| 1. |
- Dahne, Joel, 1994-
(author)
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Computer Assisted Studies in Fluid Mechanics and Spectral Geometry
- 2024
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Doctoral thesis (other academic/artistic)abstract
- This thesis contains four papers in the area of partial differential equations. The first two papers are related to spectral geometry and the last two papers to fluid mechanics. A common theme of the papers is that they make use of computer assisted methods in the proofs.Paper I concerns the computation of very precise enclosures of eigenvalues of the Laplace-Beltrami operator on spherical triangles. The interest in these eigenvalues comes from a connection with the combinatorial problem of studying discrete random walks.Paper II gives a concrete counterexample to Payne's nodal line conjecture. The conjecture is concerned with the existence of bounded planar domains for which the second eigenfunction of the Dirichlet Laplacian has a nodal line that doesn't touch the boundary. The paper gives an explicit domain for which it is proved that the nodal line doesn't touch the boundary.Paper III and IV both prove the existence of a certain type of solution known as a highest cusped traveling wave. Paper III deals with the Burgers-Hilbert equation and Paper IV with a family of fractional Korteweg-de Vries equations. The existence is asserted by constructing an explicit approximation of the solution and proving the existence of an exact solution nearby with the use of a fixed point formulation. The proof not only establishes the existence, but also determines the precise asymptotic behavior of the waves near the cusp.
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| 2. |
- Dahne, Joel, et al.
(author)
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Enclosing all zeros of a system of analytic functions
- 2019
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In: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 348, s. 513-522
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Journal article (peer-reviewed)abstract
- We present a rigorous numerical method for location of simple zeros of a system of two analytic functions in a rectangular cuboid domain based on the logarithmic integral. We compare this to a simpler, also rigorous, method based on bisection. The latter is determined to be more efficient in the examples considered. This is mainly due to inefficient methods for computing the logarithmic integral occurring in the former method. (C) 2018 Elsevier Inc. All rights reserved.
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| 3. |
- Galias, Zbigniew, et al.
(author)
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On the Existence of the Double Scroll Attractor for the Chua's Circuit with a Smooth Nonlinearity
- 2018
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In: 2018 IEEE INTERNATIONAL SYMPOSIUM ON CIRCUITS AND SYSTEMS (ISCAS). - : IEEE. - 9781538648810
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Conference paper (peer-reviewed)abstract
- In simulations of the Chua's circuit with a smooth nonlinearity for certain parameter values one observes the double scroll attractor. This attractor contains an unstable equilibrium, and typical trajectories belonging to the attractor may pass arbitrarily close to this equilibrium. In consequence, it is impossible to compute trajectories over the whole attractor using standard rigorous numerical integration procedures. This is due to the existence of trajectories which spend arbitrarily long time in a neighborhood of the equilibrium. In this work, a method to find enclosures of trajectories passing arbitrarily close to an unstable fixed point of spiral type is presented. This method is used to prove the existence of a trapping region enclosing the double scroll attractor for the Chua's circuit with a cubic nonlinearity.
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| 4. |
- Galias, Zbigniew, et al.
(author)
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Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor
- 2019
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In: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 266:5, s. 2408-2434
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Journal article (peer-reviewed)abstract
- In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow.
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| 5. |
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| 6. |
- Gennemark, Peter, 1974, et al.
(author)
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Optimal Design in Population Kinetic Experiments by Set-Valued Methods
- 2011
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In: AAPS Journal. - : Springer Science and Business Media LLC. - 1550-7416. ; 13:4, s. 495-507
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Journal article (peer-reviewed)abstract
- We propose a new method for optimal experimental design of population pharmacometric experiments based on global search methods using interval analysis; all variables and parameters are represented as intervals rather than real numbers. The evaluation of a specific design is based on multiple simulations and parameter estimations. The method requires no prior point estimates for the parameters, since the parameters can incorporate any level of uncertainty. In this respect, it is similar to robust optimal design. Representing sampling times and covariates like doses by intervals gives a direct way of optimizing with rigorous sampling and dose intervals that can be useful in clinical practice. Furthermore, the method works on underdetermined problems for which traditional methods typically fail.
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| 7. |
- Johnson, Tomas, 1979-, et al.
(author)
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A note on the convergence of parametrised non-resonant invariant manifolds
- 2011
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In: Qualitative Theory of Dynamical Systems. - : Springer Science and Business Media LLC. - 1575-5460 .- 1662-3592. ; 10:1, s. 107-121
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Journal article (peer-reviewed)abstract
- Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds.
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| 8. |
- Johnson, Tomas, 1979-, et al.
(author)
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A rigorous study of possible configurations of limit cycles bifurcating from a hyper-elliptic Hamiltonian of degree five
- 2009
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In: Dynamical systems. - : Informa UK Limited. - 1468-9367 .- 1468-9375. ; 24:2, s. 237-247
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Journal article (peer-reviewed)abstract
- We consider a hyper-elliptic Hamiltonian of degree five, chosen from a generic set of parameters, and study what configurations of limit cycles can bifurcate from the corresponding differential system under quartic perturbations. Perturbations of Lienard type are considered separately. Several different configurations with seven (four) limit cycles, bifurcating from the given system for general (Lienard type) quartic perturbations, are constructed. We also discuss how to construct perturbations yielding a given configuration, and how to validate the correctness of such a candidate perturbation.
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| 9. |
- Johnson, Tomas, 1979-, et al.
(author)
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An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7
- 2010
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In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - : World Scientific Publishing. - 0218-1274. ; 20:5, s. 1451-1458
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Journal article (peer-reviewed)abstract
- The limit cycle bifurcations of a Z(2) equivariant planar Hamiltonian vector field of degree 7 under Z(2) equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert's 16th problem for degree 7, i.e. on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation.
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| 10. |
- Johnson, Tomas, 1979-, et al.
(author)
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An improved lower bound on the number of limit cycles bifurcating from a quintic Hamiltonian planar vector field under quintic perturbation
- 2010
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In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering. - : World Scientific Publishing. - 0218-1274. ; 20:1, s. 63-70
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Journal article (peer-reviewed)abstract
- The limit cycle bifurcations of a equivariant quintic planar Hamiltonian vector field under equivariant quintic perturbation is studied. We prove that the given system can have at least 27 limit cycles. This is an improved lower bound on the possible number of limit cycles that can bifurcate from a quintic planar Hamiltonian system under quintic perturbation.
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| 11. |
- Johnson, Tomas, 1979-, et al.
(author)
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Automated computation of robust normal forms of planar analytic vector fields
- 2009
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In: Discrete and continuous dynamical systems. Series B. - : American Institute of Mathematical Sciences (AIMS). - 1531-3492 .- 1553-524X. ; 12:4, s. 769-782
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Journal article (peer-reviewed)abstract
- We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighbourhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method.
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| 12. |
- Johnson, Tomas, 1979-, et al.
(author)
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Enclosing all zeros of an analytic function : a rigorous approach
- 2009
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In: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427 .- 1879-1778. ; 228:1, s. 418-423
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Journal article (peer-reviewed)abstract
- We present a method to find all zeros of an analytic function in a rectangular domain. The approach is based on finding guaranteed enclosures rather than approximations of the zeros. Well-isolated simple zeros are determined fast and with high accuracy. Clusters of zeros can in many cases be distinguished from multiple zeros by applying the argument principle to sufficiently high-order derivatives of the function. We illustrate the proposed method through five examples of varying levels of complexity.
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| 13. |
- Johnson, Tomas, 1979-, et al.
(author)
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On a computer-aided approach to the computation of Abelian integrals
- 2011
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In: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 51:3, s. 653-667
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Journal article (peer-reviewed)abstract
- An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is applied to the study of bifurcations of limit cycles arising from a cubic perturbation of an elliptic Hamiltonian of degree four.
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| 14. |
- Johnson, Tomas, 1979-, et al.
(author)
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Rigorous parameter reconstruction for differential equations with noisy data
- 2008
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In: Automatica. - : Elsevier BV. - 0005-1098 .- 1873-2836. ; 44:9, s. 2422-2426
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Journal article (peer-reviewed)abstract
- We present a method that–given a data set, a finitely parametrized system of ordinary differential equations (ODEs), and a search space of parameters–discards portions of the search space that are inconsistent with the model ODE and data. The method is completely rigorous as it is based on validated integration of the vector field. As a consequence, no consistent parameters can be lost during the pruning phase. For data sets with moderate levels of noise, this yields a good reconstruction of the underlying parameters. Several examples are included to illustrate the merits of the method.
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| 15. |
- Mitrea, Irina, et al.
(author)
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Invertibility Properties of Singular Integral Operators Associated with the Lam, and Stokes Systems on Infinite Sectors in Two Dimensions
- 2017
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In: Integral equations and operator theory. - : Springer Science and Business Media LLC. - 0378-620X .- 1420-8989. ; 89:2, s. 151-207
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Journal article (peer-reviewed)abstract
- In this paper we establish sharp invertibility results for the elastostatics and hydrostatics single and double layer potential type operators acting on , , whenever is an infinite sector in . This analysis is relevant to the layer potential treatment of a variety of boundary value problems for the Lam, system of elastostatics and the Stokes system of hydrostatics in the class of curvilinear polygons in two dimensions, such as the Dirichlet, the Neumann, and the Regularity problems. Mellin transform techniques are used to identify the critical integrability indices for which invertibility of these layer potentials fails. Computer-aided proofs are produced to further study the monotonicity properties of these indices relative to parameters determined by the aperture of the sector and the differential operator in question.
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| 16. |
- Ryeznik, Yevgen, 1979-
(author)
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Optimal adaptive designs and adaptive randomization techniques for clinical trials
- 2019
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Doctoral thesis (other academic/artistic)abstract
- In this Ph.D. thesis, we investigate how to optimize the design of clinical trials by constructing optimal adaptive designs, and how to implement the design by adaptive randomization. The results of the thesis are summarized by four research papers preceded by three chapters: an introduction, a short summary of the results obtained, and possible topics for future work.In Paper I, we investigate the structure of a D-optimal design for dose-finding studies with censored time-to-event outcomes. We show that the D-optimal design can be much more efficient than uniform allocation design for the parameter estimation. The D-optimal design obtained depends on true parameters of the dose-response model, so it is a locally D-optimal design. We construct two-stage and multi-stage adaptive designs as approximations of the D-optimal design when prior information about model parameters is not available. Adaptive designs provide very good approximations to the locally D-optimal design, and can potentially reduce total sample size in a study with a pre-specified stopping criterion.In Paper II, we investigate statistical properties of several restricted randomization procedures which target unequal allocation proportions in a multi-arm trial. We compare procedures in terms of their operational characteristics such as balance, randomness, type I error/power, and allocation ratio preserving (ARP) property. We conclude that there is no single “best” randomization procedure for all the target allocation proportions, but the choice of randomization can be done through computer-intensive simulations for a particular target allocation.In Paper III, we combine the results from the papers I and II to implement optimal designs in practice when the sample size is small. The simulation study done in the paper shows that the choice of randomization procedure has an impact on the quality of dose-response estimation. An adaptive design with a small cohort size should be implemented with a procedure that ensures a “well-balanced” allocation according to the D-optimal design at each stage.In Paper IV, we obtain an optimal design for a comparative study with unequal treatment costs and investigate its properties. We demonstrate that unequal allocation may decrease the total study cost while having the same power as traditional equal allocation. However, a larger sample size may be required. We suggest a strategy on how to choose a suitable randomization procedure which provides a good trade-off between balance and randomness to implement optimal allocation. If there is a strong linear trend in observations, then the ARP property is important to maintain the type I error and power at a certain level. Otherwise, a randomization-based inference can be a good alternative for non-ARP procedures.
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| 17. |
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| 18. |
- Tucker, Warwick, 1970-
(author)
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Computer-Aided Proofs via Interval Analysis
- 2015
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In: The Princeton Companion to Applied Mathematics. - : Princeton University Press. - 9780691150390 ; , s. 790-795
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Book chapter (peer-reviewed)
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| 19. |
- Tucker, Warwick, 1970-
(author)
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Interval Analysis
- 2015
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In: The Princeton Companion to Applied Mathematics. - : Princeton University Press. - 9780691150390 ; , s. 105-106
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Book chapter (peer-reviewed)
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| 20. |
- Tucker, Warwick, 1970-
(author)
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Interval Methods
- 2015
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In: <em>Uncertainty in Biology</em>. - Cham : Springer Publishing Company. - 9783319212968 - 9783319212951 ; , s. 99-211
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Book chapter (peer-reviewed)
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| 21. |
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| 22. |
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