12,409 research outputs found
Solving the Monge-Amp\`ere Equations for the Inverse Reflector Problem
The inverse reflector problem arises in geometrical nonimaging optics: Given
a light source and a target, the question is how to design a reflecting
free-form surface such that a desired light density distribution is generated
on the target, e.g., a projected image on a screen. This optical problem can
mathematically be understood as a problem of optimal transport and equivalently
be expressed by a secondary boundary value problem of the Monge-Amp\`ere
equation, which consists of a highly nonlinear partial differential equation of
second order and constraints. In our approach the Monge-Amp\`ere equation is
numerically solved using a collocation method based on tensor-product
B-splines, in which nested iteration techniques are applied to ensure the
convergence of the nonlinear solver and to speed up the calculation. In the
numerical method special care has to be taken for the constraint: It enters the
discrete problem formulation via a Picard-type iteration. Numerical results are
presented as well for benchmark problems for the standard Monge-Amp\`ere
equation as for the inverse reflector problem for various images. The designed
reflector surfaces are validated by a forward simulation using ray tracing.Comment: 28 pages, 8 figures, 2 tables; Keywords: Inverse reflector problem,
elliptic Monge-Amp\`ere equation, B-spline collocation method, Picard-type
iteration; Minor revision: reference [59] to a recent preprint has been added
and a few typos have been correcte
A fractional spline collocation method for the fractional order logistic equation
We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method.We construct a collocation method based on the fractional B-splines to solve a nonlinear differential problem that involves fractional derivative, i.e. the fractional order logistic equation. The use of the fractional B-splines allows us to express the fractional derivative of the approximating function in an analytic form. Thus, the fractional collocation method is easy to implement, accurate and efficient. Several numerical tests illustrate the efficiency of the proposed collocation method
Maximum-principle preserving space-time isogeometric analysis
In this work we propose a nonlinear stabilization technique for
convection-diffusion-reaction and pure transport problems discretized with
space-time isogeometric analysis. The stabilization is based on a
graph-theoretic artificial diffusion operator and a novel shock detector for
isogeometric analysis. Stabilization in time and space directions are performed
similarly, which allow us to use high-order discretizations in time without any
CFL-like condition. The method is proven to yield solutions that satisfy the
discrete maximum principle (DMP) unconditionally for arbitrary order. In
addition, the stabilization is linearity preserving in a space-time sense.
Moreover, the scheme is proven to be Lipschitz continuous ensuring that the
nonlinear problem is well-posed. Solving large problems using a space-time
discretization can become highly costly. Therefore, we also propose a
partitioned space-time scheme that allows us to select the length of every time
slab, and solve sequentially for every subdomain. As a result, the
computational cost is reduced while the stability and convergence properties of
the scheme remain unaltered. In addition, we propose a twice differentiable
version of the stabilization scheme, which enjoys the same stability properties
while the nonlinear convergence is significantly improved. Finally, the
proposed schemes are assessed with numerical experiments. In particular, we
considered steady and transient pure convection and convection-diffusion
problems in one and two dimensions
Un método Wavelet-Galerkin para ecuaciones diferenciales parciales parabólicas
In this paper an Adaptive Wavelet-Galerkin method for the solution ofparabolic partial differential equations modeling physical problems withdifferent spatial and temporal scales is developed. A semi-implicit timedifference scheme is applied andB-spline multiresolution structure on theinterval is used. As in many cases these solutions are known to presentlocalized sharp gradients, local error estimators are designed and an ef-ficient adaptive strategy to choose the appropriate scale for each time isdeveloped. Finally, experiments were performed to illustrate the applica-bility and efficiency of the proposed method.En este trabajo se desarrolla un método Wavelet-Galerkin Adaptativopara la resolución de ecuaciones diferenciales parabólicas que modelanproblemas fÃsicos, con diferentes escalas en el espacio y en el tiempo. Seutiliza un esquema semi-implÃcito en diferencias temporales y la estructuramultirresolución de las B-splines sobre intervalo.Como es sabido que enmuchos casos las soluciones presentan gradientes localmente altos, se handiseñado estimadores locales de error y una estrategia adaptativa eficientepara elegir la escala apropiada en cada tiempo. Finalmente, se realizaronexperimentos que ilustran la aplicabilidad y la eficiencia del método pro-puestoFil: Vampa, Victoria Cristina. Universidad Nacional de La Plata. Facultad de IngenierÃa; ArgentinaFil: MartÃn, MarÃa Teresa. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Nacional de La Plata. Facultad de IngenierÃa; Argentin
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