PDF superior Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces / Antonio Falcó and Anthony Nouy

Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces / Antonio Falcó and Anthony Nouy

Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces / Antonio Falcó and Anthony Nouy

A rst family of numerical methods based on classical constructions of tensor approximations [18,23,35] have been recently investigated for the solution of high- dimensional partial dierential equations [3,19,21,22,28]. They are based on the sys- tematic use of tensor approximations inside classical iterative solvers. Another family of methods, called Proper Generalized Decomposition (PGD) methods [9,16,27,33,34], have been introduced for the direct construction of representations of type (1). PGD methods introduce alternative denitions of tensor approximations, not based on nat- ural best approximation problems, for the approximation to be computable without a priori information on the solution u. The particular structure of approximation sets allows the interpretation of PGDs as generalizations of Proper Orthogonal Decompo- sition (or Singular Value Decomposition, or Karhunen-Loève Decomposition) for the a priori construction of a separated representation u m of the solution. They can also
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A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart? Young approach / A  Falcó and A  Nouy

A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart? Young approach / A Falcó and A Nouy

of operator A and right-hand side l. This can be achieved by introducing new definitions of optimal decompositions (2). The Proper Generalized Decomposi- tion (PGD) method have been first introduced under the name of “Radial-type approximation” for the solution of time dependent partial differential equations (PDE), by separating space and time variables, and used in the context of the LATIN method in computational solid mechanics [15, 10, 16, 24, 17, 23]. It has been also introduced for the separation of coordinate in multidimensional PDEs [1, 2], with many applications in kinetic theory of complex fluids, finan- cial mathematics, computational chemistry. . . It has also been introduced in the context of stochastic or parametrized PDEs by introducing a separation of phys- ical variables (space, time. . . ) and (random) parameters [19, 20, 21]. Still in the context of stochastic PDEs, a further separation of parameters have also been introduced, by exploiting the tensor product structure of stochastic func- tion spaces [9, 22]. In this context, it leads to a representation of functionals of random variables alternative to classical chaos expansions [28, 12, 27, 26, 29]. Of course, separated representations constitute an effective alternative only for functionals of random variables that admit a low rank representation.
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Geometric structures in tensor representations : final release / Antonio Falcó, Wolfgang Hackbusch and Anthony Nouy

Geometric structures in tensor representations : final release / Antonio Falcó, Wolfgang Hackbusch and Anthony Nouy

Tensor approximation methods play a central role in the numerical solution of high dimensional problems arising in a wide range of applications. Low-rank tensor formats based on subspaces are widely used for complexity reduction in the representation of high-order tensors. The construction of these formats are usually based on a hierarchy of tensor product subspaces spanned by orthonormal bases, because in most cases a hierarchical representation fits with the structure of the mathematical model and facilitates its computational implementation. Two of the most popular formats are the Tucker format and the Hierarchical Tucker format [18] (HT for short). It is possible to show that the Tensor Train format [29] (TT for short), introduced originally by Vidal [35], is a particular case of the HT format (see e.g. Chapter 12 in [19]). An important feature of these formats, in the framework of topological tensor spaces, is the existence of a best approximation in each fixed set of tensors with bounded rank [11]. In particular, it allows to construct, on a theoretical level, iterative minimisation methods for nonlinear convex problems over reflexive tensor Banach spaces [12].
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Implementation of user element subroutines for frequency domain analysis of wave scattering problems with commercial finite element codes

Implementation of user element subroutines for frequency domain analysis of wave scattering problems with commercial finite element codes

In the case of infinite or semi-infinite media problems-as is the usual scenario in earthquake engineering and geophysics-one of the challenges in the numerical solution, is the proper impo- sition of radiation boundary conditions. This problem can be dealt with in a very natural way, using integral equations formulations and its discretization in terms of different versions of the boundary element method. In that approach the radiation condition is implicit in the specific problem Green’s function. Solutions through different versions of the direct and indirect boundary element method and corresponding to problems in different physical contexts are reported in Wong & Jennings (1975), citesills1978scattering, S´ anchez-Sesma & Rosenblueth (1979), S´ anchez-Sesma (1983), Dravinski & Mossessian (1987), Zhang & Achenbach (1988), Kawase (1988), Manolis & Beskos (1988), Kawase & Aki (1989), Mossessian & Dravinski (1989), S´ anchez-Sesma & Campillo (1991), Kim & Papageorgiou (1993), Papageorgiou & Pei (1998), Janod & Coutant (2000), Itur- rar´ an-Viveros et al. (2005), Sohrabi-Bidar et al. (2010). Although BE-based solutions are very accurate, its effective application to real size problems is very limited due to large computing requirements. As a result, the method is currently being used as a tool to perform parametric analysis in conceptual problems.
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Ethics, psychiatry and mental health

Ethics, psychiatry and mental health

Moving this view to the medicine field, the to- tal circumstance of the physician in the Medical Act is the patient, whom, therefore, the physician must “take care of” in order to “save” him/her. If doing it, the physician will be saved as well, as a person and as a professional. On the contrary, if failing this moral and scientific obligation, the physician will harm himself affecting thus his own humane and professional flourishing. A poorly treated patient may sue the physician; a well-treated patient may love, thank the physi- cian, and decisively contribute to his professional prestige. Let us remind ourselves that with our science we diagnose, heal and/or relieve; with our art, we bring care and support. And that art, that attitude and disposition to service settled in our medical vocation and instrumented by the right technique, constitute, in essence, the dynamic core of all medical act, particularly in psychiatry and mental health.
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A Grüss Type Operator Inequality

A Grüss Type Operator Inequality

Abstract. In [P. Renaud, A matrix formulation of Gr¨ uss inequality, Linear Al- gebra Appl. 335 (2001), 95–100] it was proved an operator inequality involving the usual trace functional. In this article, we give a refinement of such result and we answer positively the Renaud’s open problem.

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Photon-conserving generalized nonlinear Schrödinger equation for frequency-dependent nonlinearities

Photon-conserving generalized nonlinear Schrödinger equation for frequency-dependent nonlinearities

is the delayed Raman response. As shown in Ref. [30], Eq. (1) does not conserve the number of photons in general, a fact inconsistent with the underlying photon-conserving physical processes, namely, dispersion, four-wave mixing, and Raman scattering. Moreover, although Raman scattering involves the annihilation of a photon at frequency ω and the creation of a photon at frequency ω − µ and a phonon of frequency µ , lead- ing to a transfer of energy toward lower frequencies (red shift), an unphysical soliton blue shift is predicted when a negative nonlinearity is considered [23].
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BraunMurdoch-AFirstCourseinStatisticalProgrammingwithR

BraunMurdoch-AFirstCourseinStatisticalProgrammingwithR

Programming involves writing relatively complex systems of instructions. There are two broad styles of programming: the imperative style (used in R, for example) involves stringing together instructions telling the com- puter what to do. The declarative style (used in HTML in web pages, for example) involves writing a description of the end result, without giving the details about how to get there. Within each of these broad styles, there are many subdivisions, and a given program may involve aspects of sev- eral of them. For example, R programs may be procedural (describing what steps to take to achieve a task), modular (broken up into self- contained packages), object-oriented (organized to describe operations on complex objects), functional (organized as a collection of functions which do specific calculations without having external side-effects), among other possibilities. In this book we will concentrate on the procedural aspects of programming.
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Extending polynomials on Banach spaces—A survey

Extending polynomials on Banach spaces—A survey

Surely the simplest kind of polynomials over a Banach space are the polyno- mials of finite type, those which may be written as P (x) = n i =1 γ i (x) k , where each γ i is a continuous linear form over E. Slightly larger is the class of nu- clear polynomials, made up of those which may be represented (non-uniquely) as P (x) = ∞ i =1 γ i (x) k , with ∞ i =1 γ i k < ∞. These form a non-closed vector sub- space of P ( k E) (although they do form a Banach space in the ‘nuclear’ norm, i.e. the infimum of ∞ i =1 γ i k over all possible representations of P ). The closure of the space of nuclear (or also of the finite-type polynomials) in the uniform norm is the space of approximable polynomials. Larger still are the spaces of weakly con- tinuous polynomials (those which are weakly continuous over all bounded subsets of E) and weakly sequentially continuous polynomials. We thus have the chain of vector subspaces of P( k E)
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Five basic lemmas for symmetric tensor products
            of normed spaces

Five basic lemmas for symmetric tensor products of normed spaces

In the theory of full 2-fold tensor norms, “The Five Basic Lemmas” (see Sec- tion 13 in Defant and Floret’s book [14]) are rather simple results which turn out to be “basic for the understanding and use of tensor norms”. Namely, they are the Approximation Lemma, the Extension Lemma, the Embedding Lemma, the Density Lemma and the Local Technique Lemma. Applications of these lemmas can be seen throughout the book. We present here the analogous results for the symmetric setting. We also exhibit some applications as example of their potential. In order to obtain our five basic lemmas and their applications, we combine new and known results in a methodic way, following the lines of [14]. Although some proofs are similar to the 2-fold case, the symmetric nature of our tensor products introduces some difficulties, as we can see, for example, in the symmetric version of the Extension Lemma 2.2, whose proof is much more complicated than that of its full 2-fold version.
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Algebraic elementary operators

Algebraic elementary operators

Proposition 2.2. If A, B ∈ B(X) are algebraic operators such that [A, B] = AB − BA = 0, then A + B is algebraic. A proof of the proposition (in a certain sense, a more direct proof) may be obtained as a consequence of the easily proved fact that if A and B are commuting algebraic elements of an algebra, then each polynomial p(A, B) is also algebraic: In keeping with the spirit of this paper, in the following we draw upon local spectral theory to prove the proposition.

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Regularización de ecuaciones integrales no lineales en un parámetro. Aplicación a la determinación de la distribución de tamaños y del índice de refracción de látex por dispersión de luz.

Regularización de ecuaciones integrales no lineales en un parámetro. Aplicación a la determinación de la distribución de tamaños y del índice de refracción de látex por dispersión de luz.

Regularization is the approximation of an ill-posed inverse problem by a family of closed well-posed problems. The theory of regularization methods is well developed and the properties of the regularized solutions are established rigorously for linear problems. To solve an inverse linear problem usually requires the selection of a regularization parameter and the determination of a function that optimizes a least square problem.

13 Lee mas

Generalized sampling: from shift-invariant to U-invariant spaces

Generalized sampling: from shift-invariant to U-invariant spaces

Pérez-Villalón, Dual frames in L (0,1) connected with general- ized sampling in shift-invariant spaces, Appl.. Pérez-Villalón, Generalized irregular sampling in shift-invariant spaces, [r]

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A finite Benders decomposition algorithm for mixed integer problems, resolution  through parametric Branch and Bound

A finite Benders decomposition algorithm for mixed integer problems, resolution through parametric Branch and Bound

Previous section implicitly assumed that master dual problem ( MD ) was a bounded problem, so each resolution would give a new multiplier proposal λ . It did also assume problem ( PR λ ) was bounded for each value λ . However, this is not the general situation and a family of cuts is necessary to guarantee master dual problem boundness. It is also necessary to obtain a bounding cut that excludes the proposed multiplier value in case problem ( PR λ ) turns out to be unbounded. From hereafter it is assumed that the objective function is linear and the constraints are affine. We will exhaustively use the Farkas’ law results.
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Topologies of Llocp type for Carathéodory functions with applications in non-autonomous differential equations

Topologies of Llocp type for Carathéodory functions with applications in non-autonomous differential equations

Remark 2.5. As regards Definitions 2.1, 2.2 and 2.4, we identify the functions which lay in the same set and only differ on a negligible subset of R 1+N . The con- straint about belonging to the same set is crucial. Indeed, without any additional constraint, a function in SC p R M could in fact be identified with a function which is not in SC p R M . Furthermore, such a rule implies that LC p R M ⊂ SC p R M but SC p R M is not included in ΘC p R M . Nevertheless, a continuous injection (which is not a bijection, as we will show in section 6) of SC p R M in ΘC p R M is straightforward. Thus, the following chain can be sketched
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28 Lee mas

Anti-stokes Raman gain enabled by modulation instability in mid-IR waveguides

Anti-stokes Raman gain enabled by modulation instability in mid-IR waveguides

Complete models of MI in waveguides, see, e.g., [5, 6] and references therein, can be used to analyze the interplay between high-order dispersion, Raman scattering, and self-steepening. This last effect, arising from a first-order expansion of the waveg- uide nonlinear-coefficient dependence on frequency, is often associated with the break-up of ultrashort pulses upon prop- agation. However, it also has profound consequences in the case of continuous-wave (CW) pumping of the waveguide, as its inclusion leads to an optimum pump power that maximizes the MI gain [7] and a power cutoff [8] above which the MI gain vanishes, leaving behind only the Raman contribution. Recent work [9] focusing on emerging phenomena beyond this cutoff showed that the remnant gain has a Raman, and frequency- tunable, profile in the Stokes (low-frequency) band. As it is well known, Raman scattering transfers energy from high to low
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Climate change in Spain's media | A deficient answer

Climate change in Spain's media | A deficient answer

Media coverage on these events is proof that climate change is one of the problems whose magnitude and repercussions in the environment and in even world peace we started to be aware of thanks to media. in the Uni- ted kingdom, for instance, there are studies that confirm that information about climate change is mainly transmitted to the citizens by the mass media (Meira 2008: 55-6). There are also some investigations in the United States that show knowledge of science matters gained through the media is particularly obvious in the case of climate change (wilson 2000: 201-17). Spain, however, does not have specific data on this. The overview looks similar, although, based on the fact that Spanish media has been increasing the attention paid to climate change. Specifically, there has been an incre- ase in news frequency and extension, editorials and opinion articles rela- ted to this. an increasingly diverse and plural approach is also noticeable. Moreover, it is moving from Social, Science or environmental sections to national, international, economy and even Society (Meira 2008: 58).
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Spaces for gender and power

Spaces for gender and power

In the field of political participation, within the framework of fighting for recognition and sexual and reproductive rights of sexual dissidences, sociologists Catalina Otárola and Catalina Tapia present a research based on the analysis of motivations and experiences, as mutually implicated dimensions, which encour- age lesbians to participate in different organizations of the LG- BTI+ movement in Chile. The article begins from the assumption that asymmetrical power relations unfold from the gender and influence experiences of subjects and, in this sense, the object of study is implied in a double record: that of the life experience and that of the lesbians’ political participation, both in a world his- torically led by men. In this way, through a qualitative approach, the article provides the possibility of knowing motivations from a logic that goes beyond rationality to emphasize the importance of emotions and values as vectors of political participation.
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A linear Uzawa type FEM BEM solver for nonlinear transmission problems

A linear Uzawa type FEM BEM solver for nonlinear transmission problems

Section 3 is the mathematical core of the manuscript. We discretize each step of the Uzawa iteration by conforming BEM resp. FEM with piecewise polynomials of order p − 1 resp. p . Algorithm 4 formulates the outer Uzawa iteration for the discretized problem, and Theorem 5 proves linear convergence. The inner iteration with adaptive FEM (resp. adaptive BEM) is the topic of Section 3.3. In the spirit of [13], we give an abstract analysis of an adaptive mesh-refining algorithm (Algorithm 9) which also allows the inexact solution of the arising linear systems by means of the preconditioned conjugate gradient method (PCG). Proposition 11 proves that the algorithm reaches any prescribed tolerance in finite computational time. Moreover, for properly chosen preconditioners, the number of CG iterations in each step of the adaptive algorithm is uniformly bounded (Remark 10). We apply this adaptive algorithm in each step of the outer Uzawa iteration for the BEM part (Section 3.4) and for the FEM part (Section 3.5), where we employ a weighted-residual error estimator for the BEM and the standard residual error estimator for the FEM. Theorem 13 resp. Theorem 16 prove that the number of adaptive mesh- refinement steps (Algorithm 9) in each step of the discrete Uzawa iteration (Algorithm 4) is generically uniformly bounded.
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T1 theorems on generalized Besov and Triebel Lizorkin
spaces over spaces of homogeneous type

T1 theorems on generalized Besov and Triebel Lizorkin spaces over spaces of homogeneous type

[ HS ] . We consider, as a control of the 'local regularity', functions 'IjJ(t) more gen­ eral than the potentials tet used in their case. We also state Tl-type theorems in these spaces. Our appro a ch yields some new results for kernels satisfying integral regularity conditions.

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