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Minimization of functionals. Addition
Published in Simon Serovajsky, Optimization and Differentiation, 2017
Dirichlet integral. Consider the minimization problem for the functional I(v)=∫Ω[∑i=1n(∂v(x)∂xi)2+2v(x)f(x)]dx $$ \begin{aligned} I(v)\,=\,\int \limits _\Omega \Big [\sum \limits _{i=1}^n\Big (\frac{\partial v(x)}{\partial x_i}\Big )^2+2v(x)f(x)\Big ]dx \end{aligned} $$
Numerical Solutions of the Cauchy Problem in Potential and Elastostatics
Published in G Nakamura, S Saitoh, J K Seo, M Yamamoto, Inverse problems and related topics, 2019
Here we insist J:H1/2(Γid)∍ω↦R+=[0,+∞). The Dirichlet integral added in Eq.(2) as a regularizer guarantees unique existence of the minimum of the functional J(ω).
A review on consolidation theories and its application
Published in International Journal of Geotechnical Engineering, 2020
Bhandary P. Radhika, A. Krishnamoorthy, A. U. Rao
When construction is made on a consolidating soil, differences can occur in settlement due to different loading intensities. These differential settlements are of more interest since it causes more damage than uniform settlement. The settlement on the edge of the loaded area is affected by the settlement and restraint on flow of water of the unloaded area. The general theory for three-dimensional consolidation was used by Biot (1941b) to determine the settlement of an infinite strip of constant width particularly at the edge of the loaded area laid on consolidating soil loaded uniformly. The process of consolidation was attained due to the free flow of water from top surface both below and outside the loaded area. The solution was obtained by first calculating the settlement produced by a suddenly applied load with sinusoidal distribution. For the discontinuous loading, the settlement below the footing and surrounding the footing was determined by using Dirichlet integral and the principle of superposition. The settlement immediately after loading was less affected by the unloaded regions on both sides, whereas in the last phase the restraining effect of the unloaded regions due to elastic stresses reduced the settlement considerably.
A Fully Bayesian Inference with Gibbs Sampling for Finite and Infinite Discrete Exponential Mixture Models
Published in Applied Artificial Intelligence, 2022
Xuanbo Su, Nuha Zamzami, Nizar Bouguila
where we have . From Equation (12), we have the prior distribution for the Z parameter that corresponds to multinomial distribution. Using the standard Dirichlet integral, we could marginalize out the parameter to get the following probability for the prior directly in terms of the indicators (Rasmussen 1999):
On the stationary Navier–Stokes problem in 3D exterior domains
Published in Applicable Analysis, 2020
Vincenzo Coscia, Remigio Russo, Alfonsina Tartaglione
Starting from the classical paper of Odqvist [1] and Leray [2] in the early 30s of the last century, the study of Equation (1) has been of interest for many researchers until today (see Chapter I and the bibliography in [3]) and there are still several intriguing open problems (see Chapter I of [3]). To point out the main steps in our history, let us recall that the first general existence theorem is due to Leray [2] who proved that if Ω, a are sufficiently regular and there are not sinks or sources, i.e. then (1) has a regular solution , with which satisfies (1) uniformly for and in weak sense for . In this last case, the uniform attainability of u to was proved by Finn [4] (see also [5], [6,p.136]). The behavior at large distance for (see Theorem 1.1) was established by Babenko [7]. The results of Leray were recovered by Fujita [8] by a different method; however, due to a lack of a uniqueness theorem, the two solutions, at least for , are not comparable and many efforts have been devoted to establish whether the properties of the solution obtained by Leray still hold for an arbitrary solution with finite Dirichlet integral, the so-called D-solution (see, e.g. [9] for Babenko's results). It is worth mentioning that Leray's hypothesis (3) is extraneous to the mathematical theory of hydrodynamics and it was first observed by Finn [5] that it can be weakened by requiring to be sufficiently small. The possibility to delete (or not) (3) in the hypotheses for existence is one of the most outstanding problem posed by (1) (see [3], (iv) p.13). Recently, a first step toward a positive answer to this question has been made in [10], where it has been proved that existence can be obtained without (3) for Ω and a axially symmetric. Another research path associated with (1) and of interest in the applications is to find the minimal regularity assumptions on and a ensuring existence of a solution satisfying (3). For Ω Lipschitz, a sufficient condition is [3]. For Equation (1) in bounded domains, starting from a classical paper of Serre [11], this problem has been investigated by several authors (see [3,p.648] and the reference therein for regular domains, and [12–14] for Lipschitz domains). The same problem in exterior domains has been considered in [15], where the Leray existence theorem for has been extended to Ω Lipschitz and , and to Ω regular and .