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A new weak Galerkin (WG) method is introduced and analyzed for the second order elliptic equation formulated as a system of two first order linear equations. This method, called WG-MFEM, is designed by using discontinuous piecewise polynomials on finite element partitions with arbitrary shape of polygons/polyhedra. The WG-MFEM is capable of providing very accurate numerical approximations for both the primary and flux variables. Allowing the use of discontinuous approximating functions on arbitrary shape of polygons/polyhedra makes the method highly flexible in practical computation. Optimal order error estimates in both discrete H1H^1 and L2L^2 norms are established for the corresponding weak Galerkin mixed finite element solutions.

We present a method for learning sparse representations shared across multiple tasks. This method is a generalization of the well-known single-task 1-norm regularization. It is based on a novel non-convex regularizer which controls the number of learned features common across the tasks. We prove that the method is equivalent to solving a convex optimization problem for which there is an iterative algorithm which converges to an optimal solution. The algorithm has a simple interpretation: it alternately performs a supervised and an unsupervised step, where in the former step it learns task-specific functions and in the latter step it learns common-across-tasks sparse representations for these functions. We also provide an extension of the algorithm which learns sparse nonlinear representations using kernels. We report experiments on simulated and real data sets which demonstrate that the proposed method can both improve the performance relative to learning each task independently and lead to a few learned features common across related tasks. Our algorithm can also be used, as a special case, to simply select—not learn—a few common variables across the tasks.

We discuss the application of virtual elements to linear elasticity problems, for both the compressible and the nearly incompressible case. Virtual elements are very close to mimetic finite differences (see, for linear elasticity, [L. Beirão da Veiga, M2AN Math. Model. Numer. Anal., 44 (2010), pp. 231-250]) and in particular to higher order mimetic finite differences. As such, they share the good features of being able to represent in an exact way certain physical properties (conservation, incompressibility, etc.) and of being applicable in very general geometries. The advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity.

In this paper, we prove Ekeland's type of variational principle for a vector equilibrium problem, and present a Caristi-Kirk type fixed point theorem and an existence result for vector equilibrium solution.

Let (Ω, ∑, μ) be a σ—finite measure space, 1 ≤ p < ∞, X be a Banach space X and B : X x Y → Z be a bounded bilinear map. We say that an X-valued function f is p—integrable with respect to B whenever sup{∫Ω∥B(f(w),y)∥pdμ : ∥y∥ = 1} is finite. We identify the spaces of functions integrable with respect to the bilinear maps arising from Hölder's and Young's inequalities. We apply the theory to give conditions on X-valued kernels for the boundedness of integral operators TB(f)(w) = ∫Ω,B(k(w,w′), f(w′))dμ′(w′) from Lp(Y) into Lp(Z), extending the results known in the operator-valued case, corresponding to B : L(X, Y) × X → Y given by B(T, x) = Tx.

The aim of this paper is to characterize in terms of scalar quasiconvexity the vector-valued functions which are KK-quasiconvex with respect to a closed convex cone KK in a Banach space. Our main result extends a well-known characterization of KK-quasiconvexity by means of extreme directions of the polar cone of KK, obtained by Dinh The Luc in the particular case when KK is a polyhedral cone generated by exactly nn linearly independent vectors in the Euclidean space Rn\\mathbb {R}^n.

Bayesian optimization with Gaussian process (GP) surrogates is a popular approach for optimizing expensive-to-evaluate functions in terms of time, energy, or computational resources. Typically, a GP models a scalar objective derived from observed data. However, in many real-world applications, the objective is a combination of multiple outputs from physical experiments or simulations. Converting these multidimensional observations into a single scalar can lead to information loss, slowing convergence and yielding suboptimal results. To address this, we propose to use multi-output GPs to learn the full vector of observations directly, before mapping them to the scalar objective via an inexpensive analytical function. This physics-informed approach retains more information from the underlying physical processes, improving surrogate model accuracy. As a result, the approach accelerates optimization and produces better final designs compared to standard implementations.

In this study we present a geometric approach to proxy economic uncertainty. We design a positional indicator of disagreement among survey-based agents' expectations about the state of the economy. Previous dispersion-based uncertainty indicators derived from business and consumer surveys exclusively make use of the two extreme pieces of information: the percentage of respondents expecting a variable to rise and to fall. With the aim of also incorporating the information coming from the share of respondents expecting a variable to remain constant, we propose a geometrical framework and use a barycentric coordinate system to generate a measure of disagreement, referred to as a discrepancy indicator. We assess its performance both empirically and experimentally by comparing it to the standard deviation of the share of positive and negative responses. When applied in sixteen European countries, we find that both time-varying metrics co-evolve in most countries for expectations about the country's overall economic situation in the present, but not in the future. Additionally, we obtain their simulated sampling distributions and we find that the proposed indicator gravitates uniformly towards the three vertices of the simplex representing the three answering categories, as opposed to the standard deviation, which tends to overestimate the level of uncertainty as a result of ignoring the no-change responses. Consequently, we find evidence that the information coming from agents expecting a variable to remain constant has an effect on the measurement of disagreement.

For X a compact Hausdorff topological space and F a strictly convex and reflexive Banach space we characterize the surjective isometries of certain subspaces of C(X, F) . It follows from this characterization that surjective isometries on spaces of vector valued Lipschitz functions equipped with a p -norm are also weighted composition operators.

Let A be a commutative semisimple Banach algebra, X be a locally compact Hausdorff topological space and G be a locally compact topological group. In this paper, we investigate several properties of vector valued Banach algebras C 0 ( X , A ) , L p ( G , A ) , ℓ p ( X , A ) and ℓ ∞ ( X , A ) . We prove that these algebras are isomorphic with a C ∗ -algebra if and only if A is so.