Open Source Development in a Differentiated Duopoly

Open source software is released under an open source license giving individuals the right to use, modify, and redistribute freely the programs. This paper proposes a model of differentiated duopoly in which firms invest in the development of proprietary or open source software. The main findings are: (i) firms invest more when the products are substitutes; (ii) for substitute products, firms’ investment in software development is greatest when the software is open source; (iii) for close to perfect complements, firms’ investment in software development is greatest when the software is proprietary; and (iv) for substitute products, investment in open source software yields higher profits than investment in proprietary software.

Topology optimization of multiple anisotropic materials, with application to self-assembling diblock copolymers

We propose a solution strategy for a multimaterial minimum compliance topology optimization problem, which consists in finding the optimal allocation of a finite number of candidate (possibly anisotropic) materials inside a reference domain, with the aim of maximizing the stiffness of the body. As a relevant and novel application we consider the optimization of self-assembled structures obtained by means of diblock copolymers. Such polymers are a class of self-assembling materials which spontaneously synthesize periodic microstructures at the nanoscale, whose anisotropic features can be exploited to build structures with optimal elastic response, resembling biological tissues exhibiting microstructures, such as bones and wood. For this purpose we present a new generalization of the classical Optimality Criteria algorithm to encompass a wider class of problems, where multiple candidate materials are considered, the orientation of the anisotropic materials is optimized, and the elastic properties of the materials are assumed to depend on a scalar parameter, which is optimized simultaneously to the material allocation and orientation. Well-posedness of the optimization problem and well-definition of the presented algorithm are narrowly treated and proved. The capabilities of the proposed method are assessed through several numerical tests

Contraction and optimality properties of an adaptive Legendre-Galerkin method: the multi-dimensional case

We analyze the theoretical properties of an adaptive Legendre-Galerkin method in the multidimensional case. After the recent investigations for Fourier-Galerkin methods in a periodic box and for Legendre-Galerkin methods in the one dimensional setting, the present study represents a further step towards a mathematically rigorous understanding of adaptive spectral/$hp$ discretizations of elliptic boundary-value problems. The main contribution of the paper is a careful construction of a multidimensional Riesz basis in $H^1$, based on a quasi-orthonormalization procedure. This allows us to design an adaptive algorithm, to prove its convergence by a contraction argument, and to discuss its optimality properties (in the sense of non-linear approximation theory) in certain sparsity classes of Gevrey type

Optimal control in ink-jet printing via instantaneous control

This paper concerns the optimal control of a free surface flow with moving contact line, inspired by an application in ink-jet printing. Surface tension, contact angle and wall friction are taken into account by means of the generalized Navier boundary condition. The time-dependent differential system is discretized by an arbitrary Lagrangian-Eulerian finite element method, and a control problem is addressed by an instantaneous control approach, based on the time discretization of the flow equations. The resulting control procedure is computationally highly efficient and its assessment by numerical tests show its effectiveness in deadening the natural oscillations that occur inside the nozzle and reducing significantly the duration of the transient preceding the attainment of the equilibrium configuration

On the decay of the inverse of matrices that are sum of Kronecker products

Decay patterns of matrix inverses have recently attracted considerable interest, due to their relevance in numerical analysis, and in applications requiring matrix function approximations. In this paper we analyze the decay pattern of the inverse of banded matrices in the form $S=M \otimes I_n + I_n \otimes M$ where $M$ is tridiagonal, symmetric and positive definite, $I_n$ is the identity matrix, and $\otimes$ stands for the Kronecker product. It is well known that the inverses of banded matrices exhibit an exponential decay pattern away from the main diagonal. However, the entries in $S^{-1}$ show a non-monotonic decay, which is not caught by classical bounds. By using an alternative expression for $S^{-1}$, we derive computable upper bounds that closely capture the actual behavior of its entries. We also show that similar estimates can be obtained when $M$ has a larger bandwidth, or when the sum of Kronecker products involves two different matrices. Numerical experiments illustrating the new bounds are also reported

Adaptive Fourier-Galerkin Methods

We study the performance of adaptive Fourier-Galerkin methods in a periodic box in $\mathbb{R}^d$ with dimension $d\ge 1$. These methods offer unlimited approximation power only restricted by solution and data regularity. They are of intrinsic interest but are also a first step towards understanding adaptivity for the $hp$-FEM. We examine two nonlinear approximation classes, one classical corresponding to algebraic decay of Fourier coefficients and another associated with exponential decay. We study the sparsity classes of the residual and show that they are the same as the solution for the algebraic class but not for the exponential one. This possible sparsity degradation for the exponential class can be compensated with coarsening, which we discuss in detail. We present several adaptive Fourier algorithms, and prove their contraction and optimal cardinality properties.Comment: 48 page