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  • 1. Agram, Nacira
    et al.
    Øksendal, Bernt
    Stochastic Control of Memory Mean-Field Processes2017Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 79, nr 1, s. 181-204Artikkel i tidsskrift (Fagfellevurdert)
  • 2.
    Andersson, Daniel
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    Djehiche, Boualem
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    A maximum principle for SDEs of mean-field type2011Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 63, nr 3, s. 341-356Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study the optimal control of a stochastic differential equation (SDE) of mean-field type, where the coefficients are allowed to depend on some functional of the law as well as the state of the process. Moreover the cost functional is also of mean-field type, which makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. Under the assumption of a convex action space a maximum principle of local form is derived, specifying the necessary conditions for optimality. These are also shown to be sufficient under additional assumptions. This maximum principle differs from the classical one, where the adjoint equation is a linear backward SDE, since here the adjoint equation turns out to be a linear mean-field backward SDE. As an illustration, we apply the result to the mean-variance portfolio selection problem.

  • 3. Bahlali, Khaled
    et al.
    Djehiche, Boualem
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    Mezerdi, Brahim
    On the stochastic maximum principle in optimal control of degenerate diffusions with lipschitz coefficients2007Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 56, nr 3, s. 364-378Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We establish a stochastic maximum principle in optimal control of a general class of degenerate diffusion processes with global Lipschitz coefficients, generalizing the existing results on stochastic control of diffusion processes. We use distributional derivatives of the coefficients and the Bouleau Hirsh flow property, in order to define the adjoint process on an extension of the initial probability space.

  • 4. Buckdahn, Rainer
    et al.
    Djehiche, Boualem
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    Li, Juan
    A General Stochastic Maximum Principle for SDEs of Mean-field Type2011Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 64, nr 2, s. 197-216Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study the optimal control for stochastic differential equations (SDEs) of mean-field type, in which the coefficients depend on the state of the solution process as well as of its expected value. Moreover, the cost functional is also of mean-field type. This makes the control problem time inconsistent in the sense that the Bellman optimality principle does not hold. For a general action space a Peng's-type stochastic maximum principle (Peng, S.: SIAM J. Control Optim. 2(4), 966-979, 1990) is derived, specifying the necessary conditions for optimality. This maximum principle differs from the classical one in the sense that here the first order adjoint equation turns out to be a linear mean-field backward SDE, while the second order adjoint equation remains the same as in Peng's stochastic maximum principle.

  • 5.
    Djehiche, Boualem
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Hamadene, Said
    Univ Maine, LMM, Ave Olivier Messiaen, F-72085 Le Mans 9, France..
    Optimal Control and Zero-Sum Stochastic Differential Game Problems of Mean-Field Type2020Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 81, nr 3, s. 933-960Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We establish existence of nearly-optimal controls, conditions for existence of an optimal control and a saddle-point for respectively a control problem and zero-sum differential game associated with payoff functionals of mean-field type, under dynamics driven by weak solutions of stochastic differential equations of mean-field type.

  • 6.
    Djehiche, Boualem
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematisk statistik.
    Hamadene, Said
    Hdhiri, Ibtissam
    Stochastic Impulse Control of Non-Markovian Processes2010Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 61, nr 1, s. 1-26Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider a class of stochastic impulse control problems of general stochastic processes i.e. not necessarily Markovian. Under fairly general conditions we establish existence of an optimal impulse control. We also prove existence of combined optimal stochastic and impulse control of a fairly general class of diffusions with random coefficients. Unlike, in the Markovian framework, we cannot apply quasi-variational inequalities techniques. We rather derive the main results using techniques involving reflected BSDEs and the Snell envelope.

  • 7. Doll, Jim
    et al.
    Dupuis, Paul
    Nyquist, Pierre
    A large deviation analysis of certain qualitative properties of parallel tempering and infinite swapping algorithms2018Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 78, nr 1, s. 103-144Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Parallel tempering, or replica exchange, is a popular method for simulating complex systems. The idea is to run parallel simulations at different temperatures, and at a given swap rate exchange configurations between the parallel simulations. From the perspective of large deviations it is optimal to let the swap rate tend to infinity and it is possible to construct a corresponding simulation scheme, known as infinite swapping. In this paper we propose a novel use of large deviations for empirical measures for a more detailed analysis of the infinite swapping limit in the setting of continuous time jump Markov processes. Using the large deviations rate function and associated stochastic control problems we consider a diagnostic based on temperature assignments, which can be easily computed during a simulation. We show that the convergence of this diagnostic to its a priori known limit is a necessary condition for the convergence of infinite swapping. The rate function is also used to investigate the impact of asymmetries in the underlying potential landscape, and where in the state space poor sampling is most likely to occur.

  • 8. Lang, L. F.
    et al.
    Neumayer, S.
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Centra, Strategiskt centrum för industriell och tillämpad matematik, CIAM. KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Schönlieb, C. -B
    Template-Based Image Reconstruction from Sparse Tomographic Data2019Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We propose a variational regularisation approach for the problem of template-based image reconstruction from indirect, noisy measurements as given, for instance, in X-ray computed tomography. An image is reconstructed from such measurements by deforming a given template image. The image registration is directly incorporated into the variational regularisation approach in the form of a partial differential equation that models the registration as either mass- or intensity-preserving transport from the template to the unknown reconstruction. We provide theoretical results for the proposed variational regularisation for both cases. In particular, we prove existence of a minimiser, stability with respect to the data, and convergence for vanishing noise when either of the abovementioned equations is imposed and more general distance functions are used. Numerically, we solve the problem by extending existing Lagrangian methods and propose a multilevel approach that is applicable whenever a suitable downsampling procedure for the operator and the measured data can be provided. Finally, we demonstrate the performance of our method for template-based image reconstruction from highly undersampled and noisy Radon transform data. We compare results for mass- and intensity-preserving image registration, various regularisation functionals, and different distance functions. Our results show that very reasonable reconstructions can be obtained when only few measurements are available and demonstrate that the use of a normalised cross correlation-based distance is advantageous when the image intensities between the template and the unknown image differ substantially.

  • 9.
    Strömqvist, Martin H.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Optimal Control of the Obstacle Problem in a Perforated Domain2012Inngår i: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 66, nr 2, s. 239-255Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study the problem of optimally controlling the solution of the obstacle problem in a domain perforated by small periodically distributed holes. The solution is controlled by the choice of a perforated obstacle which is to be chosen in such a fashion that the solution is close to a given profile and the obstacle is not too irregular. We prove existence, uniqueness and stability of an optimal obstacle and derive necessary and sufficient conditions for optimality. When the number of holes increase indefinitely we determine the limit of the sequence of optimal obstacles and solutions. This limit depends strongly on the rate at which the size of the holes shrink.

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