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  • 151.
    Andersson, Joakim
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Wiesinger, Jakob
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Counting words avoiding patterns of length three with generating functions2015Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The set of 123-avoiding words with exactly r occurrences of each letter was recently enumerated by N. Shar and D. Zeilberger. This paper enumerates more complicated sets of pattern avoiding words, such as those allowing 1, 2, ... or r occurrences of each letter, or those for which the numbers of occurrences of each individual letter follow a repeating sequence. The results are also generalized to apply to all patterns of length three with distinct letters. The generating functions enumerating the words are shown to be algebraic, for all investigated sets of words. A notable number of coefficients for the relevant generating functions have been found and the first few confirmed by independent methods. The asymptotic behaviour of these coefficients has been established as exponential. The employed strategy involves partitioning words into subwords which allow for the construction of equations relating the generating functions.

  • 152.
    Andersson, Joel
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On Invertibility of the Radon Transform and Compressive Sensing2014Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis contains three articles. The first two concern inversion andlocal injectivity of the weighted Radon transform in the plane. The thirdpaper concerns two of the key results from compressive sensing.In Paper A we prove an identity involving three singular double integrals.This is then used to prove an inversion formula for the weighted Radon transform,allowing all weight functions that have been considered previously.Paper B is devoted to stability estimates of the standard and weightedlocal Radon transform. The estimates will hold for functions that satisfy an apriori bound. When weights are involved they must solve a certain differentialequation and fulfill some regularity assumptions.In Paper C we present some new constant bounds. Firstly we presenta version of the theorem of uniform recovery of random sampling matrices,where explicit constants have not been presented before. Secondly we improvethe condition when the so-called restricted isometry property implies the nullspace property.

  • 153.
    Andersson, Joel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Boman, Jan
    Stockholm University.
    Stability estimates with a priori bound for the inverse local Radon transformManuscript (preprint) (Other academic)
    Abstract [en]

    We consider the inverse problem for the 2-dimensional weighted local Radon transform , where  is supported in  and is defined near . For weight functions satisfying a certain differential equation we give weak estimates of in terms of  for functions  that satisfies an a priori bound.

  • 154.
    Andersson, Joel
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Strömberg, Jan-Olov
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    On the Theorem of Uniform Recovery of Random Sampling Matrices2014In: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 60, no 3, p. 1700-1710Article in journal (Refereed)
    Abstract [en]

    We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an s-sparse signal from linear measurements (with high probability) is known to be m greater than or similar to s(ln s)(3) ln N. We present new and improved constants together with what we consider to be a more explicit proof. A proof that also allows for a slightly larger class of m x N-matrices, by considering what is called effective sparsity. We also present a condition on the so-called restricted isometry constants, delta s, ensuring sparse recovery via l(1)-minimization. We show that delta(2s) < 4/root 41 is sufficient and that this can be improved further to almost allow for a sufficient condition of the type delta(2s) < 2/3.

  • 155.
    Andersson, Johan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Locating Multiple Change-Points Using a Combination of Methods2014Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    The aim of this study is to find a method that is able to locate multiple change-points in a time series with unknown properties. The methods that are investigated are the CUSUM and CUSUM of squares test, the CUSUM test with OLS residuals, the Mann-Whitney test and Quandt’s log likelihood ratio. Since all methods are detecting single change-points, the binary segmentation technique is used to find multiple change-points. The study shows that the CUSUM test with OLS residuals, Mann-Whitney test and Quandt’s log likelihood ratio work well on most samples while the CUSUM and CUSUM of squares are not able to detect the location of the change-points. Furthermore the study shows that the binary segmentation technique works well with all methods and is able to detect multiple change-points in most circumstances. The study also shows that the results can, most of the time, be improved by using a combination of the methods.

  • 156.
    Andersson, John
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Free boundary problems and global solutions in half spaces2005Doctoral thesis, comprehensive summary (Other scientific)
  • 157.
    Andersson, John
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    On the De Giorgi conjecture in half spacesManuscript (preprint) (Other academic)
  • 158.
    Andersson, John
    Max Planck Institute, Leipzig.
    On the regularity of a free boundary near contact points with a fixed boundary2007In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 232, no 1, p. 285-302Article in journal (Refereed)
    Abstract [en]

    We investigate the regularity of a free boundary near contact points with a fixed boundary, with C boundary data, for an obstacle-like free boundary problem. We will show that under certain assumptions on the solution, and the boundary function, the free boundary is uniformly C 1 up to the fixed boundary. We will also construct some examples of irregular free boundaries.

  • 159.
    Andersson, John
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal regularity for the Signorini problem and its free boundary2015In: Inventiones Mathematicae, ISSN 0020-9910, E-ISSN 1432-1297Article in journal (Refereed)
    Abstract [en]

    We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if (Formula presented.) minimizes (Formula presented.)in the convex set (Formula presented.)where (Formula presented.) say. Then (Formula presented.). Moreover the free boundary, given by (Formula presented.)will be a (Formula presented.) graph close to points where (Formula presented.) is not degenerate. Similar results have been know before for scalar partial differential equations (see for instance [5, 6]). The novelty of this approach is that it does not rely on maximum principle methods and is therefore applicable to systems of equations.

  • 160.
    Andersson, John
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    The free boundary near the fixed boundary for the heat equationManuscript (preprint) (Other academic)
  • 161. Andersson, John
    et al.
    Lindgren, Erik
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal Regularity for the No-Sign Obstacle Problem2013In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 66, no 2, p. 245-262Article in journal (Refereed)
    Abstract [en]

    In this paper we prove the optimal C-1,C-1(B-1/2)-regularity for a general obstacle-type problem Delta u = f chi({u not equal 0}) in B-1, under the assumption that f * N is C-1,C-1(B-1), where N is the Newtonian potential. This is the weakest assumption for which one can hope to get C-1,C-1-regularity. As a by-product of the C-1,C-1-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point x(0), the free boundary is locally a C-1-graph close to x(0) provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades.

  • 162.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal regularity for the obstacle problem for the p-Laplacian2015In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 259, no 6, p. 2167-2179Article in journal (Refereed)
    Abstract [en]

    In this paper we discuss the obstacle problem for the p-Laplace operator. We prove optimal growth results for the solution. Of particular interest is the point-wise regularity of the solution at free boundary points. The most surprising result we prove is the one for the p-obstacle problem: Find the smallest u such thatdiv(|∇u|p-2∇u)≤0,u≥ϕ,in B1, with ϕ∈C1,1(B1) and given boundary datum on ∂B1. We prove that the solution is uniformly C1,1 at free boundary points. Similar results are obtained in the case of an inhomogeneity belonging to L∞. When applied to the corresponding parabolic problem, these results imply that any solution which is Lipschitz in time is C1,1p-1 in the spatial variables.

  • 163.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Lindgren, Erik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Optimal regularity for the parabolic no-sign obstacle type problem2013In: Interfaces and free boundaries (Print), ISSN 1463-9963, E-ISSN 1463-9971, Vol. 15, no 4, p. 477-499Article in journal (Refereed)
    Abstract [en]

    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.

  • 164.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Matevosyan, Norayr
    Mikayelyan, Hayk
    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)
    Abstract [en]

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

  • 165. Andersson, John
    et al.
    Shah Gholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    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)
  • 166.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Shahgholian, Henrik
    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)
    Abstract [en]

    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.

  • 167.
    Andersson, John
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Uraltseva, Nina N.
    Weiss, Georg S.
    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)
    Abstract [en]

    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.

  • 168. Andersson, John
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    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)
    Abstract [en]

    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.

  • 169.
    Andersson, John
    et al.
    Mathematics Institute, University of Warwick.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Weiss, Georg S.
    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)
    Abstract [en]

    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.

  • 170.
    Andersson, Lars
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Backdahl, Thomas
    Blue, Pieter
    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)
    Abstract [en]

    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.

  • 171.
    Andersson, Lars
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Max Planck Institute for Gravitational Physics, United Kingdom.
    Blue, P.
    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)
    Abstract [en]

    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.

  • 172. Andersson, Lars
    et al.
    Dahl, Mattias
    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)
    Abstract [en]

    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.

  • 173.
    Andersson, Lars
    et al.
    KTH, Superseded Departments, Mathematics.
    Dahl, Mattias
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Howard, Ralph
    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)
    Abstract [en]

    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.

  • 174.
    Andersson, Markus
    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
    Abstract [en]

    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.

  • 175.
    Andersson, Martin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Constructing Multidimensional Dynamical Systems with Positive Lyapunov Exponents2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    In this thesis we adapt Marcelo Viana's general construction of smooth transformations exhibiting non-uniform expansion in several dimensions. In our construction we, instead of coupling with a quadratic map, as Viana did, derive the expansion from a mapping with a cubic critical point. Further, we discuss how the argument can be extended and expansion derived from certain mappings with a critical point of arbitrarily high (odd) degree. We also discuss the issues in trying to use Viana's method with the double standard map as our source of non-uniform expansion.

  • 176.
    Andersson, Nicole
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
    Söderberg, Petra
    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
    Abstract [en]

    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. 

  • 177. Andersson, O.
    et al.
    Argenton, C.
    Weibull, Jörgen W.
    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)
    Abstract [en]

    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.

  • 178. Andersson, Ola
    et al.
    Argenton, Cedric
    Weibull, Jörgen W.
    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)
    Abstract [en]

    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.

  • 179.
    Andersson, Sofia
    et al.
    AstraZeneca R and D.
    Rydén, Tobias
    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)
    Abstract [en]

    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.

  • 180.
    Andreasson, Håkan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). Univ Gothenburg.
    Ringstrom, H.
    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)
    Abstract [en]

    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.

  • 181.
    André, Léo
    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
    Abstract [en]

    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.

  • 182.
    Anisi, David A.
    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)
    Abstract [en]

    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.

  • 183.
    Anisi, David A.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Adaptive Node Distribution for Online Trajectory PlanningManuscript (Other academic)
  • 184.
    Anisi, David A.
    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)
    Abstract [en]

    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.

  • 185.
    Anisi, David A.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Online trajectory planning and observer based control2006Licentiate thesis, comprehensive summary (Other scientific)
    Abstract [en]

    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.

  • 186.
    Anisi, David A.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Hamberg, Johan
    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)
    Abstract [en]

    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.

  • 187.
    Anisi, David A.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Hu, Xiaoming
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Active Observers for Mobile Robotic SystemsManuscript (Other academic)
    Abstract [en]

    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.

  • 188.
    Anisi, David A.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Ögren, Petter
    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)
    Abstract [en]

    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.

  • 189. Anisi, David A.
    et al.
    Ögren, Petter
    Swedish Defence Research Agency (FOI), Sweden.
    Hu, Xiaoming
    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)
    Abstract [en]

    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.

  • 190.
    Anisi, David A.
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Ögren, Petter
    Department of Autonomous Systems Swedish Defence Research Agency.
    Robinson, John W. C.
    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)
    Abstract [en]

    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.

  • 191.
    Anisi, David
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
    Robinson, John W.C.
    Dept. of Autonomous Systems, Swedish Defence Research Agency (FOI), Stockholm, Sweden.
    Ögren, Petter
    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)
    Abstract [en]

    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. 

  • 192.
    Ansanay-Alex, Guillaume
    et al.
    Institut de Sûreté et de Radioprotection Nucléaire.
    Babik, Fabrice
    Institut de Sûreté et de Radioprotection Nucléaire.
    Gastaldo, Laura
    Institut de Sûreté et de Radioprotection Nucléaire.
    Larcher, Aurélien
    Institut de Sûreté et de Radioprotection Nucléaire.
    Lapuerta, Céline
    Institut de Sûreté et de Radioprotection Nucléaire.
    Latché, Jean-Claude
    Institut de Sûreté et de Radioprotection Nucléaire.
    Vola, Didier
    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)
    Abstract [en]

    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.

  • 193.
    Appelö, Daniel
    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)
    Abstract [en]

    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.

  • 194.
    Appelö, Daniel
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Hagstrom, Thomas
    Construction of stable PMLs for general 2 x 2 symmetric hyperbolic systems2004In: Proceedings of the HYP2004 conference, 2004, p. 1-8Conference paper (Refereed)
    Abstract [en]

    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.

  • 195.
    Appelö, Daniel
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Hagstrom, Thomas
    Department of Mathematics and Statistics, University of New Mexico.
    Kreiss, Gunilla
    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)
    Abstract [en]

    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.

  • 196.
    Appelö, Daniel
    et al.
    KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
    Kreiss, Gunilla
    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)
    Abstract [en]

    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.

  • 197.
    Appelö, Daniel
    et al.
    KTH, Superseded Departments, Numerical Analysis and Computer Science, NADA.
    Kreiss, Gunilla
    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)
    Abstract [en]

    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.

  • 198. Apushkinskaya, D. E.
    et al.
    Ural’Tseva, N. N.
    Shahgholian, Henrik
    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)
    Abstract [en]

    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.

  • 199. Arakelyan, A.
    et al.
    Shah Gholian, Henrik
    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)
    Abstract [en]

    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.

  • 200. Arakelyan, A.
    et al.
    Shahgholian, Henrik
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Prajapat, J. V.
    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)
    Abstract [en]

    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.

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