only minor experience of the notion of mathematical modelling in the mathematics class, but in physics class modelling is used as a common activity.

• 43.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
A Provable Stable and Accurate Davies-like Relaxation Procedure Using Multiple Penalty Terms for Lateral Boundaries in Weather Prediction2014Report (Other academic)

A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method, that we refer to as the multiple penalty technique, is similar to Davies relaxation and have similar areas of application. The method is proven, by energy methods, to be stable. We show how to apply this technique on the linearized Euler equations in two space dimensions, and that it reduces the errors in the computational domain.

• 44.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
A stable and accurate data assmimilation technique using multiple penalty terms in space and time2016Report (Other academic)

A new method for data assimilation based on weak imposition of external data is introduced. The technique is simple, easy to implement, and the resulting numerical scheme is unconditionally stable. Numerical experiments show that the error growth naturally present in long term simulations can be prevented by using the new technique.

• 45.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Constructing non-reflecting boundary conditions using summation-by-parts in time2016Report (Other academic)

In this paper we provide a new approach for constructing non-reflecting boundary conditions. The boundary conditions are based on summation-by-parts operators and derived without Laplace transformation in time. We prove that the new non-reflecting boundary conditions yield a well-posed problem and that the corresponding numerical approximation is unconditionally stable. The analysis is demonstrated on a hyperbolic system in two space dimensions, and the theoretical results are confirmed by numerical experiments.

• 46.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
Spurious solutions for the advection-diffusion equation using wide stencils for approximating the second derivative.2014Report (Other academic)

A one dimensional steady-state advection-diffusion problem using summation-by-parts operators has been investigated. For approximating the second derivative, a wide stencil has been used, which has spurious, oscillating, modes for all mesh-sizes. We show that the size of the spurious modes are equal to the size of the truncation error for a stable approximation. The theoretical results are veried with numerical experiments.

• 47.
Indian Statistical Institute, Bangalore, India.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Conditional Independence Models which are Totally Ordered2018Report (Other academic)

The totally ordered conditional independence (TOCI) model N(K) is defined to be the set of all normal distributions on RI such that for each adjacent pair (Ki, Ki+1) $\in$ K, the components of a multivariate normal vector x $\in$ RI, indexed by the set difference { Ki+1 \ Ki } are mutually conditionally independent given the variables indexed by Ki. Here K = {K1 $\subset$ … $\subset$ Kq } is a totally ordered set of subsets of a finite index set I. It is shown that TOCI models constitute a proper subset of lattice conditional independence (LCI) models. It follows that like LCI models, for the TOCI models the likelihood function and parameter space can be factored into the products of conditional likelihood functions and disjoint parameter spaces, respectively, where each conditional likelihood function corresponds to an ordinary multivariate normal regression model.

• 48.
Department of Space Physics‎, ‎Institute of Geophysics‎, ‎University of Tehran, Tehran, Iran.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
High-order compact finite difference schemes for the spherical shallow water equations2013Report (Other academic)

This work is devoted to the application of the super compact finite difference (SCFDM) and the combined compact finite difference (CCFDM) methods for spatial differencing of the spherical shallow water equations in terms of vorticity, divergence and height. Five high-order schemes including the fourth-order compact, the sixth-order and eighth-order SCFDM and the sixth-order and eighth-order CCFDM schemes are used for spatial differencing of the spherical shallow water equations. To advance the solution in time, a semi-implicit Runge-Kutta method is used. In addition, to control the nonlinear instability and avoiding the polar problem a high-order spatial filter is proposed. An unstable barotropic mid-latitude zonal jet is employed as an initial condition. For the numerical solution of the elliptic equations in the problem, a direct hybrid method which consists of using a high-order compact scheme for spatial differencing in the latitude coordinate and a fast Fourier transform in longitude coordinate is utilized. The convergence rate for all methods is studied and veried. Qualitative and quantitative assessment of the results, such as measures of maximum vorticity gradient, power spectrum of total energy, relative change in potential enstrophy and potential palinstrophy, reveal that the sixth-order and eighth-order CCFDM and the sixth-order and eighth-order SCFDM methods lead to a remarkable improvement of the solution over the fourth-order compact method. It is also shown that the performance of the sixth-order and eighth-order CCFDM methods are superior to the sixth-order and eighth-order SCFDM methods. Copyright c 2013 John Wiley & Sons, Ltd.

• 49.
Department of Space Physics, Institute of Geophysics, University of Tehran, Tehran, Iran.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, The Institute of Technology.
Well-posed boundary conditions for the shallow water equations2013Report (Other academic)

We derive well-posed boundary conditions for the two-dimensional shallow water equations by using the energy method. Both the number and the type of boundary conditions are presented for subcritical and supercritical flows on a general domain. Then, as an example, the boundary conditions are discussed for a rectangular domain.

• 50.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Coupling requirements for multi-physics problems2016Report (Other academic)

We consider two hyperbolic systems onfirst order form of different size posed on two domains. Our ambition is to derive general conditions for when the two systems can and cannot be coupled.

The adjoint equations are derived and well-posedness of the primal and dual problems are discussed. By applying the energy method, interface conditions for the primal and dual problems are derived such that the continuous problems are well posed.

The equations are discretized using a high order finite difference method on summation-by-parts form and the interface conditions are imposed weakly in a stable way, using penalty formulations. It is shown that one specific choice of penalty matrices leads to a dual consistent scheme and superconverging functionals.

By considering an example, it is shown that the correct physical coupling conditions are contained in the set of well posed coupling conditions. It is also shown that the correct convergence rates are obtained for both the solutions and functionals.

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