| 1. |
- Berglund, Tomas, et al.
(författare)
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Epi-convergence of minimum curvature variation B-splines
- 2003
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We study the curvature variation functional, i.e., the integral over the square of arc-length derivative of curvature, along a planar curve. With no other constraints than prescribed position, slope angle, and curvature at the endpoints of the curve, the minimizer of this functional is known as a cubic spiral. It remains a challenge to effectively compute minimizers or approximations to minimizers of this functional subject to additional constraints such as, for example, for the curve to avoid obstacles such as other curves. In this paper, we consider the set of smooth curves that can be written as graphs of three times continuously differentiable functions on an interval, and, in particular, we consider approximations using quartic uniform B- spline functions. We show that if quartic uniform B-spline minimizers of the curvature variation functional converge to a curve, as the number of B-spline basis functions tends to infinity, then this curve is in fact a minimizer of the curvature variation functional. In order to illustrate this result, we present an example of sequences of B-spline minimizers that converge to a cubic spiral.
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| 2. |
- Kjellmert, Bo, et al.
(författare)
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Time-dependent electromagnetic waves in a cavity
- 2009
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Ingår i: Applications of Mathematics. - : Institute of Mathematics, Czech Academy of Sciences. - 0862-7940 .- 1572-9109. ; 54:1, s. 17-45
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Tidskriftsartikel (refereegranskat)abstract
- The electromagnetic initial-boundary value problem for a cavity enclosed by perfectly conducting walls is considered. The cavity medium is defined by its permittivity and permeability which vary continuously in space. The electromagnetic field comes from a source in the cavity. The field is described by a magnetic vector potential A satisfying a wave equation with initial-boundary conditions. This description through A is rigorously shown to give a unique solution of the problem and is the starting point for numerical computations. A Chebyshev collocation solver has been implemented for a cubic cavity, and it has been compared to a standard finite element solver. The results obtained are consistent while the collocation solver performs substantially faster. Some time histories and spectra are computed.
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| 3. |
- Choquard, Philippe, et al.
(författare)
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A one-dimensional inviscid and compressible fluid in a harmonic potential well
- 2007
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Ingår i: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications. - : Springer Science and Business Media LLC. - 0167-8019 .- 1572-9036. ; 99:2, s. 161-183
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Tidskriftsartikel (refereegranskat)abstract
- A Hamiltonian model is analyzed for a one-dimensional inviscid compressible fluid. The space-time evolution of the fluid is governed by the following system of the Hamilton-Jacobi and the continuity equations: S-t + 1/2(S-x(2) + omega(2)chi(2)) =0, S(x, 0) = S-0(x); rho(t) + (rho S-x)(x) =0, rho(x, 0) =rho(0)(x).Here S and rho designate the velocity potencial and the mass density, respectively. Unless S-0 is convex, shocks form and the velocity S (x) becomes discontinuous in {0 < omega t < pi/2}. It is demonstrated that there nevertheless exists a unique viscosity-measure solution (S,rho) when S-0 is globally Lipschitz continuous and locally semi-concave while rho(0) is a finite Borel measure. The structure of the velocity and the density is exhibited. For initial data correlated in a certain sense, a class of classical solutions (S,rho) is given. Negative time is also considered, and illustrating examples are given.
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| 4. |
- Maad, Sara, et al.
(författare)
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Surface fitting with boundary data
- 2004
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Ingår i: 2004 43rd IEEE Conference on Decision and Control. - Piscataway, NJ : IEEE Communications Society. ; , s. 3649-3653
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Konferensbidrag (refereegranskat)abstract
- The problem of fitting surfaces to data is a well studied problem in statistics. However when there is prior information the theory is not developed. It often happens in toxicology and in medicine that the effect of a single drug is well understood.. However if a pair of drugs is delivered in tandem or if two toxins are interacting the effect is not understood. In this paper we attack the problem of fitting a surface to a data set contained in a square when two of the boundaries are known. Our approach generalizes the concept of smoothing splines.
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| 5. |
- Persson, Lars-Erik, et al.
(författare)
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Green's method applied to the plate equation in mechanics
- 1993
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Ingår i: Annales Societatis Mathematicae Polonae. Series 1: Commentationes Mathematicae - Prace Matematyczne. - 0373-8299. ; 33, s. 119-133
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Tidskriftsartikel (refereegranskat)abstract
- This paper demonstrates existence and uniqueness of Green functions for the general fourth-order linear elliptic operator modeling the deflection of an anisotropic elastic plate with the linear boundary conditions that model the most common edge conditions. The existence theory is presented in the context of Sobolev spaces. Included is the derivation of the boundary value problem from the principles of elasticity and of the Green function for a homogeneous rectangular plate. The paper concludes with a brief discussion of qualitative properties of the Green functions. The techniques and results are classic, and the exposition is accessible to a broad audience. This paper could serve as an admirable introduction to the boundary value problems of anisotropic plates.
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| 6. |
- Strömberg, Thomas
(författare)
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A counterexample to uniqueness of generalized characteristics in Hamilton-Jacobi theory
- 2011
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Ingår i: Nonlinear Analysis. - : Elsevier BV. - 0362-546X .- 1873-5215. ; 74:7, s. 2758-2762
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Tidskriftsartikel (refereegranskat)abstract
- The notion of generalized characteristics plays a pivotal role in the study of propagation of singularities for Hamilton{Jacobi equations. This note gives an example of nonuniqueness of forward generalized characteristics emanating from a given point.
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| 7. |
- Strömberg, Thomas
(författare)
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A new proof of indefinite propagation of singularities for a Hamilton-Jacobi equation
- 2016
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Ingår i: Journal of evolution equations (Printed ed.). - : Springer Science and Business Media LLC. - 1424-3199 .- 1424-3202. ; 16:4, s. 895-903
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Tidskriftsartikel (refereegranskat)abstract
- We study propagation of singularities for the Hamilton–Jacobi equation S t +H(∇S)=0,(t,x)∈(0,T)×R n , St+H(∇S)=0,(t,x)∈(0,T)×Rn,where H(p)=12 ⟨p,Ap⟩ H(p)=12⟨p,Ap⟩ is a positive definite quadratic form. Each viscosity solution S S is semiconcave, and it is known that its singularities move along generalized characteristics. We give a new proof of the recent result by Cannarsa et al. (Discrete Contin Dyn Syst 35:4225–4239, 2015), namely that the singularities propagate along generalized characteristics indefinitely forward in time.
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| 8. |
- Strömberg, Thomas
(författare)
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A note on the differentiability of conjugate functions
- 2009
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Ingår i: Archiv der Mathematik. - : Springer Science and Business Media LLC. - 0003-889X .- 1420-8938. ; 93:5, s. 481-485
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Tidskriftsartikel (refereegranskat)abstract
- For a proper, lower semicontinuous and convex function f with Legendre-Fenchel conjugate f *, it is well-known that differentiability properties of f * are equivalent to strict convexity properties of f. In this note a result of this kind is obtained without a convexity assumption on f.
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| 9. |
- Strömberg, Thomas
(författare)
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A study of the operation of infimal convolution
- 1994
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of five papers (A-E), which examine the operation of infimal convolution and discuss its close connections to unilateral analysis, convex analysis, inequalities, approximation, and optimization. In particular, we attempt to provide a detailed investigation for both the convex and the non-convex case, including several examples. Paper (A) is both a survey of and a self-contained introduction to the operation of infimal convolution. In particular, we discuss the infimal value and minimizers of an infimal convolute, infimal convolution on subadditive functions, sufficient conditions for semicontinuity or continuity of an infimal convolute, "exactness," regularizing effects, continuity of the operation of infimal convolution, and approximation methods based on infimal convolution. A Young-type inequality, closely connected to the operation of infimal convolution, is studied in paper (B). The main results obtained are an equivalence theorem and a representation formula. In paper (C) we consider coercive, convex, proper, and lower sernicontinuous functions on a reflexive Banach space. For the infimal convolution of such functions we establish, in particular, different formulae. Moreover, we demonstrate the possibility of using the formulae obtained for solving special types of Hamilton-Jacobi equations. Furthermore, the operation of infimal convolution is interpreted from a physical viewpoint. Paper (D) presents properties of infimal convolution of functions that are uniformly continuous on bounded sets. In particular, we present regularization procedures by means of infimal convolution. The role of growth conditions on the functions under consideration is essential. Finally, in paper (E) we study semicontinuity, continuity, and differentiability of the infimal convolute of two convex functions. Moreover, under certain geometric conditions, the classical Moreau-Yosida approximation process is, roughly speaking, extended to the non-convex case.
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| 10. |
- Strömberg, Thomas
(författare)
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A system of the Hamilton-Jacobi and the continuity equations in the vanishing viscosity limit
- 2011
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Ingår i: Communications on Pure and Applied Analysis. - : American Institute of Mathematical Sciences (AIMS). - 1534-0392 .- 1553-5258. ; 10:2, s. 479-506
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Tidskriftsartikel (refereegranskat)abstract
- We study the following system of the viscous Hamilton-Jacobi and the continuity equations in the limit as epsilon down arrow 0: S-t(epsilon) + 1/2 vertical bar DS epsilon vertical bar(2) + V(x) - epsilon Delta S-epsilon = 0 in Q(T), S-epsilon(0, x) = S-0(x) in R-n; rho(epsilon)(t) + div(rho(epsilon) DS epsilon) = 0 in Q(T), rho(epsilon)(0, x) = rho(0)(x) in R-n. Here Q(T) = (0, T] x R-n. The potential V and the initial function S-0 are allowed to grow quadratically while rho(0) is a Borel measure. The paper justifies and describes the vanishing viscosity transition to the corresponding inviscid system. The notion of weak solution employed for the inviscid system is that of a viscosity-measure solution (S, rho).
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