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• 1.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, University of Dar Es Salaam Tanzania.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Approximation of misclassification probabilities in linear discriminant analysis with repeated measurements2020Rapport (Övrigt vetenskapligt)

In this paper, we propose approximations for the probabilities of misclassification in linear discriminant analysis when means follow a growth curve structure. The discriminant function can classify a new observation vector of p repeated measurements into one of two multivariate normal populations with equal covariance matrix. We derive certain relations of the statistics under consideration in order to obtain approximations of the misclassification errors. Finally, we perform Monte Carlo simulations to evaluate the performance of proposed results.

fulltext
• 2.
Exploiting parallelization and synergy in derivative free optimization2020Rapport (Övrigt vetenskapligt)

Real life optimization often concerns difficult objective functions, in two aspects, namely that gradients are unavailable, and that evaluation of the objective function takes a long time. Such problems are often attacked with model building algorithms, where an approximation of the function is constructed and solved, in order to find a new promising point to evaluate. We study several ways of saving time by using parallel calculations in the context of model building algorithms, which is not trivial, since such algorithms are inherently sequential. We present a number of ideas that has been implemented and tested on a large number of known test functions, and a few new ones. The computational results reveal that some ideas are quite promising.

fulltext
• 3.
Department of Mechanical Engineering and Institute for Computational Mathematical Engineering, Stanford University, Stanford, California, USA. Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
Learning to Differentiate2020Rapport (Övrigt vetenskapligt)

Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-byparts known as summation-by-parts. Furthermore we demonstrate the benefits of building the summation-by-parts property into the network by weight restriction, rather than enforcing it through a regularizer. We conclude that, if possible, known structural elements of an operation are best implemented as innate—rather than learned—properties of the network. The strategy developed in this work also opens the door for constructing stable differential operators on general meshes.

Learning to Differentiate
• 4.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, University of Dar Es Salaam, Tanzania.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Linear discriminant analysis via the Growth Curve model and restrictions on the mean space2020Rapport (Övrigt vetenskapligt)

A linear classification function is applied when the means follow a Growth Curve model with restriction on the mean space. If the underlying assumption is that different groups in the experimental design follow different growth proles, a bilinear restriction on the mean space gives an Extended Growth Curve model. Given this restriction the approximations for the probability of misclassifications are derived. Moreover, a discriminant function is also derived when there exist rank restrictions on the mean parameters.

fulltext
• 5.
Optimal proportional representation2020Rapport (Övrigt vetenskapligt)

In a democratic proportional election system, it is vital that the mandates in the parliament are allocated as proportionally as possible to the number of votes the parties got in the election. We formulate an optimization model for allocation of seats in a parliament so as to minimize the disproportionality. By applying separable programming techniques, we obtain an easily solvable problem, and present a method for solving it optimally. The obtained solution is the feasible solution that has the minimal disproportionality (with the measure chosen), even in the presence of a parliament threshold, which is not always the case for the practical procedures used in many countries. We apply the approach to real life data from the last three elections in Sweden, and show that the result is better, i.e. more proportional, than what was obtained with the modified Sainte-Laguë method, which is presently used. A natural suggestion would be to use our method instead.

We also consider the issue about constituencies, and suggest a procedure, based on the same kind of optimization problem, for allocating mandates in the constituencies, without changing the overall allocation with respect to parties. The numbers of mandates for the constituencies are based on the number of votes given, not on estimated numbers of inhabitants entitled to vote. This removes the need for compensatory mandates, and makes the question about sizes of the constituencies less important.

fulltext
• 6.
Department of Mathematics, Southern Methodist University, Dallas, TX 75275 USA.
The number of boundary conditions for initial boundary value problems2020Rapport (Övrigt vetenskapligt)

Both the energy method and the Laplace transform method are frequently used for determining the number of boundary conditions required for a well posed initial boundary value problem.We show that these two distinctly dierent methods yield the same results.

The Number of Boundary Conditions for Initial Boundary Value Problems
• 7.
A matheuristic approach to large-scale avionic scheduling2019Rapport (Övrigt vetenskapligt)

Pre-runtime scheduling of avionic systems is used to ensure that the systems provide the desired functionality at the correct time. This paper considers scheduling of an integrated modular avionic system which from a more general perspective can be seen as a multiprocessor scheduling problem that includes a communication network. The addressed system is practically relevant and the computational evaluations are made on large-scale instances developed together with the industrial partner Saab. A subset of the instances is made publicly available.

Our contribution is a matheuristic for solving these large-scale instances and it is obtained by improving the model formulations used in a previously suggested constraint generation procedure and by including an adaptive large neighbourhood search to extend it into a matheuristic. Characteristics of our adaptive large neighbourhood search are that it is made over both discrete and continuous variables and that it needs to balance the search for feasibility and profitable objective value. The repair operation is to apply a mixed-integer programming solver on a model where most of the constraints are treated as soft and a violation of them is instead penalised in the objective function. The largest solved instance, with respect to the number of tasks, has 45 988 tasks and 2 011 communication messages.

• 8.
A new method for optimal proportional representation2019Rapport (Övrigt vetenskapligt)

In a democratic proportional election system, it is vital that the mandates in the parliament are allocated as proportionally as possible to the number of votes the parties got in the election. We formulate an optimization model for allocation of seats in a parliament so as to minimize the disproportionality. By applying separable programming techniques, we obtain an easily solvable problem, and present a method for solving it optimally. The obtained solution is thus the feasible solution that has the minimal disproportionality (with the measure chosen), in contrast to the heuristic procedures used in many countries. We apply the approach to real life data from the last three elections in Sweden, and show that the result is better, i.e. more proportional, than what was obtained with the “adjusted odd number rule”, which is presently used. A natural suggestion would be to use our method instead.

We also consider the issue about constituencies, and suggest a procedure, based on the same kind of optimization problem, for allocating mandates in the constituencies, without changing the overall allocation with respect to parties. In our approach, the numbers of mandates for the constituencies are based on the number of votes given, not on estimated numbers of inhabitants. This removes the need for fixed and equalization mandates, and also makes the question about sizes of the constituencies less important.

A new method for optimal proportional representation
presentationsbild
• 9.
An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations2019Rapport (Övrigt vetenskapligt)

The coupling of the compressible and incompressible Navier-Stokes equations is considered. Our ambition is to take a first step towards a provably well posed and stable coupling procedure. We study a simplified setting with a stationary planar interface and small disturbances from a steady background ow with zero velocity normal to the interface.

The simplified setting motivates the use of the linearized equations, and we derive interface conditions such that the continuous problem satisfy an energy estimate. The interface conditions can be imposed both strongly and weakly. It is shown that the weak and strong interface imposition produce similar continuous energy estimates.

We discretize the problem in time and space by employing finite difference operators that satisfy a summation-by-parts rule. The interface and initial conditions are imposed weakly using a penalty formulation. It is shown that the results obtained for the weak interface conditions in the continuous case, lead directly to stability of the fully discrete problem.

An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations
• 10.
An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations2019Rapport (Övrigt vetenskapligt)

The coupling of the compressible and incompressible Navier-Stokes equations is considered. Our ambition is to take a first step towards a provably well posed and stable coupling procedure. We study a simplified setting with a stationary planar interface and small disturbances from a steady background ow with zero velocity normal to the interface.

The simplified setting motivates the use of the linearized equations, and we derive interface conditions such that the continuous problem satisfy an energy estimate. The interface conditions can be imposed both strongly and weakly. It is shown that the weak and strong interface imposition produce similar continuous energy estimates.

We discretize the problem in time and space by employing finite difference operators that satisfy a summation-by-parts rule. The interface and initial conditions are imposed weakly using a penalty formulation. It is shown that the results obtained for the weak interface conditions in the continuous case, lead directly to stability of the fully discrete problem.

An Energy Stable Coupling Procedure for the Compressible and Incompressible Navier-Stokes Equations
• 11.
Facultyof, Mechanica, Engineering,Technion, Israel Institute of Technology, Haifa, Israel.
Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten. Facultyof, Mechanica, Engineering,Technion, Israel Institute of Technology, Haifa, Israel.
Convergence and stability properties of summation-by-parts in time2019Rapport (Övrigt vetenskapligt)

We extend the list of stability properties satisfied by Summation-By-Parts (SBP) in time to include strong S-stability, dissipative stability and stiff accuracy. Further, it is shown that SBP in time is B-convergent for strictly contractive non-linear problems and weakly convergent for non-linear problems that are both contractive and dissipative

Convergence and stability properties of summation-by-parts in time
• 12.
Dual Time-Stepping Using Second Derivatives2019Rapport (Övrigt vetenskapligt)

We present a modied formulation of the dual time-stepping technique which makes use of two derivatives in pseudo-time. This new technique retains and improves the convergence properties to the stationary solution. When compared with the conventional dual time-stepping, the method with two derivatives reduces the stiness of the problem and requires fewer iterations for full convergence to steady-state. In the current formulation, these positive eects require that an approximation of the square root of the spatial operator is available and inexpensive.

Dual Time-Stepping Using Second Derivatives
• 13.
Eigenvalue analysis for summation-by-parts finite difference time discretizations2019Rapport (Övrigt vetenskapligt)

Diagonal norm finite-difference based time integration methods in summation-by-parts form are investigated. The second, fourth and sixth order accurate discretizations are proven to have eigenvalues with strictly positive real parts. This leads to provably invertible fully-discrete approximations of initial boundary value problems.

Our findings also allow us to conclude that the second, fourth and sixth order time discretizations are stiffly accurate, strongly S-stable and dissipatively stable Runge-Kutta methods. The procedure outlined in this article can be extended to even higher order summation-by-parts approximations with repeating stencil.

Eigenvalue analysis for summation-by-parts finite difference time discretizations
• 14.
Energy versus entropy estimates for nonlinear hyperbolic systems of equations2019Rapport (Övrigt vetenskapligt)

We compare and contrast information provided by the energy analysis of Kreiss and the entropy theory of Tadmor for systems of nonlinear hyperbolic conservation laws. The two-dimensional nonlinear shallow water equations are used to highlight the similarities and differences since the total energy of the system is a mathematical entropy function. We demonstrate that the classical energy method is consistent with the entropy analysis, but significantly more fundamental as it guides proper boundary treatments. In particular, the energy analysis provides information on what type of and how many boundary conditions are required, which is lacking in the entropy analysis. For the shallow water system we determine the number and the type of boundary conditions needed for subcritical and supercritical flows on a general domain. As eigenvalues are augmented in the nonlinear analysis, we find that a flow may be classified as subcritical, but the treatment of the boundary resembles that of a supercritical flow. Because of this, we show that the nonlinear energy analysis leads to a different number of boundary conditions compared with the linear energy analysis. We also demonstrate that the entropy estimate leads to erroneous boundary treatments by over specifying and/or under specifying boundary data causing the loss of existence and/or energy bound, respectively. Our analysis reveals that the nonlinear energy analysis is the only one that provides an estimate for open boundaries. Both the entropy and linear energy analysis fail.

Energy versus entropy estimates for nonlinear hyperbolic systems of equations
• 15.
Universitat Autònoma de Barcelona, Bellaterra, Spain.
Universitat Autònoma de Barcelona, Bellaterra, Spain. Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. Helsingin Yliopisto, Helsinki, Finland. Oxford Brookes University, Oxford, United Kingdom. Universitat Autònoma de Barcelona, Bellaterra, Spain.
Fundamental Mathematical Knowledge: progressing its specification2019Rapport (Övrigt vetenskapligt)

In this paper, we elaborate on the notion of Fundamental Mathematical Knowledge (FMK) which we understand as the minimum mathematical content knowledge essential to enter a degree in primary teacher education. We propose that FMK can be assessed in terms of competency in different areas of primary mathematics. The aim of this paper is to present our on-going work to specify FMK based on the common content of mathematics as a primary school subject in Catalonia, England, Finland, and Sweden.

Fundamental Mathematical Knowledge: progressing itsspecification
• 16.
On conservation and dual consistency for summation-by-parts based approximations of parabolic problems2019Rapport (Övrigt vetenskapligt)

We consider the coupling of parabolic problems discretized using difference operators on summation-by-parts (SBP) form with interface conditions imposed weakly. In, it was shown that conservation and dual consistency are equivalent concepts for linear conservation laws. Here, we show that these concepts are equivalent also for symmetric or symmetrizable parabolic problems, exemplified by the heat equation. We rewrite the heat equation as first order system as in the local discontinuous Galerkin method and show the equivalence of dual consistency and conservation for both the first and second order forms.

On conservation and dual consistency for summation-by-parts based approximations of parabolic problems
• 17. Köp publikationen >>
Graduate School of Science, Hiroshima University, Japan.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Sweden.
On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix2019Rapport (Övrigt vetenskapligt)

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. We complement the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart  matrix are close to the degrees of freedom of the distribution.

On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix
• 18.
The Relation Between Primal and Dual Boundary Conditions for Hyperbolic Systems of Equations2019Rapport (Övrigt vetenskapligt)

In this paper we study boundary conditions for linear hyperbolic systems of equations and the corresponding dual problems. In particular, we show that the primal and dual boundary conditions are related by a simple scaling relation. It is also shown that the weak dual problem can be derived directly from the weak primal problem.

Based on the continuous analysis, we discretize and perform computations with a high-order finite difference scheme on summation-by-parts form with weak boundary conditions. It is shown that the results obtained in the continuous analysis lead directly to stability results for the primal and dual discrete problems. Numerical experiments corroborate the theoretical results.

The Relation Between Primal and Dual Boundary Conditions for Hyperbolic Systems of Equations
• 19.
Linköpings universitet, Institutionen för ekonomisk och industriell utveckling, Mekanisk värmeteori och strömningslära. Linköpings universitet, Tekniska fakulteten. Linköpings universitet, Centrum för medicinsk bildvetenskap och visualisering, CMIV. Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. St Petersburg State University, Russia; St Petersburg State Polytech University, Russia.
A Modification to the Kirchhoff Conditions at a Bifurcation and Loss Coefficients2018Rapport (Övrigt vetenskapligt)

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.

fulltext
• 20.
Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten. St Petersburg State University, Russia; St Petersburg State Polytech University, Russia. Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
A Two Dimensional Model of the Thin Laminar Wall of a Curvilinear Flexible Pipe2018Rapport (Övrigt vetenskapligt)

We present a two dimensional model describing the elastic behaviour of the wall of a curved pipe to model blood vessels in particular. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the vessel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with the surrounding material and the fluid flowing inside into account and is obtained via a dimension reduction procedure. The curvature and twist of the vessel axis as well as the anisotropy of the laminate wallpresent the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of vessels and their walls are supplied with explicit systems of dierential equations at the end.

fulltext
• 21.
An efficient hybrid method for uncertainty quantification2018Rapport (Övrigt vetenskapligt)

A technique for coupling an intrusive and non-intrusive uncertainty quantification method is proposed. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. A strongly stable coupling procedure between the two methods at an interface is constructed. The efficiency of the hybrid method is exemplified using a hyperbolic system of equations, and verified by numerical experiments.

An efficient hybrid method for uncertainty quantification
• 22.
Department of Statistics, Stockholm University.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences.
Bilinear regression with random effects and reduced rank restrictions2018Rapport (Övrigt vetenskapligt)

Bilinear models with three types of effects are considered: fixed effects, random effects and latent variable effects. Explicit estimators are proposed.

Bilinear regression with random effects and reduced rank restrictions
• 23.
Department of Statistics, Stockholm University.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences.
Bilinear regression with rank restrictions on the mean and the dispersion matrix2018Rapport (Övrigt vetenskapligt)

A bilinear regression model with rank restrictions imposed on the mean-parameter matrix and on the dispersion matrix is studied. Maximum likelihood inspired estimates are derived. The approach generalizes classical reduced rank regression analysis and principal component analysis. It is shown via a simulation study and a real example that even for small dimensions the method works as well as reduced rank regression analysis whereas the approach in this article also can be used when the dimension is large.

fulltext
• 24.
Indian Statistical Institute, Bangalore, India.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Conditional Independence Models which are Totally Ordered2018Rapport (Övrigt vetenskapligt)

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.

fulltext
• 25.
Finite difference schemes with transferable interfaces for parabolic problems2018Rapport (Övrigt vetenskapligt)

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.

Finite difference schemes with transferable interfaces for parabolic problems
• 26.
Prepared Test Instances Extracted from OpenStreetMapData Using Different Network Reductions2018Rapport (Övrigt vetenskapligt)

We investigate the effect of different reductions when importing networksfrom OpenStreetMap data. We describe the network reductions and report computationaltests for doing the network extraction and reduction. We also show the effect ofthe reductions by solving a few standard optimization problems in the resulting networks.Computational tests show that the reductions have a dramatic effect on thenetwork size and the time needed for solving the optimization problems. In many cases,the reductions are necessary in order to be able to solve the optimization problem inreasonable time. A practical result of this work is a set of networks that will be usedas benchmarks in future research, and are publically available for other researchers.

fulltext
• 27.
St Petersburg State University, Russia; St. Petersburg State Polytechnical University, St. Petersburg, Russia. St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia.
Pressure drop matrix for a bifuration of an artery with defects2018Rapport (Övrigt vetenskapligt)

We consider a bifurcation of an artery. The influence of defects of the vessel's wall near the bifurcation point on the pressure drop matrix is analyzed. The elements of this matrix are included in the modified Kirchhoff transmission conditions, which were introduced earlier in [1], [2], and which describe adequately the total pressure loss at the bifurcation point of the flow passed through it.

Pressure drop matrix for a bifuration of an artery with defects
• 28.
Stochastic Galerkin Projection and Numerical Integration for Stochastic Investigations of the Viscous Burgers’ Equation2018Rapport (Övrigt vetenskapligt)

We consider a stochastic analysis of the non-linear viscous Burgers’ equation and focus on the comparison between intrusive and non-intrusive uncer- tainty quantification methods. The intrusive approach uses a combination of polynomial chaos and stochastic Galerkin projection. The non-intrusive method uses numerical integration by combining quadrature rules and the probability density functions of the prescribed uncertainties. The two methods are applied to a provably stable formulation of the viscous Burgers’ equation, and compared. As measures of comparison: variance size, computational efficiency and accuracy are used.

Stochastic Galerkin Projection and Numerical Integration for Stochastic Investigations of the Viscous Burgers’ Equation
• 29.
Trapped modes in armchair graphene nanoribbons2018Rapport (Övrigt vetenskapligt)

We study scattering on an ultra-low potential in armchair graphene nanorib bon. Using the continuous Dirac model and including a couple of articial waves in the scattering process, described by an augumented scattering matrix, we derive a condition for the existence of a trapped mode. We consider the threshold energies, where the the multiplicity of the continuous spectrum changes and show that a trapped mode may appear for energies slightly less than a thresold and its multiplicity does not exceed one. For energies which are higher than a threshold, there are no trapped modes, provided that the potential is suciently small.

Publikationen är tillgänglig i fulltext från 2020-07-01 08:00
• 30.
A new multigrid formulation for high order finite difference methods on summation-by-parts form2017Rapport (Övrigt vetenskapligt)

Multigrid schemes for high order finite difference methods on summation-by-parts form are studied by comparing the effect of different interpolation operators. By using the standard linear prolongation and restriction operators, the Galerkin condition leads to inaccurate coarse grid discretizations. In this paper, an alternative class of interpolation operators that bypass this issue and preserve the summation-by-parts property on each grid level is considered. Clear improvements of the convergence rate for relevant model problems are achieved.

A New Multigrid Formulation for High Order Finite Difference Methods on Summation-by-Parts Form
• 31.
A Note on the Rechargeable Polya Urn Scheme2017Rapport (Övrigt vetenskapligt)

A very simple specific case of a Polya urn scheme is as follows. At each trial one draws a ball from an urn with balls of two different colours. Then, one looks at the ball, and returns the ball to the urn together with another ball of the same colour. Then one makes another draw. Et cetera. At the first draw there is one ball of each colour.

The rechargeable Polya urn scheme is essentially the same except that between each draw there is a fixed probability that the process starts over with two balls in the urn having different colours.

Now, for n = 1,2, ..., let B(n) and G(n) denote respectively the number of blue and yellow balls in the urn and let Y(n) denote the colour of the ball drawn at the nth draw. Further let Z(n) denote the probability distribution of (B(n),G(n)) given that we have observed Y(m), from m = 1 to m = n. In this note we prove that the sequence Z(1),Z(2),.... converges in distribution.

A Note on the Rechargeable Polya Urn Scheme
• 32.
An analysis of non-conforming grid techniques for high order summation-by-parts metods2017Rapport (Övrigt vetenskapligt)

We derive a bound on the order of accuracy of interpolation operators for energy stable summation-by-parts discretizations on non-conforming multiblock meshes. The new theoretical result, which corroborate with experience from previous work, implies a local reduction in the formal accuracy of summation-by-parts discretizations based on diagonal norms. Numerical results confirm a corresponding reduction in convergence rate in both the maximum norm and the discrete L2 norm for a hyperbolic model problem.

An analysis of non-conforming grid techniques for high order summation-by-parts metods
• 33.
Saab AB, Sweden.
An Optimisation Approach for Pre-Runtime Scheduling of Tasks and Communication in an Integrated Modular Avionic System2017Rapport (Övrigt vetenskapligt)

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.

fulltext
• 34.
Energy Stable Boundary Conditions for the Nonlinear Incompressible Navier-Stokes Equations2017Rapport (Övrigt vetenskapligt)

The nonlinear incompressible Navier-Stokes equations with boundary conditions at far fields and solid walls is considered. Two different formulations of boundary conditions are derived using the energy method. Both formulations are implemented in both strong and weak form and lead to an estimate of the velocity field. Equipped with energy bounding boundary conditions, the problem is approximated by using difference operators on summation-by-parts form and weak boundary and initial conditions. By mimicking the continuous analysis, the resulting semi-discrete as well as fully discrete scheme are shown to be provably stable, divergence free and high-order accurate.

fulltext
• 35.
Long time error bounds for the wave equation on second order form2017Rapport (Övrigt vetenskapligt)

Temporal error bounds for the wave equation expressed on second order form is investigated. By using the energy method, we show that, with appropriate choices of boundary condition, the time and space derivative of the error is bounded even for long times. No long time bound on the actual error can be obtained, although numerical experiments indicate that such a bound exist.

Long time error bounds for the wave equation on second order form
• 36.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, College of Science and Technology, University of Rwanda, Kigali, Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden.
Mean-squared errors of small area estimators under a multivariate linear model for repeated measures data2017Rapport (Övrigt vetenskapligt)

In this paper, we discuss the derivation of the first and second moments for the proposed small area estimators under a multivariate linear model for repeated measures data. The aim is to use these moments to estimate the mean-squared errors (MSE) for the predicted small area means as a measure of precision. A two stage estimator of MSE is obtained. At the first stage, we derive the MSE when the covariance matrices are known. To obtain an unbiased estimator of the MSE, at the second stage, a method based on parametric bootstrap is  proposed for bias correction and for prediction error that reects the uncertainty when the unknown covariance is replaced by its suitable estimator.

Mean-Squared errors of small area estimators under a multivariate linear model for repeated measures data
• 37.
More on Estimation of Banded and Banded Toeplitz Covariance Matrices2017Rapport (Övrigt vetenskapligt)

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.

fulltext
• 38.
Dept. of Mathematics, University of Bergen, Bergen, Norway.
ON THE CONVERGENCE RATES OF ENERGY-STABLE FINITE-DIFFERENCE SCHEMES2017Rapport (Övrigt vetenskapligt)
fulltext
• 39.
University of Cape Town, South Africa. Linköpings universitet, Matematiska institutionen, Beräkningsmatematik. Linköpings universitet, Tekniska fakulteten.
On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-By-Parts Form2017Rapport (Övrigt vetenskapligt)

In this paper we generalise results regarding the order of accuracy of finite difference operators on Summation-By-Parts (SBP) form, previously known to hold on uniform grids, to grids with arbitrary point distributions near domain boundaries. We give a definite proof that the order of accuracy in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. Additionally, we prove that if the order of accuracy in the interior is precisely twice that of the boundary, then the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid point distribution near the domain boundaries.

On the order of Accuracy of Finite Difference Operators on Diagonal Norm Based Summation-By-Parts Form
• 40.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, College of Science and Technology, University of Rwanda, Kigali, Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences, Uppsala, Sweden. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Small area estimation under a multivariate linear model for incomplete repeated measures data2017Rapport (Övrigt vetenskapligt)

In this paper, the issue of analysis of multivariate repeated measures data that follow a monotonic sample pattern for small area estimation is addressed. Random effects growth curve models with covariates for both complete and incomplete data are formulated. A conditional likelihood based approach is proposed for estimation of the mean parameters and covariances. Further, the prediction of random effects and predicted small area means are also discussed. The proposed techniques may be useful for small area estimation under longitudinal surveys with grouped response units and drop outs.

Small area estimation under a multivariate linear model for incomplete repeated measures data
• 41.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Mathematics, College of Science and Technology, University of Rwanda.
Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten. Department of Energy and Technology, Swedish University of Agricultural Sciences. Linköpings universitet, Matematiska institutionen, Matematisk statistik. Linköpings universitet, Tekniska fakulteten.
Small area estimation with missing data using a multivariate linear random effects model2017Rapport (Övrigt vetenskapligt)

In this article small area estimation with multivariate data that follow a monotonic missing sample pattern is addressed. Random effects growth curve models with covariates are formulated. A likelihood based approach is proposed for estimation of the unknown  parameters. Moreover, the prediction of random effects and predicted small area means are also discussed.

Small area estimation with missing data using a multivariate linear random effects model
• 42.
Spectral analysis of the incompressible Navier-Stokes equations with different boundary conditions2017Rapport (Övrigt vetenskapligt)

The influence of boundary conditions on the spectrum of the incompressible Navier-Stokes equations is studied. The spectra associated to different types of boundary conditions are derived using the Fourier-Laplace technique. In particular, the effect of various combinations of generalized in- and outgoing variables on the convergence to the steady state is investigated. The boundary conditions are analysed in both the continuous and semi-discrete problems. In the latter, high-order schemes in summation-by-parts form with weakly imposed boundary conditions are used to approximate the equations. Numerical calculations are performed and show that the discrete behaviour agrees with the theoretical analysis.

fulltext
• 43.
Weather derivatives pricing using regim switching models2017Rapport (Övrigt vetenskapligt)

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.

fulltext
• 44.
Well-posed and Stable Transmission Problems2017Rapport (Övrigt vetenskapligt)

We introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability is analysed for continuous and discrete problems using both strong and weak formulations, and a general transmission condition is obtained. The theory is applied to several examples including the coupling of fluid flow models, multi-grid implementations, multi-block formulations and numerical filtering.

Well-posed and Stable Transmission Problems
• 45.
A dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains2016Rapport (Övrigt vetenskapligt)

In this article, well-posedness and dual consistency of the linearized incompressible Navier-Stokes equations posed on time-dependent spatial domains are studied. To simplify the derivation of the dual problem, the second order formulation is transformed to rst order form. Boundary conditions that simultaneously lead to well-posedness of the primal and dual problems are derived.

We construct fully discrete nite di erence schemes on summation-byparts form, in combination with the simultaneous approximation technique. We prove energy stability and discrete dual consistency. Moreover, we show how to construct the penalty operators such that the scheme automatically adjusts to the variations of the spatial domain, and as a result, stability and discrete dual consistency follow simultaneously.

The method is illustrated by considering a deforming time-dependent spatial domain in two dimensions. The numerical calculations are performed using high order operators in space and time. The results corroborate the stability of the scheme and the accuracy of the solution. We also show that linear functionals are superconverging. Additionally, we investigate the convergence of non-linear functionals and the divergence of the solution.

Dual consistent summation-by-parts formulation for the linearized incompressible Navier-Stokes equations posed on deforming domains
• 46.
A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces2016Rapport (Övrigt vetenskapligt)

We introduce an interface/coupling procedure for hyperbolic problems posedon time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions.

Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected.

The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.

Version 1. Fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces
• 47.
A provably stable, non-iterative domain decomposition technique for the advection-diffusion equation2016Rapport (Övrigt vetenskapligt)

We describe an efficient, non-iterative domain decomposition approach for the onedimensional advection–diffusion equation based on the Summation-by-Parts technique in both time and space. A fully discrete multidomain analogue of the continuous equation is formulated and a linear system consisting only of the solution components involved in the coupling between the subdomain interfaces is isolated. Once the coupling system is solved, the full solution is found by computing linear combinations of known vectors, weighted by the coupling components. Both stability and invertibility of the discrete scheme is proved using standard Summation-by-Parts procedures.

In a numerical study we show that perfunctory implementations of monodomain Summation-by-Parts based time integration can be improved upon significantly. Using our proposed method we are able to reduce execution time and memory footprint by up to 80% and 95% respectively. Similar improvements in execution time is shown also when compared against explicit Runge–Kutta time integration.

Provably Stable, Non-iterative Domain Decomposition Technique for the Advection-Diffusion Equation
• 48.
A Smoothed Monotonic Regression via L2 Regularization2016Rapport (Övrigt vetenskapligt)

Monotonic Regression (MR) is a standard method for extracting a monotone function from non-monotonic data, and it is used in many applications. However, a known drawback of this method is that its fitted response is a piecewise constant function, while practical response functions are often required to be continuous. The method proposed in this paper achieves monotonicity and smoothness of the regression by introducing an L2 regularization term, and it is shown that the complexity of this method is O(n2). In addition, our simulations demonstrate that the proposed method normally has higher predictive power than some commonly used alternative methods, such as monotonic kernel smoothers. In contrast to these methods, our approach is probabilistically motivated and has connections to Bayesian modeling.

Smoothed Monotonic Regression via L2 Regularization
• 49.
A stable and accurate data assmimilation technique using multiple penalty terms in space and time2016Rapport (Övrigt vetenskapligt)

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.

A stable and accurate data assmimilation technique using multiple penalty terms in space and time
• 50.
An Energy Stable Summation-by-parts Formulation for General Multi-block and Hybrid Meshes2016Rapport (Övrigt vetenskapligt)

Most high order methods for solving conservation laws can be shown to satisfy a summation-by-parts rule. In this work we present a general framework for multi-block and multi-element summation-by-parts implementations in several dimensions that includes most, if not all of the previously known schemes on summation-by-parts form. This includes finite volume, spectral and nodal discontinuous Galerkin methods, as well as high order multi-block finite difference schemes on curvilinear domains. We use the framework to derive general conditions for conservation and stability, and formulate extended representations of conservative and energy stable couplings between completely general multi-block, multi-element or hybrid meshes.

An Energy Stable Summation-by-parts Formulation for General Multi-block and Hybrid Meshes
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