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• 151.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem2006In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 44, no 1, p. 1-15Article in journal (Refereed)

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The singular set of higher dimensional unstable obstacle type problems2013In: Rendiconti Lincei - Matematica e Applicazioni, ISSN 1120-6330, E-ISSN 1720-0768, Vol. 24, no 1, p. 123-146Article in journal (Refereed)
• 153.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Global solutions of the obstacle problem in half-spaces, and their impact on local stability2005In: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 23, no 3, p. 271-279Article in journal (Refereed)

We show that there are an abundance of non-homogeneous global solutions to the obstacle problem, in the half-space, Delta u = X-{u > 0}, u >= 0 inR(+)(2), with a (fixed) homogeneous boundary condition u(0, x(2)) = lambda(2) (x(2)(+))(2) (0 < lambda < 1/root 2) As a consequence we obtain local instability of the free boundary under C-1,C-1 perturbation, of the Dirichlet data.

• 154.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Equilibrium points of a singular cooperative system with free boundary2015In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 280, p. 743-771Article in journal (Refereed)

In this paper we initiate the study of maps minimising the energy integral(D)(vertical bar del u vertical bar(2) + 2 vertical bar u vertical bar) dx, which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations Delta u = u / vertical bar u vertical bar chi({vertical bar u vertical bar >0}), u = (u(1,) ... ,u(m)) Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Double Obstacle Problems with Obstacles Given by Non-C-2 Hamilton-Jacobi Equations2012In: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 206, no 3, p. 779-819Article in journal (Refereed)

We prove optimal regularity for double obstacle problems when obstacles are given by solutions to Hamilton-Jacobi equations that are not C (2). When the Hamilton-Jacobi equation is not C (2) then the standard Bernstein technique fails and we lose the usual semi-concavity estimates. Using a non-homogeneous scaling (different speeds in different directions) we develop a new pointwise regularity theory for Hamilton-Jacobi equations at points where the solution touches the obstacle. A consequence of our result is that C (1)-solutions to the Hamilton-Jacobi equation <Equation ID="Equa"> <MediaObject> </MediaObject> </Equation>, are, in fact, C (1,alpha/2), provided that . This result is optimal and, to the authors' best knowledge, new.

• 156.
Mathematics Institute, University of Warwick.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). Mathematical Institute of the Heinrich Heine University.
On the singularities of a free boundary through Fourier expansion2012In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 187, no 3, p. 535-587Article in journal (Refereed)

In this paper we are concerned with singular points of solutions to the unstable free boundary problem Delta u = -chi({u>0}) in B-1. The problem arises in applications such as solid combustion, composite membranes, climatology and fluid dynamics. It is known that solutions to the above problem may exhibit singularities-that is points at which the second derivatives of the solution are unbounded-as well as degenerate points. This causes breakdown of by-now classical techniques. Here we introduce new ideas based on Fourier expansion of the non-linearity chi({u>0}). The method turns out to have enough momentum to accomplish a complete description of the structure of the singular set in R-3. A surprising fact in R-3 is that although u(rx)/sup(B1) vertical bar u(rx)vertical bar can converge at singularities to each of the harmonic polynomials xy, x(2) + y(2)/2 - z(2) and z(2) - x(2) + y(2)/2, it may not converge to any of the non-axially-symmetric harmonic polynomials alpha((1 + delta)x(2) + (1 - delta)y(2) - 2z(2)) with delta not equal 1/2. We also prove the existence of stable singularities in R-3.

• 157.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
A NEW TENSORIAL CONSERVATION LAW FOR MAXWELL FIELDS ON THE KERR BACKGROUND2017In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 105, no 2, p. 163-176Article in journal (Refereed)

A new, conserved, symmetric tensor field for a source-free Maxwell test field on a four-dimensional spacetime with a conformal Killing-Yano tensor, satisfying a certain compatibility condition, is introduced. In particular, this construction works for the Kerr spacetime.

• 158.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Max Planck Institute for Gravitational Physics, United Kingdom.
Hidden symmetries and decay for the wave equation on the Kerr spacetime2015In: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 182, no 3, p. 787-853Article in journal (Refereed)

Energy and decay estimates for the wave equation on the exterior region of slowly rotating Kerr spacetimes are proved. The method used is a generalisation of the vector-field method that allows the use of higher-order symmetry operators. In particular, our method makes use of the second-order Carter operator, which is a hidden symmetry in the sense that it does not correspond to a Killing symmetry of the spacetime.

KTH, Superseded Departments, Mathematics.
scalar curvature rigidity for asymptotically locally hyperbolic manifolds1998In: Annals of Global Analysis and Geometry, ISSN 0232-704X, E-ISSN 1572-9060, Vol. 16, p. 1-27Article in journal (Refereed)

Rigidity results for asymptotically locally hyperbolic manifoldswith lower bounds on scalar curvature are proved using spinor methodsrelated to the Witten proof of the positive mass theorem. The argument isbased on a study of the Dirac operator defined with respect to the Killingconnection. The existence of asymptotic Killing spinors is related to thespin structure on the end. The expression for the mass is calculated andproven to vanish for conformally compact Einstein manifolds with conformalboundary a spherical space form, giving rigidity. In the 4-dimensional case,the signature of the manifold is related to the spin structure on the end andexplicit formulas for the relevant invariants are given.

• 160.
KTH, Superseded Departments, Mathematics.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Boundary and lens rigidity of Lorentzian surfaces1996In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 348, p. 2307-2329Article in journal (Refereed)

Let g be a Lorentzian metric on the plane ℝ2 that agrees with the standard metric g0 = -dx2 + dy2 outside a compact set and so that there are no conjugate points along any time-like geodesic of (ℝ2, g). Then (ℝ2, g) and (ℝ2, g0) are isometric. Further, if (M*, g*) and (M*, p*) are two dimensional compact time oriented Lorentzian manifolds with space-like boundaries and so that all time-like geodesies of (M, g) maximize the distances between their points and (M, g) and (M*, g*) are "boundary isometric", then there is a conformal diffeomorphism between (M, g) and (M*, g*) and they have the same areas. Similar results hold in higher dimensions under an extra assumption on the volumes of the manifolds.

• 161.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
Multivariate Financial Time Series and Volatility Models with Applications to Tactical Asset Allocation2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

The financial markets have a complex structure and the modelling techniques have recently been more and more complicated. So for a portfolio manager it is very important to find better and more sophisticated modelling techniques especially after the 2007-2008 banking crisis. The idea in this thesis is to find the connection between the components in macroeconomic environment and portfolios consisting of assets from OMX Stockholm 30 and use these relationships to perform Tactical Asset Allocation (TAA). The more specific aim of the project is to prove that dynamic modelling techniques outperform static models in portfolio theory.

• 162.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
A study of the most important volume drivers for sparkling wine in Sweden2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

This paper examines the factors that affect volume of sparkling wine sold at licensed Swedish liquor stores. Regression analysis is used to model the relationship between the identified parameters believed to have an impact on the volume sold.

Social sustainability is also covered, and will be examined in the context of a deregulated alcohol market in the future. A 5C analysis and an analysis based on both WACOSS Social Sustainability framework and Social Life’s framework have been conducted.

Results from the regression analysis show that the parameters which affected the volume sold fell into three categories: Country, what kind of grape, and price. Most parameters affected the volume sold in a positive way; except for wines from New Zealand and higher priced wine which reduce the volume sold.

This thesis arrives at the conclusion that a regulated alcohol market is favorable in Sweden since the aim is to keep the alcohol consumption low and the overall public health high.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Stockholm School of Economics, Sweden.
Robustness to strategic uncertainty in the Nash demand game2018In: Mathematical Social Sciences, ISSN 0165-4896, E-ISSN 1879-3118, Vol. 91, p. 1-5Article in journal (Refereed)

This paper studies the role of strategic uncertainty in the Nash demand game. A player's uncertainty about another player's strategy is modeled as an atomless probability distribution over that player's strategy set. A strategy profile is robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others (Andersson et al., 2014). In the context of the Nash demand game, we show that robustness to symmetric (asymmetric) strategic uncertainty singles out the (generalized) Nash bargaining solution. The least uncertain party obtains the bigger share.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Robustness to strategic uncertainty2014In: Games and Economic Behavior, ISSN 0899-8256, E-ISSN 1090-2473, Vol. 85, no 1, p. 272-288Article in journal (Refereed)

We introduce a criterion for robustness to strategic uncertainty in games with continuum strategy sets. We model a player's uncertainty about another player's strategy as an atomless probability distribution over that player's strategy set. We call a strategy profile robust to strategic uncertainty if it is the limit, as uncertainty vanishes, of some sequence of strategy profiles in which every player's strategy is optimal under his or her uncertainty about the others. When payoff functions are continuous we show that our criterion is a refinement of Nash equilibrium and we also give sufficient conditions for existence of a robust strategy profile. In addition, we apply the criterion to Bertrand games with convex costs, a class of games with discontinuous payoff functions and a continuum of Nash equilibria. We show that it then selects a unique Nash equilibrium, in agreement with some recent experimental findings.

• 165.
AstraZeneca R and D.
Lund University.
Subspace estimation and prediction methods for hidden Markov models2009In: Annals of Statistics, ISSN 0090-5364, E-ISSN 2168-8966, Vol. 37, no 6B, p. 4131-4152Article in journal (Refereed)

Hidden Markov models (HMMs) are probabilistic functions of finite Markov chains, or, put in other words, state space models with finite state space. In this paper, we examine subspace estimation methods for HMMs whose output lies a finite set as well. In particular, we study the geometric structure arising from the nonminimality of the linear state space representation of HMMs, and consistency of a subspace algorithm arising from a certain factorization of the singular value decomposition of the estimated linear prediction matrix, For this algorithm, we show that the estimates of the transition and emission probability matrices are consistent up to a similarity transformation, and that the in-step linear predictor Computed from the estimated system matrices is consistent, i.e., converges to the true optimal linear m-step predictor.

• 166.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Univ Gothenburg.
Proof of the cosmic no-hair conjecture in the T-3-Gowdy symmetric Einstein-Vlasov setting2016In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 18, no 7, p. 1565-1650Article in journal (Refereed)

The currently preferred models of the universe undergo accelerated expansion induced by dark energy. One model for dark energy is a positive cosmological constant. It is consequently of interest to study Einstein's equations with a positive cosmological constant coupled to matter satisfying the ordinary energy conditions: the dominant energy condition etc. Due to the difficulty of analysing the behaviour of solutions to Einstein's equations in general, it is common to either study situations with symmetry, or to prove stability results. In the present paper, we do both. In fact, we analyse, in detail, the future asymptotic behaviour of T-3-Gowdy symmetric solutions to the Einstein-Vlasov equations with a positive cosmological constant. In particular, we prove the cosmic no-hair conjecture in this setting. However, we also prove that the solutions are future stable (in the class of all solutions). Some of the results hold in a more general setting. In fact, we obtain conclusions concerning the causal structure of T-2-symmetric solutions, assuming only the presence of a positive cosmological constant, matter satisfying various energy conditions and future global existence. Adding the assumption of T-3-Gowdy symmetry to this list of requirements, we obtain C-0-estimates for all but one of the metric components. There is consequently reason to expect that many of the results presented in this paper can be generalised to other types of matter.

• 167.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
Prediction of French day-ahead electricity prices: Comparison between a deterministic and a stochastic approach2015Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis

This thesis deals with the new flow-based computation method used in the Central Western Europe Area. This is done on the financial side. The main aim is to produce some robust methods for predicting. Two approaches are used: the first one is based on a deterministic and algorithmic method involving the study of the interaction between the fundamentals and the prices. The other one is a more statistical approach based on a time series modeling of the French flow-based prices. Both approaches have advantages and disadvantages which will be discussed in the following. The work is mainly based on global simulated data provided by CASC in their implementation phase of the flow-base in Western Europe.

• 168.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
Adaptive node distribution for on-line trajectory planning2006In: ICAS-Secretariat - 25th Congress of the International Council of the Aeronautical Sciences 2006, Curran Associates, Inc., 2006, p. 3150-3157Conference paper (Refereed)

Direct methods for trajectory optimization are traditionally based on a priori temporal dis- cretization and collocation methods. In this work, the problem of node distribution is for- mulated as an optimization problem, which is to be included in the underlying non-linear mathematical programming problem (NLP). The benefits of utilizing the suggested method for on-line trajectory optimization are illustrated by a missile guidance example.

• 169.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
• 170.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
On Cooperative Surveillance, Online Trajectory Planning and Observer Based Control2009Doctoral thesis, comprehensive summary (Other academic)

The main body of this thesis consists of six appended papers. In the  first two, different  cooperative surveillance problems are considered. The second two consider different aspects of the trajectory planning problem, while the last two deal with observer design for mobile robotic and Euler-Lagrange systems respectively.In Papers A and B,  a combinatorial optimization based framework to cooperative surveillance missions using multiple Unmanned Ground Vehicles (UGVs) is proposed. In particular, Paper A  considers the the Minimum Time UGV Surveillance Problem (MTUSP) while Paper B treats the Connectivity Constrained UGV Surveillance Problem (CUSP). The minimum time formulation is the following. Given a set of surveillance UGVs and a polyhedral area, find waypoint-paths for all UGVs such that every point of the area is visible from  a point on a waypoint-path and such that the time for executing the search in parallel is minimized.  The connectivity constrained formulation  extends the MTUSP by additionally requiring the induced information graph to be  kept recurrently connected  at the time instants when the UGVs  perform the surveillance mission.  In these two papers, the NP-hardness of  both these problems are shown and decomposition techniques are proposed that allow us to find an approximative solution efficiently in an algorithmic manner.Paper C addresses the problem of designing a real time, high performance trajectory planner for an aerial vehicle that uses information about terrain and enemy threats, to fly low and avoid radar exposure on the way to a given target. The high-level framework augments Receding Horizon Control (RHC) with a graph based terminal cost that captures the global characteristics of the environment.  An important issue with RHC is to make sure that the greedy, short term optimization does not lead to long term problems, which in our case boils down to two things: not getting into situations where a collision is unavoidable, and making sure that the destination is actually reached. Hence, the main contribution of this paper is to present a trajectory planner with provable safety and task completion properties. Direct methods for trajectory optimization are traditionally based on a priori temporal discretization and collocation methods. In Paper D, the problem of adaptive node distribution is formulated as a constrained optimization problem, which is to be included in the underlying nonlinear mathematical programming problem. The benefits of utilizing the suggested method for  online  trajectory optimization are illustrated by a missile guidance example.In Paper E, the problem of active observer design for an important class of non-uniformly observable systems, namely mobile robotic systems, is considered. The set of feasible configurations and the set of output flow equivalent states are defined. It is shown that the inter-relation between these two sets may serve as the basis for design of active observers. The proposed observer design methodology is illustrated by considering a  unicycle robot model, equipped with a set of range-measuring sensors. Finally, in Paper F, a geometrically intrinsic observer for Euler-Lagrange systems is defined and analyzed. This observer is a generalization of the observer proposed by Aghannan and Rouchon. Their contractivity result is reproduced and complemented  by  a proof  that the region of contraction is infinitely thin. Moreover, assuming a priori bounds on the velocities, convergence of the observer is shown by means of Lyapunov's direct method in the case of configuration manifolds with constant curvature.

• 171.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Online trajectory planning and observer based control2006Licentiate thesis, comprehensive summary (Other scientific)

The main body of this thesis consists of four appended papers. The first two consider different aspects of the trajectory planning problem, while the last two deal with observer design for mobile robotic and Euler-Lagrange systems respectively.

The first paper addresses the problem of designing a real time, high performance trajectory planner for aerial vehicles. The main contribution is two-fold. Firstly, by augmenting a novel safety maneuver at the end of the planned trajectory, this paper extends previous results by having provable safety properties in a 3D setting. Secondly, assuming initial feasibility, the planning method is shown to have finite time task completion. Moreover, in the second part of the paper, the problem of simultaneous arrival of multiple aerial vehicles is considered. By using a time-scale separation principle, one is able to adopt standard Laplacian control to this consensus problem, which is neither unconstrained, nor first order.

Direct methods for trajectory optimization are traditionally based on a priori temporal discretization and collocation methods. In the second paper, the problem of adaptive node distribution is formulated as a constrained optimization problem, which is to be included in the underlying nonlinear mathematical programming problem. The benefits of utilizing the suggested method for online trajectory optimization are illustrated by a missile guidance example.

In the third paper, the problem of active observer design for an important class of non-uniformly observable systems, namely mobile robotics systems, is considered. The set of feasible configurations and the set of output flow equivalent states are defined. It is shown that the inter-relation between these two sets may serve as the basis for design of active observers. The proposed observer design methodology is illustrated by considering a unicycle robot model, equipped with a set of range-measuring sensors.

Finally, in the fourth paper, a geometrically intrinsic observer for Euler-Lagrange systems is defined and analyzed. This observer is a generalization of the observer recently proposed by Aghannan and Rouchon. Their contractivity result is reproduced and complemented by a proof that the region of contraction is infinitely thin. However, assuming a priori bounds on the velocities, convergence of the observer is shown by means of Lyapunov's direct method in the case of configuration manifolds with constant curvature.

• 172.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
Riemannian Observers for Euler-Lagrange Systems2005In: Proceedings of the 16th IFAC World Congress: Prague, Czech Republic, July 3-8, 2005, 2005, p. 115-120Conference paper (Refereed)

In this paper, a geometrically intrinsic observer for Euler-Lagrange systems is defined and analysed. This observer is an generalization of the observer recently proposed by Aghannan and Rouchon. Their contractivity result is reproduced and complemented by a proof that the region of contractivity is infinitely thin. However, assuming a priori bounds on the velocities, convergence of the observer is shown by means of Lyapunov's direct method in the case of configuration manifolds with constant curvature. The convergence properties of the observer are illustrated by an example where the configuration manifold is the three-dimensional sphere, S3.

• 173.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
Active Observers for Mobile Robotic SystemsManuscript (Other academic)

An important class of non-uniformly observable systems come from applications in mobile robotics. In this paper, the problem of active observer design for such systems is considered. The set of feasible configurations and the set of output flow equivalent states is defined. It is shown that the inter-relation between these two sets serves as the basis for design of active observers. The proposed observer design method is illustrated by considering a unicycle robot model, equipped with a set of range-measuring sensors.

• 174.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
Minimum time multi-UGV surveillance2008In: OPTIMIZATION AND COOPERATIVE CONTROL STRATEGIES / [ed] Hirsch MJ; Commander CW; Pardalos PM; Murphey R, Berlin: Springer Verlag , 2008, p. 31-45Conference paper (Refereed)

This paper addresses the problem of concurrent task- and path planning for a number of  surveillance Unmanned Ground Vehicles (UGVs) such that a user defined area of interest is covered by the UGVs' sensors in minimum time. We first formulate the problem, and show that it is in fact  a generalization of the Multiple Traveling Salesmen Problem (MTSP), which is known to be NP-hard. We then propose a solution that decomposes the problem into three subproblems. The first is to find a maximal convex covering of the search area. Most results on static coverage  use disjoint partitions of the search area, e.g. triangulation, to convert the continuous sensor positioning problem into a  discrete one. However, by a simple example, we show that a highly overlapping set of maximal convex sets is better suited for  minimum time coverage. The second subproblem is a combinatorial assignment and ordering of the sets in the cover.  Since Tabu search algorithms are known to perform well on various routing problems,  we use it as a part of our proposed solution. Finally, the third subproblem utilizes a particular shortest path sub-routine in order to find the vehicle paths, and calculate the overall objective function used in the Tabu search. The proposed algorithm is illustrated by a number of simulation examples.

• 175. Anisi, David A.
Swedish Defence Research Agency (FOI), Sweden. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory. KTH, School of Computer Science and Communication (CSC), Centres, Centre for Autonomous Systems, CAS. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.
Cooperative Minimum Time Surveillance With Multiple Ground Vehicles2010In: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 55, no 12, p. 2679-2691Article in journal (Refereed)

In this paper, we formulate and solve two different minimum time problems related to unmanned ground vehicle (UGV) surveillance. The first problem is the following. Given a set of surveillance UGVs and a polyhedral area, find waypoint-paths for all UGVs such that every point of the area is visible from a point on a path and such that the time for executing the search in parallel is minimized. Here, the sensors' field of view are assumed to have a limited coverage range and be occluded by the obstacles. The second problem extends the first by additionally requiring the induced information graph to be connected at the time instants when the UGVs perform the surveillance mission, i.e., when they gather and transmit sensor data. In the context of the second problem, we also introduce and utilize the notion of recurrent connectivity, which is a significantly more flexible connectivity constraint than, e.g., the 1-hop connectivity constraints and use it to discuss consensus filter convergence for the group of UGVs.

• 176.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
Department of Autonomous Systems Swedish Defence Research Agency. Department of Autonomous Systems Swedish Defence Research Agency.
Safe receding horizon control of an aerial vehicle2006In: PROCEEDINGS OF THE 45TH IEEE CONFERENCE ON DECISION AND CONTROL, VOLS 1-14, IEEE , 2006, p. 57-62Conference paper (Refereed)

This paper addresses the problem of designing a real time high performance controller and trajectory generator for air vehicles. The control objective is to use information about terrain and enemy threats to fly low and avoid radar exposure on the way to a given target. The proposed algorithm builds on the well known approach of Receding Horizon Control (RHC) combined with a terminal cost, calculated from a graph representation of the environment. Using a novel safety maneuver, and under an assumption on the maximal terrain inclination, we are able to prove safety as well as task completion. The safety maneuver is incorporated in the short term optimization, which is performed using Nonlinear Programming (NLP). Some key characteristics of the trajectory planner are highlighted through simulations.

• 177.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
Dept. of Autonomous Systems, Swedish Defence Research Agency (FOI), Stockholm, Sweden. Dept. of Autonomous Systems, Swedish Defence Research Agency (FOI), Stockholm, Sweden.
Online Trajectory Planning for Aerial Vehicle: A Safe Approach with Guaranteed Task CompletionManuscript (Other academic)

On-line trajectory optimization in three dimensional space is the main topic of the paper at hand. The high-level framework augments on-line receding horizon control with an off-line computed terminal cost that captures the global characteristics of the environment, as well as any possible mission objectives. The first part of the paper is devoted to the single vehicle case while the second part considers the problem of simultaneous arrival of multiple aerial vehicles. The main contribution of the first part is two-fold. Firstly, by augmenting a so called safety maneuver at the end of the planned trajectory, this paper extends previous results by addressing provable safety properties in a 3D setting. Secondly, assuming initial feasibility, the planning method presented is shown to have finite time task completion. Moreover, a quantitative comparison between the two competing objectives of optimality and computational tractability is made. Finally, some other key characteristics of the trajectory planner, such as ability to minimize threat exposure and robustness, are highlighted through simulations. As for the simultaneous arrival problem considered in the second part, by using a time-scale separation principle, we are able to adopt standard Laplacian control to a consensus problem which is neither unconstrained, nor first order.

• 178.
Institut de Sûreté et de Radioprotection Nucléaire.
Institut de Sûreté et de Radioprotection Nucléaire. Institut de Sûreté et de Radioprotection Nucléaire. Institut de Sûreté et de Radioprotection Nucléaire. Institut de Sûreté et de Radioprotection Nucléaire. Institut de Sûreté et de Radioprotection Nucléaire. Institut de Sûreté et de Radioprotection Nucléaire.
A finite volume stability result for the convection operator in compressible flows. . . and some finite element applications2008In: Finite Volumes for Complex Applications V: Problems & Perspectives / [ed] Robert Eymard and Jean-Marc Hérard, Hermes Science Publications, 2008, p. 185-192Conference paper (Refereed)

In this paper, we build a L2-stable discretization of the non-linear convection termin Navier-Stokes equations for non-divergence-free flows, for non-conforming low order Stokesfinite elements. This discrete operator is obtained by a finite volume technique, and its stability relies on a result interesting for its own sake: the L2-stability of the natural finite volume convection operator in compressible flows, under some compatibility condition with the discrete mass balance. Then, this analysis is used to derive a boundary condition to cope with physical situations where the velocity cannot be prescribed on inflow parts of the boundary of the computational domain. We finally collect these ingredients in a pressure correction scheme for low Mach number flows, and assess the capability of the resulting algorithm to compute a natural convection flow with artificial (open) boundaries.

• 179.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
Absorbing Layers and Non-Reflecting Boundary Conditions for Wave Propagation Problems2005Doctoral thesis, comprehensive summary (Other scientific)

The presence of wave motion is the defining feature in many fields of application,such as electro-magnetics, seismics, acoustics, aerodynamics,oceanography and optics. In these fields, accurate numerical simulation of wave phenomena is important for the enhanced understanding of basic phenomenon, but also in design and development of various engineering applications.

In general, numerical simulations must be confined to truncated domains, much smaller than the physical space were the wave phenomena takes place. To truncate the physical space, artificial boundaries, and corresponding boundary conditions, are introduced. There are four main classes of methods that can be used to truncate problems on unbounded or large domains: boundary integral methods, infinite element methods, non-reflecting boundary condition methods and absorbing layer methods.

In this thesis, we consider different aspects of non-reflecting boundary conditions and absorbing layers. In paper I, we construct discretely non-reflecting boundary conditions for a high order centered finite difference scheme. This is done by separating the numerical solution into spurious and physical waves, using the discrete dispersion relation.

In paper II-IV, we focus on the perfectly matched layer method, which is a particular absorbing layer method. An open issue is whether stable perfectly matched layers can be constructed for a general hyperbolic system.

In paper II, we present a stable perfectly matched layer formulation for 2 x 2 symmetric hyperbolic systems in (2 + 1) dimensions. We also show how to choose the layer parameters as functions of the coefficient matrices to guarantee stability.

In paper III, we construct a new perfectly matched layer for the simulation of elastic waves in an anisotropic media. We present theoretical and numerical results, showing that the stability properties of the present layer are better than previously suggested layers.

In paper IV, we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters which is applicable to all hyperbolic systems, and which we prove is well-posed and perfectly matched. We also use an automatic method, derived in paper V, for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell s equations, the linearized Euler equations, as well as arbitrary 2 x 2 systems in (2 + 1) dimensions.

In paper V, we use the method of Sturm sequences for bounding the real parts of roots of polynomials, to construct an automatic method for checking Petrowsky well-posedness of a general Cauchy problem. We prove that this method can be adapted to automatically symmetrize any well-posed problem, producing an energy estimate involving only local quantities.

• 180.
KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
Construction of stable PMLs for general 2 x 2 symmetric hyperbolic systems2004In: Proceedings of the HYP2004 conference, 2004, p. 1-8Conference paper (Refereed)

The perfectly matched layer (PML) has emerged as animportant tool for accurately solving certain hyperbolic systems onunbounded domains. An open issue is whether stable PMLs can beconstructed in general. In this work we consider the specializationof our general PML formulation to 2 × 2 symmetric hyperbolicsystems in 2 + 1 dimensions. We show how to choose the layerparameters as functions of the coefficient matrices to guaranteestability.

• 181.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
Department of Mathematics and Statistics, University of New Mexico. KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
Perfectly matched layers for hyperbolic systems: General formulation, well-posedness and stability2006In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 67, no 1, p. 1-23Article in journal (Refereed)

Since its introduction the perfectly matched layer (PML) has proven to be an accurate and robust method for domain truncation in computational electromagnetics. However, the mathematical analysis of PMLs has been limited to special cases. In particular, the basic question of whether or not a stable PML exists for arbitrary wave propagation problems remains unanswered. In this work we develop general tools for constructing PMLs for first order hyperbolic systems. We present a model with many parameters, which is applicable to all hyperbolic systems and which we prove is well-posed and perfectly matched. We also introduce an automatic method for analyzing the stability of the model and establishing energy inequalities. We illustrate our techniques with applications to Maxwell's equations, the linearized Euler equations, and arbitrary 2 x 2 systems in (2 + 1) dimensions.

• 182.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
A New Absorbing Layer for Elastic Waves2006In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 215, no 2, p. 642-660Article in journal (Refereed)

A new perfectly matched layer (PML) for the simulation of elastic waves in anisotropic media on an unbounded domain is constructed. Theoretical and numerical results, showing that the stability properties of the present layer are better than previously suggested layers, are presented. In addition, the layer can be formulated with fewer auxiliary variables than the split-field PML.

• 183.
KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
Discretely nonreflecting boundary conditions for higher order centered schemes for wave equations2003In: Proceedings of the WAVES-2003 conference, Berlin: Springer Verlag , 2003, p. 130-135Chapter in book (Other academic)

Using the framework introduced by Rawley and Colonius [2] we construct a nonreflecting boundary condition for the one-way wave equation spatially discretized with a fourth order centered difference scheme. The boundary condition, which can be extended to arbitrary order accuracy, is shown to be well posed. Numerical simulations have been performed showing promising results.

• 184. Apushkinskaya, D. E.
KTH, Superseded Departments, Mathematics.
Lipschitz property of the free boundary in the parabolic obstacle problem2004In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 15, no 3, p. 375-391Article in journal (Refereed)

A parabolic obstacle problem with zero constraint is considered. It is proved, without any additional assumptions on a free boundary, that near the fixed boundary where the homogeneous Dirichlet condition is fulfilled, the boundary of the noncoincidence set is the graph of a Lipschitz function.

• 185. Arakelyan, A.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Multi-Phase Quadrature Domains and a Related Minimization Problem2016In: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, p. 1-21Article in journal (Refereed)

In this paper we introduce the multi-phase version of the so-called Quadrature Domains (QD), which refers to a generalized type of mean value property for harmonic functions. The well-established and developed theory of one-phase QD was recently generalized to a two-phase version, by one of the current authors (in collaboration). Here we introduce the concept of the multi-phase version of the problem, and prove existence as well as several properties of such solutions. In particular, we discuss possibilities of multi-junction points.

• 186. Arakelyan, A.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Two-and multi-phase quadrature surfaces2017In: Communications on Pure and Applied Analysis, ISSN 1534-0392, E-ISSN 1553-5258, Vol. 16, no 6, p. 2023-2045Article in journal (Refereed)

In this paper we shall initiate the study of the two-and multi-phase quadrature surfaces (QS), which amounts to a two/multi-phase free boundary problems of Bernoulli type. The problem is studied mostly from a potential theoretic point of view that (for two-phase case) relates to integral representation where dsx is the surface measure, μ = μ+-μ-is given measure with support in (a priori unknown domain) ω = ω+ [ω-, g is a given smooth positive function, and the integral holds for all functions h, which are harmonic on ω. Our approach is based on minimization of the corresponding two-and multiphase functional and the use of its one-phase version as a barrier. We prove several results concerning existence, qualitative behavior, and regularity theory for solutions. A central result in our study states that three or more junction points do not appear.

• 187.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The Finite Difference Method for Two-Phase Parabolic Obstacle-Like Problem: Like ProblemArticle in journal (Other academic)

In this paper for two-phase parabolic obstacle-like problem, $\Delta u -u_t=\lambda^+\cdot\chi_{\{u>0\}}-\lambda^-\cdot\chi_{\{u<0\}},\quad (t,x)\in (0,T)\times\Omega,$ where $T < \infty, \lambda^+ ,\lambda^- > 0$ are Lipschitz continuous functions, and $\Omega\subset\mathbb{R}^n$ is a bounded domain, we will introduce a certain variational form, which allows us to define a notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed. Although the paper is devoted to the parabolic version of the two-phase obstacle-like problem, we prove convergence of discretized scheme to a unique viscosity solution for both two-phase parabolic obstacle-like and standard two-phase membrane problem. Numerical simulations are also presented.

• 188.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
The Finite Difference Methods for Multi-phase Free Boundary Problems2011Doctoral thesis, comprehensive summary (Other academic)

This thesis consist of an introduction and four research papers concerning numerical analysis for a certain class of free boundary problems.

Paper I is devoted to the numerical analysis of the so-called two-phase membrane problem. Projected Gauss-Seidel method is constructed. We prove general convergence of the algorithm as well as obtain the error estimate for the finite difference scheme.

In Paper II we have improved known results on the error estimates for a Classical Obstacle (One-Phase) Problem with a finite difference scheme.

Paper III deals with the parabolic version of the two-phase obstacle-like problem. We introduce a certain variational form, which allows us to definea notion of viscosity solution. The uniqueness of viscosity solution is proved, and numerical nonlinear Gauss-Seidel method is constructed.

In the last paper, we study a numerical approximation for a class of stationary states for reaction-diffusion system with m densities having disjoint support. The proof of convergence of the numerical method is given in some particular cases. We also apply our numerical simulations for the spatial segregation limit of diffusive Lotka-Volterra models in presence of high competition and inhomogeneous Dirichlet boundary conditions.

• 189.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Numerical Solution of the Two-Phase Obstacle Problem by Finite Difference MethodManuscript (preprint) (Other academic)
• 190.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
An Error Estimate for the Finite Difference Scheme for One-Phase Obstacle Problem2011In: Journal of Contemporary Mathematical Analysis, ISSN 1068-3623, Vol. 46, no 3, p. 131-141Article in journal (Refereed)

In this paper we consider the finite difference scheme approximation for one-phase obstacle problem and obtain an error estimate for this approximation.

• 191.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
Numerical Algorithms for a Variational problem of the Spatial Segregation of Reaction-diffusion SystemsArticle in journal (Other academic)
• 192. Araujo-Cabarcas, J. C.
KTH, Centres, SeRC - Swedish e-Science Research Centre. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map2018In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, p. 177-192Article in journal (Refereed)

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

• 193. Ariel, G.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA (closed 2012-06-30). University of Texas at Austin, United States .
Multiscale computations for highly oscillatory problems2009In: Multiscale Modeling and Simulation in Science, Springer Berlin/Heidelberg, 2009, p. 237-287Conference paper (Refereed)

We review a selection of essential techniques for constructing computational multiscale methods for highly oscillatory ODEs. Contrary to the typical approaches that attempt to enlarge the stability region for specialized problems, these lecture notes emphasize how multiscale properties of highly oscillatory systems can be characterized and approximated in a truly multiscale fashion similar to the settings of averaging and homogenization. Essential concepts such as resonance, fast-slow scale interactions, averaging, and techniques for transformations to non-stiff forms are discussed in an elementary manner so that the materials can be easily accessible to beginning graduate students in applied mathematics or computational sciences.

• 194. Ariel, G.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Parareal multiscale methods for highly oscillatory dynamical systems2016In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, no 6, p. A3540-A3564Article in journal (Refereed)

We introduce a new strategy for coupling the parallel in time (parareal) iterative methodology with multiscale integrators. Following the parareal framework, the algorithm computes a low-cost approximation of all slow variables in the system using an appropriate multiscale integrator, which is refined using parallel fine scale integrations. Convergence is obtained using an alignment algorithm for fast phase-like variables. The method may be used either to enhance the accuracy and range of applicability of the multiscale method in approximating only the slow variables, or to resolve all the state variables. The numerical scheme does not require that the system is split into slow and fast coordinates. Moreover, the dynamics may involve hidden slow variables, for example, due to resonances. We propose an alignment algorithm for almost-periodic solutions, in which case convergence of the parareal iterations is proved. The applicability of the method is demonstrated in numerical examples.

• 195.
Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
Department of Mathematics, The University of Texas at Austin, Austin, USA. Department of Mathematics, The University of Texas at Austin, Austin, USA. Department of Mathematics, The University of Texas at Austin, Austin, USA.
Gaussian Beam Decomposition of High Frequency Wave Fields Using Expectation-Maximization2011In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, no 6, p. 2303-2321Article in journal (Refereed)

A new numerical method for approximating highly oscillatory wave fields as a superposition of Gaussian beams is presented. The method estimates the number of beams and their parameters automatically. This is achieved by an expectation–maximization algorithm that fits real, positive Gaussians to the energy of the highly oscillatory wave fields and its Fourier transform. Beam parameters are further refined by an optimization procedure that minimizes the difference between the Gaussian beam superposition and the highly oscillatory wave field in the energy norm.

• 196.
Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
Department of Mathematics, The University of Texas at Austin, Austin, USA. Department of Mathematics, The University of Texas at Austin, Austin, USA.
A multiscale method for stiff ordinary differential equations with resonance2009In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 78, no 266, p. 929-956Article in journal (Refereed)

A multiscale method for computing the effective behavior of a class of stiff and highly oscillatory ordinary differential equations (ODEs) is presented. The oscillations may be in resonance with one another and thereby generate hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables whose dynamics is closed up to perturbation, and is sufficient to approximate any variable and functional that are slow under the dynamics of the ODE. This set of variables is detected numerically as a preprocessing step in the numerical methods. Error and complexity estimates are obtained. The advantages of the method is demonstrated with a few examples, including a commonly studied problem of Fermi, Pasta, and Ulam.

• 197.
Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
Department of Mathematics, The University of Texas at Austin, Austin, USA. Department of Mathematics, The University of Texas at Austin, Austin, USA.
A reversible multiscale integration method2009In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 7, no 3, p. 595-610Article in journal (Refereed)

A multiscale, time reversible method for computing the effective slow behavior of systems of highly oscillatory ordinary differential equations is presented. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the system. The algorithm follows the framework of the heterogeneous multiscale method. The notion of time reversibility in the multiple time-scale setting is discussed. The algorithm requires nontrivial matching between the microscopic state variables and the macroscopic slow ones. Numerical examples show the efficiency of the multiscale method and the advantages of time reversibility.

• 198.
Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
Department of Mathematics, The University of Texas at Austin, Austin, USA. Department of Mathematics, The University of Texas at Austin, Austin, USA.
Numerical multiscale methods for coupled oscillators2009In: Multiscale Modeling & simulation, ISSN 1540-3459, E-ISSN 1540-3467, Vol. 7, no 3, p. 1387-1404Article in journal (Refereed)

A multiscale method for computing the effective slow behavior of a system of weakly coupled nonlinear planar oscillators is presented. The oscillators may be either in the form of a periodic solution or a stable limit cycle. Furthermore, the oscillators may be in resonance with one another and thereby generate some hidden slow dynamics. The proposed method relies on correctly tracking a set of slow variables that is sufficient to approximate any variable and functional that are slow under the dynamics of the ordinary differential equation. The technique is more efficient than existing methods, and its advantages are demonstrated with examples. The algorithm follows the framework of the heterogeneous multiscale method.

• 199.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Analysis and Applications of Heterogeneous Multiscale Methods for Multiscale Partial Differential Equations2015Doctoral thesis, comprehensive summary (Other academic)

This thesis centers on the development and analysis of numerical multiscale methods for multiscale problems arising in steady heat conduction, heat transfer and wave propagation in heterogeneous media. In a multiscale problem several scales interact with each other to form a system which has variations over a wide range of scales. A direct numerical simulation of such problems requires resolving the small scales over a computational domain, typically much larger than the microscopic scales. This demands a tremendous computational cost. We develop and analyse multiscale methods based on the heterogeneous multiscale methods (HMM) framework, which captures the macroscopic variations in the solution at a cost much lower than traditional numerical recipes. HMM assumes that there is a macro and a micro model which describes the problem. The micro model is accurate but computationally expensive to solve. The macro model is inexpensive but incomplete as it lacks certain parameter values. These are upscaled by solving the micro model locally in small parts of the domain. The accuracy of the method is then linked to how accurately this upscaling procedure captures the right macroscopic effects. In this thesis we analyse the upscaling error of existing multiscale methods and also propose a micro model which significantly reduces the upscaling error invarious settings. In papers I and IV we give an analysis of a finite difference HMM (FD-HMM) for approximating the effective solutions of multiscale wave equations over long time scales. In particular, we consider time scales T^ε = O(ε−k ), k =1, 2, where ε represents the size of the microstructures in the medium. In this setting, waves exhibit non-trivial behaviour which do not appear over short time scales. We use new analytical tools to prove that the FD-HMM accurately captures the long time effects. We first, in Paper I, consider T^ε =O(ε−2 ) and analyze the accuracy of FD-HMM in a one-dimensional periodicsetting. The core analytical ideas are quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic wave equations.The analysis naturally reveals the role of consistency in HMM for high order approximation of effective quantities over long time scales. Next, in paperIV, we consider T^ε = O(ε−1 ) and use the tools in a multi-dimensional settingto analyze the accuracy of the FD-HMM in locally-periodic media where fast and slow variations are allowed at the same time. Moreover, in papers II and III we propose new multiscale methods which substantially improve the upscaling error in multiscale elliptic, parabolic and hyperbolic partial differential equations. In paper II we first propose a FD-HMM for solving elliptic homogenization problems. The strategy is to use the wave equation as the micro model even if the macro problem is of elliptic type. Next in paper III, we use this idea in a finite element HMM setting and generalize the approach to parabolic and hyperbolic problems. In a spatially fully discrete a priori error analysis we prove that the upscaling error can be made arbitrarily small for periodic media, even if we do not know the exact period of the oscillations in the media.

• 200.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Numerical Analysis, NA.
Analysis and Applications of the Heterogeneous Multiscale Methods for Multiscale Elliptic and Hyperbolic Partial Differential Equations2013Licentiate thesis, comprehensive summary (Other academic)

This thesis concerns the applications and analysis of the Heterogeneous Multiscale methods (HMM) for Multiscale Elliptic and Hyperbolic Partial Differential Equations. We have gathered the main contributions in two papers.

The first paper deals with the cell-boundary error which is present in multi-scale algorithms for elliptic homogenization problems. Typical multi-scale methods have two essential components: a macro and a micro model. The micro model is used to upscale parameter values which are missing in the macro model. Solving the micro model requires, on the other hand, imposing boundary conditions on the boundary of the microscopic domain. Imposing a naive boundary condition leads to $O(\varepsilon/\eta)$ error in the computation, where $\varepsilon$ is the size of the microscopic variations in the media and $\eta$ is the size of the micro-domain. Until now, strategies were proposed to improve the convergence rate up to fourth-order in $\varepsilon/\eta$ at best. However, the removal of this error in multi-scale algorithms still remains an important open problem. In this paper, we present an approach with a time-dependent model which is general in terms of dimension. With this approach we are able to obtain $O((\varepsilon/\eta)^q)$ and $O((\varepsilon/\eta)^q + \eta^p)$ convergence rates in periodic and locally-periodic media respectively, where $p,q$ can be chosen arbitrarily large.

In the second paper, we analyze a multi-scale method developed under the Heterogeneous Multi-Scale Methods (HMM) framework for numerical approximation of wave propagation problems in periodic media. In particular, we are interested in the long time $O(\varepsilon^{-2})$ wave propagation. In the method, the microscopic model uses the macro solutions as initial data. In short-time wave propagation problems a linear interpolant of the macro variables can be used as the initial data for the micro-model. However, in long-time multi-scale wave problems the linear data does not suffice and one has to use a third-degree interpolant of the coarse data to capture the $O(1)$ dispersive effects apperaing in the long time. In this paper, we prove that through using an initial data consistent with the current macro state, HMM captures this dispersive effects up to any desired order of accuracy in terms of $\varepsilon/\eta$. We use two new ideas, namely quasi-polynomial solutions of periodic problems and local time averages of solutions of periodic hyperbolic PDEs. As a byproduct, these ideas naturally reveal the role of consistency for high accuracy approximation of homogenized quantities.

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