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Register a new userA new approach for the existence problem of minimal cubature formulas based on the Larman-Rogers-Seidel theorem
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In this paper we consider the existence problem of cubature formulas of degree 4k+1 for spherically symmetric integrals for which the equality holds in the Moller lower bound. We prove that for sufficiently large dimensional minimal formulas, any two distinct points on some concentric sphere have inner products all of which are rational numbers. By applying this result we prove that for any d > 2 there exist no d-dimensional minimal formulas of degrees 13 and 21 for some special integral.
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Positive interpolatory cubature formulas (CFs) are constructed for quite general integration domains and weight functions. These CFs are exact for general vector spaces of continuous real-valued functions that contain constants. At the same time, the number of data points -- all of which lie inside the domain of integration -- and cubature weights -- all positive -- is less or equal to the dimension of that vector space. The existence of such CFs has been ensured by Tchakaloff in 1957. Yet, to the best of the authors knowledge, this work is the first to provide a procedure to successfully construct them.
In many applications, it is impractical -- if not even impossible -- to obtain data to fit a known cubature formula (CF). Instead, experimental data is often acquired at equidistant or even scattered locations. In this work, stable (in the sense of nonnegative only cubature weights) high-order CFs are developed for this purpose. These are based on the approach to allow the number of data points N to be larger than the number of basis functions K which are integrated exactly by the CF. This yields an (N-K)-dimensional affine linear subspace from which cubature weights are selected that minimize certain norms corresponding to stability of the CF. In the process, two novel classes of stable high-order CFs are proposed and carefully investigated.
Victoir (2004) developed a method to construct cubature formulae with various combinatorial objects. Motivated by this, we generalize Victoirs method with one more combinatorial object, called regular t-wise balanced designs. Many cubature of small indices with few points are provided, which are used to update Shatalovs table (2001) of isometric embeddings in small-dimensional Banach spaces, as well as to improve some classical Hilbert identities. A famous theorem of Bajnok (2007) on Euclidean designs invariant under the Weyl group of Lie type B is extended to all finite irreducible reflection groups. A short proof of the Bajnok theorem is presented in terms of Hilbert identities.
Many applications require multi-dimensional numerical integration, often in the form of a cubature formula. These cubature formulas are desired to be positive and exact for certain finite-dimensional function spaces (and weight functions). Although there are several efficient procedures to construct positive and exact cubature formulas for many standard cases, it remains a challenge to do so in a more general setting. Here, we show how the method of least squares can be used to derive provable positive and exact formulas in a general multi-dimensional setting. Thereby, the procedure only makes use of basic linear algebra operations, such as solving a least squares problem. In particular, it is proved that the resulting least squares cubature formulas are ensured to be positive and exact if a sufficiently large number of equidistributed data points is used. We also discuss the application of provable positive and exact least squares cubature formulas to construct nested stable high-order rules and positive interpolatory formulas. Finally, our findings shed new light on some existing methods for multivariate numerical integration and under which restrictions these are ensured to be successful.
In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by eXtended Finite Element Method and studied for Poisson problem in [Renard]. The method allows computations in domains whose boundaries do not match. A mixed finite element method is used for fluid flow. The interface between the fluid and the structure is localized by a level-set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf-sup condition between the spaces for velocity and the Lagrange multiplier. Convergence analysis is given and several numerical tests are performed to illustrate the capabilities of the method.
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