Publications by authors named "V M Veliov"

7 Publications

Gradient Methods on Strongly Convex Feasible Sets and Optimal Control of Affine Systems.

Appl Math Optim 2020 6;81(3):1021-1054. Epub 2018 Oct 6.

Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria.

The paper presents new results about convergence of the gradient projection and the conditional gradient methods for abstract minimization problems on strongly convex sets. In particular, linear convergence is proved, although the objective functional does not need to be convex. Such problems arise, in particular, when a recently developed discretization technique is applied to optimal control problems which are affine with respect to the control. This discretization technique has the advantage to provide higher accuracy of discretization (compared with the known discretization schemes) and involves strongly convex constraints and possibly non-convex objective functional. The applicability of the abstract results is proved in the case of linear-quadratic affine optimal control problems. A numerical example is given, confirming the theoretical findings.
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http://dx.doi.org/10.1007/s00245-018-9528-3DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7319312PMC
October 2018

Higher-order numerical scheme for linear quadratic problems with bang-bang controls.

Comput Optim Appl 2018 6;69(2):403-422. Epub 2017 Oct 6.

2Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria.

This paper considers a linear-quadratic optimal control problem where the control function appears linearly and takes values in a hypercube. It is assumed that the optimal controls are of purely bang-bang type and that the switching function, associated with the problem, exhibits a suitable growth around its zeros. The authors introduce a scheme for the discretization of the problem that doubles the rate of convergence of the Euler's scheme. The proof of the accuracy estimate employs some recently obtained results concerning the stability of the optimal solutions with respect to disturbances.
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http://dx.doi.org/10.1007/s10589-017-9948-zDOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6566299PMC
October 2017

Set-membership estimations for the evolution of infectious diseases in heterogeneous populations.

J Math Biol 2017 04 7;74(5):1081-1106. Epub 2016 Sep 7.

ORCOS, Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstra├če 8/E105-4, 1040, Vienna, Austria.

The paper presents an approach for set-membership estimation of the state of a heterogeneous population in which an infectious disease is spreading. The population state may consist of susceptible, infected, recovered, etc. groups, where the individuals are heterogeneous with respect to traits, relevant to the particular disease. Set-membership estimations in this context are reasonable, since only vague information about the distribution of the population along the space of heterogeneity is available in practice. The presented approach comprises adapted versions of methods which are known in estimation and control theory, and involve solving parametrized families of optimization problems. Since the models of disease spreading in heterogeneous populations involve distributed systems (with non-local dynamics and endogenous boundary conditions), these problems are non-standard. The paper develops the needed theoretical instruments and a solution scheme. SI and SIR models of epidemic diseases are considered as case studies and the results reveal qualitative properties that may be of interest.
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http://dx.doi.org/10.1007/s00285-016-1050-0DOI Listing
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5388773PMC
April 2017

Modelling and estimation of infectious diseases in a population with heterogeneous dynamic immunity.

Authors:
V M Veliov A Widder

J Biol Dyn 2016 12;10(1):457-76

a ORCOS , Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology , Vienna , Austria.

Unlabelled: The paper presents a model for the evolution of an infectious disease in a population with individual-specific immunity. The immune state of an individual varies with time according to its own dynamics, depending on whether the individual is infected or not. The model involves a system of size-structured (first-order) PDEs that capture both the dynamics of the immune states and the transition between compartments consisting of infected, susceptible, etc.

Individuals: Due to the unavailability of precise data about the immune states of the individuals, the main focus in the paper is on developing a technique for set-membership estimations of aggregated quantities of interest. The technique involves solving specific optimization problems for the underlying PDE system and is developed up to a numerical method. Results of numerical simulations are presented for a benchmark model of SIS-type, potentially applicable to diseases like influenza and to various sexually transmitted diseases.
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http://dx.doi.org/10.1080/17513758.2016.1221474DOI Listing
December 2016

Aggregation and asymptotic analysis of an SI-epidemic model for heterogeneous populations.

Authors:
V M Veliov A Widder

Math Med Biol 2016 09 26;33(3):295-318. Epub 2015 May 26.

ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstrasse 8, A-1040 Vienna, Austria

The paper investigates a version of a simple epidemiological model involving only susceptible and infected individuals, where the heterogeneity of the population with respect to susceptibility/infectiousness is taken into account. A comprehensive analysis of the asymptotic behaviour of the disease is given, based on an explicit aggregation of the model. The results are compared with those of a homogeneous version of the model to highlight the influence of the heterogeneity on the asymptotics. Moreover, the performed analysis reveals in which cases incomplete information about the heterogeneity of the population is sufficient in order to determine the long-run outcome of the disease. Numerical simulation is used to emphasize that, for a given level of prevalence, the evolution of the disease under the influence of heterogeneity may in the long run qualitatively differ from the one 'predicted' by the homogeneous model. Furthermore, it is shown that, in a closed population, the indicator for the survival of the population is in the presence of heterogeneity distinct from the basic reproduction number.
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http://dx.doi.org/10.1093/imammb/dqv018DOI Listing
September 2016