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• 201.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.). KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Optimal regularity for the parabolic no-sign obstacle type problem2013Ingår i: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 15, nr 4, s. 477-499Artikel i tidskrift (Refereegranskat)

We study the parabolic free boundary problem of obstacle type Delta u - partial derivative u/partial derivative t = f chi({u not equal 0}). Under the condition that f = H nu for some function nu with bounded second order spatial derivatives and bounded first order time derivative, we establish the same regularity for the solution u. Both the regularity and the assumptions are optimal. Using this result and assuming that f is Dini continuous, we prove that the free boundary is, near so called low energy points, a C-1 graph. Our result completes the theory for this type of problems for the heat operator.

• 202.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem2006Ingår i: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 44, nr 1, s. 1-15Artikel i tidskrift (Refereegranskat)

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

KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The singular set of higher dimensional unstable obstacle type problems2013Ingår i: Rendiconti Lincei - Matematica e Applicazioni, ISSN 1120-6330, E-ISSN 1720-0768, Vol. 24, nr 1, s. 123-146Artikel i tidskrift (Refereegranskat)
• 204.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Global solutions of the obstacle problem in half-spaces, and their impact on local stability2005Ingår i: Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, E-ISSN 1432-0835, Vol. 23, nr 3, s. 271-279Artikel i tidskrift (Refereegranskat)

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.

• 205.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Equilibrium points of a singular cooperative system with free boundary2015Ingår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 280, s. 743-771Artikel i tidskrift (Refereegranskat)

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, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Double Obstacle Problems with Obstacles Given by Non-C-2 Hamilton-Jacobi Equations2012Ingår i: Archive for Rational Mechanics and Analysis, ISSN 0003-9527, E-ISSN 1432-0673, Vol. 206, nr 3, s. 779-819Artikel i tidskrift (Refereegranskat)

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.

• 207.
Mathematics Institute, University of Warwick.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Mathematical Institute of the Heinrich Heine University.
On the singularities of a free boundary through Fourier expansion2012Ingår i: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297, Vol. 187, nr 3, s. 535-587Artikel i tidskrift (Refereegranskat)

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.

KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
On the geometry and topology of initial data sets with horizons2018Ingår i: Asian Journal of Mathematics, ISSN 1093-6106, E-ISSN 1945-0036, Vol. 22, nr 5, s. 863-882Artikel i tidskrift (Refereegranskat)

We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asymptotically flat initial set (M, g,K) such that the boundary ∂M of M is a collection of Marginally Outer (or Inner) Trapped Surfaces (MOTSs and/or MITSs) and such that M \ ∂M contains no MOTSs or MITSs. This definition is meant to capture, on the level of the initial data sets, the well known notion of the domain of outer communications (DOC) as the region of spacetime outside of all the black holes (and white holes). Our main theorem establishes that in dimensions 3 ≤ n ≤ 7, a CDOC which satisfies the dominant energy condition and has a strictly stable boundary has a positive scalar curvature metric which smoothly compactifies the asymptotically flat end and is a Riemannian product metric near the boundary where the cross sectional metric is conformal to a small perturbation of the initial metric on the boundary ∂M induced by g. This result may be viewed as a generalization of Galloway and Schoen's higher dimensional black hole topology theorem [17] to the exterior of the horizon. We also show how this result leads to a number of topological restrictions on the CDOC, which allows one to also view this as an extension of the initial data topological censorship theorem, established in [10] in dimension n = 3, to higher dimensions.

• 209.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
A NEW TENSORIAL CONSERVATION LAW FOR MAXWELL FIELDS ON THE KERR BACKGROUND2017Ingår i: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 105, nr 2, s. 163-176Artikel i tidskrift (Refereegranskat)

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.

• 210.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.). Max Planck Institute for Gravitational Physics, United Kingdom.
Hidden symmetries and decay for the wave equation on the Kerr spacetime2015Ingår i: Annals of Mathematics, ISSN 0003-486X, E-ISSN 1939-8980, Vol. 182, nr 3, s. 787-853Artikel i tidskrift (Refereegranskat)

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, Tidigare Institutioner                               , Matematik.
scalar curvature rigidity for asymptotically locally hyperbolic manifolds1998Ingår i: Annals of Global Analysis and Geometry, ISSN 0232-704X, E-ISSN 1572-9060, Vol. 16, s. 1-27Artikel i tidskrift (Refereegranskat)

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.

• 212.
KTH, Tidigare Institutioner, Matematik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Boundary and lens rigidity of Lorentzian surfaces1996Ingår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 348, s. 2307-2329Artikel i tidskrift (Refereegranskat)

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.

• 213.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Multivariate Financial Time Series and Volatility Models with Applications to Tactical Asset Allocation2015Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Den finansiella marknaden är av en väldigt komplex struktur och modelleringsteknikerna har under senare tid blivit allt mer komplicerade. För en portföljförvaltare är det av yttersta vikt att finna mer sofistikerade modelleringstekniker, speciellt efter finanskrisen 2007-2008. Idéen i den här uppsatsen är att finna ett samband mellan makroekonomiska faktorer och aktieportföljer innehållande tillgångar från OMX Stockholm 30 och använda dessa för att utföra Tactial Asset Allocation (TAA). Mer specifikt är målsättningen att visa att dynamiska modelleringstekniker har ett bättre utfall än mer statiska modeller i portföljteori.

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• 214.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Constructing Multidimensional Dynamical Systems with Positive Lyapunov Exponents2018Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

I denna uppsats anpassar vi Marcelo Vianas generella konstruktion av  glatta transformationer som uppvisar icke-likformig expansion i flera  dimensioner. I vår konstruktion, istället för att koppla ihop vårt system med en kvadratisk avbildning, får vi expansionen från en avbildning med en  kubisk kritisk punkt. Dessutom så diskuterar vi hur argumentationen kan  utvidgas och expansionen fås av vissa typer av avbildningar med en kritisk punkt av godtyckligt hög (udda) grad. Vi diskuterar också problemen i att försöka använda Vianas metod med den så kallade "double standard"-avbildningen som källan till den icke-likformiga expansionen.

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• 215.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
A study of the most important volume drivers for sparkling wine in Sweden2015Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

I denna uppsats undersöks de faktorer som påverkar den sålda volymen av mousserande vin. Regressionsanalys används för att modellera sambandet mellan de identifierade parametrarna och volymen.

Den sociala hållbarheten i och med en avreglering av Sveriges alkoholmarknad kommer att undersökas med hjälp av en 5C-analys av Systembolaget, WACOSS’s Social Sustainability framework samt Social Life’s framework.

Parametrarna som påverkade den sålda volymen mest föll under tre olika kategorier: Land, vilken typ av vindruva vinet är gjort på samt pris per liter. De flesta parametrarna visade sig ha en positiv påverkan på den sålda volymen, med undantag för viner från Nya Zeeland samt högre pris.

Denna uppsats kommer fram till slutsatsen att en reglerad alkoholmarknad är gynnsam när syftet är att hålla alkoholkonsumtionen låg och den övergripande folkhälsan hög.

KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.). Stockholm School of Economics, Sweden.
Robustness to strategic uncertainty in the Nash demand game2018Ingår i: Mathematical Social Sciences, ISSN 0165-4896, E-ISSN 1879-3118, Vol. 91, s. 1-5Artikel i tidskrift (Refereegranskat)

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, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Robustness to strategic uncertainty2014Ingår i: Games and Economic Behavior, ISSN 0899-8256, E-ISSN 1090-2473, Vol. 85, nr 1, s. 272-288Artikel i tidskrift (Refereegranskat)

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.

• 218.
AstraZeneca R and D.
Lund University.
Subspace estimation and prediction methods for hidden Markov models2009Ingår i: Annals of Statistics, ISSN 0090-5364, E-ISSN 2168-8966, Vol. 37, nr 6B, s. 4131-4152Artikel i tidskrift (Refereegranskat)

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.

• 219.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Prediction of French day-ahead electricity prices: Comparison between a deterministic and a stochastic approach2015Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Denna avhandling behandlar den nya flödesbaserade beräkningsmetoden som används i Centrala Västeuropa på ekonomisidan. Målet är att producera tillförlitliga metoder för prognostisering. Två tillvägagångssätt kan användas: den första är baserad på en deterministisk och algoritmisk metod som inbegriper studier av interaktionen mellan fundamenta och priserna. Den andra är en mer statistisk metod som bygger på en tidsseriemodellering av de franska flödesbaserade priserna. Båda tillvägagångssätten har fördelar och nackdelar som kommer som diskuteras i det följande. Arbetet är främst

flödesbasen i Västeuropa.

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• 220.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Adaptive node distribution for on-line trajectory planning2006Ingår i: ICAS-Secretariat - 25th Congress of the International Council of the Aeronautical Sciences 2006, Curran Associates, Inc., 2006, s. 3150-3157Konferensbidrag (Refereegranskat)

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.

• 221.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Adaptive Node Distribution for Online Trajectory PlanningManuskript (Övrigt vetenskapligt)
• 222.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
On Cooperative Surveillance, Online Trajectory Planning and Observer Based Control2009Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

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.

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• 223.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Online trajectory planning and observer based control2006Licentiatavhandling, sammanläggning (Övrigt vetenskapligt)

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.

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• 224.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Riemannian Observers for Euler-Lagrange Systems2005Ingår i: Proceedings of the 16th IFAC World Congress: Prague, Czech Republic, July 3-8, 2005, 2005, s. 115-120Konferensbidrag (Refereegranskat)

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.

• 225.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Active Observers for Mobile Robotic SystemsManuskript (Övrigt vetenskapligt)

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.

• 226.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Minimum time multi-UGV surveillance2008Ingår i: OPTIMIZATION AND COOPERATIVE CONTROL STRATEGIES / [ed] Hirsch MJ; Commander CW; Pardalos PM; Murphey R, Berlin: Springer Verlag , 2008, s. 31-45Konferensbidrag (Refereegranskat)

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.

• 227. Anisi, David A.
Swedish Defence Research Agency (FOI), Sweden. KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori. KTH, Skolan för datavetenskap och kommunikation (CSC), Centra, Centrum för Autonoma System, CAS. KTH, Skolan för elektro- och systemteknik (EES), Centra, ACCESS Linnaeus Centre.
Cooperative Minimum Time Surveillance With Multiple Ground Vehicles2010Ingår i: IEEE Transactions on Automatic Control, ISSN 0018-9286, E-ISSN 1558-2523, Vol. 55, nr 12, s. 2679-2691Artikel i tidskrift (Refereegranskat)

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.

• 228.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
Department of Autonomous Systems Swedish Defence Research Agency. Department of Autonomous Systems Swedish Defence Research Agency.
Safe receding horizon control of an aerial vehicle2006Ingår i: PROCEEDINGS OF THE 45TH IEEE CONFERENCE ON DECISION AND CONTROL, VOLS 1-14, IEEE , 2006, s. 57-62Konferensbidrag (Refereegranskat)

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.

• 229.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Optimeringslära och systemteori.
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 CompletionManuskript (Övrigt vetenskapligt)

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.

• 230.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
Introduction of the Academic Factor Quality Minus Junk to a Commercial Factor Model and its Effect on the Explanatory Power. An OLS Regression on Stock Returns2019Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)

Förmågan att förutsäga aktiers avkastning önskar många besitta, och på ett så precist sätt som möjligt. Under många år har forskning pågått inom området för faktormodeller som förklarar avkastningar, med målet att öka modellernas förklaringsgrad. Detta är dock en komplex verksamhet eftersom faktorerna och deras förbättring av förklaringsgraden måste vara signifikanta för modellen. Då och då kommer forskare fram med nya sådana faktorer som har positiv påverkan på modeller. AQR Capital Management är inget undantag eftersom de 2013 presenterade sin faktor Quality Minus Junk som visar signifikanta riskjusterade avkastningar. Detta kandidatexamensarbete inom matematisk statistik och industriell ekonomi, ämnar att utreda huruvida den kommersiella faktormodellen som används på Fjärde AP-fonden förbättras genom tillägget av faktorn Quality Minus Junk, i förklaringsgradsmening. Metoden som används är till största delen baserad på multipel linjär regression och tre treårsperioder studeras i tidsintervallet 2010 till 2018. Resultaten från detta projekt visar på att faktorn Quality Minus Junk bidrar med signifikanta ökningar av förklaringsgraden för en av tre perioder, den senaste perioden 2016-2018. Eftersom resultaten är inkonklusiva krävs vidare studier för att bättre förstå och konkludera vad dessa resultat faktiskt innebär samt för att inkludera QMJ-faktorn i modellen eller ej.

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• 231.
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 applications2008Ingår i: Finite Volumes for Complex Applications V: Problems & Perspectives / [ed] Robert Eymard and Jean-Marc Hérard, Hermes Science Publications, 2008, s. 185-192Konferensbidrag (Refereegranskat)

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.

• 232.
KTH, Skolan för industriell teknik och management (ITM).
Macroeconomic factors in Probability of Default: A study applied to a Swedish credit portfolio2018Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)

Makroekonomiska omständigheter kan påverka hushållens betalningsförmåga och i sin tur återbetalningsförmågan hos bolånetagare. Om flertalet låntagare inom en banks retailportfölj upplever en försämrad betalningsförmåga kommer det att avspeglas på sannolikheten för fallissemang (probability of default) i den totala portföljen. Med IFRS 9 förväntas banker förfina sina beräkningar av förväntade kreditförluster, vilket kräver framåtblickande beräkningar av probability of default med makroekonomiska prognoser i åtanke. Genom att identifiera vilka makroekonomiska faktorer som har statistisk signifikans för förändringar i historisk fallissemangsfrekvens i en portfölj förväntas banker kunna integrera dessa i, och därmed förbättra, sina beräkningar av probability of default. Denna studie syftar till att utreda sambandet mellan makroekonomiska faktorer och fallissemangsfrekvensen i en svensk retailportfölj. Den kvantitativa analysen av data över historiska fallissemang och makroekonomiska faktorer kompletteras med teoretiska implikationer av makroekonomiska omständigheter för hushållens betalningsförmåga. De makroekonomiska faktorer som studeras är svensk BNP, Boprisindex, Reporänta och Arbetslöshet. Fallissemangsfrekvensen baseras på data från Nordeas svenska retailportfölj som täcker åren 2008-2015 och därmed inkluderar följdeffekter av finanskrisen 2008. En multipel linjär regressionsmodell används för att förklara relationen mellan de makroekonomiska faktorerna och fallissemangsfrekvensen. Regressionskoefficienterna estimeras med hjälp av minstakvadratmetoden och kompletteras med diagnostiska test. Resultaten visar att BNP och Reporäntan är statistiskt signifikanta makroekonomiska faktorer för påvisandet av förändringar i fallissemangsfrekvensen och följaktligen Probability of Default i en svensk retailkreditportfölj.

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• 233.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
Absorbing Layers and Non-Reflecting Boundary Conditions for Wave Propagation Problems2005Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

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.

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• 234.
KTH, Tidigare Institutioner                               , Numerisk analys och datalogi, NADA.
Construction of stable PMLs for general 2 x 2 symmetric hyperbolic systems2004Ingår i: Proceedings of the HYP2004 conference, 2004, s. 1-8Konferensbidrag (Refereegranskat)

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.

• 235.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
Department of Mathematics and Statistics, University of New Mexico. KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
Perfectly matched layers for hyperbolic systems: General formulation, well-posedness and stability2006Ingår i: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 67, nr 1, s. 1-23Artikel i tidskrift (Refereegranskat)

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.

• 236.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk Analys och Datalogi, NADA.
A New Absorbing Layer for Elastic Waves2006Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 215, nr 2, s. 642-660Artikel i tidskrift (Refereegranskat)

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.

• 237.
KTH, Tidigare Institutioner                               , Numerisk analys och datalogi, NADA.
KTH, Tidigare Institutioner                               , Numerisk analys och datalogi, NADA.
Discretely nonreflecting boundary conditions for higher order centered schemes for wave equations2003Ingår i: Proceedings of the WAVES-2003 conference, Berlin: Springer Verlag , 2003, s. 130-135Kapitel i bok, del av antologi (Övrigt vetenskapligt)

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.

• 238. Apushkinskaya, D. E.
KTH, Tidigare Institutioner, Matematik.
Lipschitz property of the free boundary in the parabolic obstacle problem2004Ingår i: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 15, nr 3, s. 375-391Artikel i tidskrift (Refereegranskat)

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.

• 239. Arakelyan, A.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
Multi-Phase Quadrature Domains and a Related Minimization Problem2016Ingår i: Potential Analysis, ISSN 0926-2601, E-ISSN 1572-929X, s. 1-21Artikel i tidskrift (Refereegranskat)

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.

• 240. Arakelyan, A.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Two-and multi-phase quadrature surfaces2017Ingår i: Communications on Pure and Applied Analysis, ISSN 1534-0392, E-ISSN 1553-5258, Vol. 16, nr 6, s. 2023-2045Artikel i tidskrift (Refereegranskat)

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.

• 241.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The Finite Difference Method for Two-Phase Parabolic Obstacle-Like Problem: Like ProblemArtikel i tidskrift (Övrigt vetenskapligt)

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.

• 242.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
The Finite Difference Methods for Multi-phase Free Boundary Problems2011Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)

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.

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• 243.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Numerical Solution of the Two-Phase Obstacle Problem by Finite Difference MethodManuskript (preprint) (Övrigt vetenskapligt)
• 244.
NAS Armenia, Inst Math, Yerevan 0019, Armenia..
Amer Univ Armenia, Inst Math, NAS Armenia, Yerevan 0019, Armenia.. KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Qatar Univ, Dept Math Stat & Phys, POB 2713, Doha, Qatar..
Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles2019Ingår i: Bulletin of the Iranian Mathematical Society, ISSN 1018-6301, E-ISSN 1017-060X, Vol. 45, nr 1, s. 59-73Artikel i tidskrift (Refereegranskat)

In this paper, we continue to study a non-local free boundary problem arising in financial bubbles. We focus on the parabolic counterpart of the bubble problem and suggest an iterative algorithm which consists of a sequence of parabolic obstacle problems at each step to be solved, that in turn gives the next obstacle function in the iteration. The convergence of the proposed algorithm is proved. Moreover, we consider the finite difference scheme for this algorithm and obtain its convergence. At the end of the paper, we present and discuss computational results.

• 245.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
An Error Estimate for the Finite Difference Scheme for One-Phase Obstacle Problem2011Ingår i: Journal of Contemporary Mathematical Analysis, ISSN 1068-3623, Vol. 46, nr 3, s. 131-141Artikel i tidskrift (Refereegranskat)

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

• 246.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
Numerical Algorithms for a Variational problem of the Spatial Segregation of Reaction-diffusion SystemsArtikel i tidskrift (Övrigt vetenskapligt)
• 247. Araujo-Cabarcas, J. C.
KTH, Centra, SeRC - Swedish e-Science Research Centre. KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map2018Ingår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 330, s. 177-192Artikel i tidskrift (Refereegranskat)

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

• 248. Ariel, G.
KTH, Skolan för datavetenskap och kommunikation (CSC), Numerisk analys, NA (stängd 2012-06-30). University of Texas at Austin, United States .
Multiscale computations for highly oscillatory problems2009Ingår i: Multiscale Modeling and Simulation in Science, Springer Berlin/Heidelberg, 2009, s. 237-287Konferensbidrag (Refereegranskat)

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.

• 249. Ariel, G.
KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Numerisk analys, NA.
Parareal multiscale methods for highly oscillatory dynamical systems2016Ingår i: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197, Vol. 38, nr 6, s. A3540-A3564Artikel i tidskrift (Refereegranskat)

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.

• 250.
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-Maximization2011Ingår i: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 230, nr 6, s. 2303-2321Artikel i tidskrift (Refereegranskat)

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.

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