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Register a new userCondenser capacity and hyperbolic perimeter
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Added by
Oona Rainio
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We apply domain functionals to study the conformal capacity of condensers $(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is a compact subset of $G$. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$. Novel computational algorithms based on implementations of the fast multipole method are combined with analytic techniques. Computational experiments are used throughout to, for instance, demonstrate sharpness of established inequalities. In the case of model problems with known analytic solutions, very high precision of computation is observed.
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Given a compact connected set $E$ in the unit disk $mathbb{B}^2$, we give a new upper bound for the conformal capacity of the condenser $(mathbb{B}^2, E),$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$ we construct a set of diameter $t$ and show by numerical computation that it has larger capacity than a hyperbolic disk with the same diameter. The set we construct is of constant hyperbolic width equal to $t$, the so called hyperbolic Reuleaux triangle.
For compact subsets $E$ of the unit disk $ mathbb{D}$ we study the capacity of the condenser ${rm cap}( mathbb{D},E)$ by means of set functionals defined in terms of hyperbolic geometry. In particular, we study experimentally the case of a hyperbolic triangle and arrive at the conjecture that of all triangles with the same hyperbolic area, the equilateral triangle has the least capacity.
We use hyperbolic wavelet regression for the fast reconstruction of high-dimensional functions having only low dimensional variable interactions. Compactly supported periodic Chui-Wang wavelets are used for the tensorized hyperbolic wavelet basis. In a first step we give a self-contained characterization of tensor product Sobolev-Besov spaces on the $d$-torus with arbitrary smoothness in terms of the decay of such wavelet coefficients. In the second part we perform and analyze scattered-data approximation using a hyperbolic cross type truncation of the basis expansion for the associated least squares method. The corresponding system matrix is sparse due to the compact support of the wavelets, which leads to a significant acceleration of the matrix vector multiplication. In case of i.i.d. samples we can even bound the approximation error with high probability by loosing only $log$-terms that do not depend on $d$ compared to the best approximation. In addition, if the function has low effective dimension (i.e. only interactions of few variables), we qualitatively determine the variable interactions and omit ANOVA terms with low variance in a second step in order to increase the accuracy. This allows us to suggest an adapted model for the approximation. Numerical results show the efficiency of the proposed method.
In this work, we determine the full expression of the local truncation error of hyperbolic partial differential equations (PDEs) on a uniform mesh. If we are employing a stable numerical scheme and the global solution error is of the same order of accuracy as the global truncation error, we make the following observations in the asymptotic regime, where the truncation error is dominated by the powers of $Delta x$ and $Delta t$ rather than their coefficients. Assuming that we reach the asymptotic regime before the machine precision error takes over, (a) the order of convergence of stable numerical solutions of hyperbolic PDEs at constant ratio of $Delta t$ to $Delta x$ is governed by the minimum of the orders of the spatial and temporal discretizations, and (b) convergence cannot even be guaranteed under only spatial or temporal refinement. We have tested our theory against numerical methods employing Method of Lines and not against ones that treat space and time together, and we have not taken into consideration the reduction in the spatial and temporal orders of accuracy resulting from slope-limiting monotonicity-preserving strategies commonly applied to finite volume methods. Otherwise, our theory applies to any hyperbolic PDE, be it linear or non-linear, and employing finite difference, finite volume, or finite element discretization in space, and advanced in time with a predictor-corrector, multistep, or a deferred correction method. If the PDE is reduced to an ordinary differential equation (ODE) by specifying the spatial gradients of the dependent variable and the coefficients and the source terms to be zero, then the standard local truncation error of the ODE is recovered. We perform the analysis with generic and specific hyperbolic PDEs using the symbolic algebra package SymPy, and conduct a number of numerical experiments to demonstrate our theoretical findings.
Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we propose a novel REXI scheme that drastically improves accuracy and efficiency. The chosen approach will also allow us to easily determine how many terms are required in the approximation in order to obtain accurate results. We provide comparative numerical simulations for a shallow water equation that highlight the efficiency of our approach and demonstrate that REXI schemes can be efficiently implemented on graphic processing units.
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