Showing papers on "Dual norm published in 2014"
TL;DR: A space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but with no restriction on coercivity of the linearized operator.
Abstract: We present a space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but with no restriction on coercivity of the linearized operator. We first consider a finite element discretization based on discontinuous Galerkin time integration and introduce associated Petrov--Galerkin space-time trial- and test-space norms that yield optimal and asymptotically mesh independent stability constants. We then employ an $hp$ Petrov--Galerkin (or minimum residual) space-time reduced basis approximation. We provide the Brezzi--Rappaz--Raviart a posteriori error bounds which admit efficient offline-online computational procedures for the three key ingredients: the dual norm of the residual, an inf-sup lower bound, and the Sobolev embedding constant. The latter are based, respectively, on a more round-off resistant residual norm evaluation procedure, a variant of the successive constraint method, and ...
107 citations
TL;DR: A general a posteriori error analysis is established for the natural norms of the DPG schemes under conditions equivalent to a priori stability estimates, proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors.
Abstract: A combination of ideas in least-squares finite element methods with those of hybridized methods recently led to discontinuous Petrov--Galerkin (DPG) finite element methods. They minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard, locally computable, dual norm. This paper establishes a general a posteriori error analysis for the natural norms of the DPG schemes under conditions equivalent to a priori stability estimates. It is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors. The presented abstract framework for a posteriori error analysis applies to known DPG discretizations of Laplace and Lame equations and to a novel DPG method for the stress-velocity formulation of Stokes flow with symmetric stress approximations. Since the error control does not rely on the discrete equations, it applies to inexactly computed or otherwise perturbed solutions within the discrete space...
89 citations
TL;DR: In this article, a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval parametrized with respect to the Peclet number is presented.
Abstract: We present a space-time interpolation-based certified reduced basis method for Burgers' equation over the spatial interval (0, 1) and the temporal interval (0, T] parametrized with respect to the Peclet number. We first introduce a Petrov–Galerkin space-time finite element discretization which enjoys a favorable inf–sup constant that decreases slowly with Peclet number and final time T. We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi–Rappaz–Raviart a posteriori error bounds. We describe computational offline–online decomposition procedures for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf–sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L2-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T, in marked contrast to the exponentially growing estimate of the classical formulation for high Peclet number cases.
64 citations
Proceedings Article•
08 Dec 2014TL;DR: This work extends the norms from the vector to the matrix setting and introduces the spectral k-support norm, and derives an efficient algorithm to compute the proximity operator of the squared norm, improving upon the original method.
Abstract: The k-support norm has successfully been applied to sparse vector prediction problems. We observe that it belongs to a wider class of norms, which we call the box-norms. Within this framework we derive an efficient algorithm to compute the proximity operator of the squared norm, improving upon the original method for the k-support norm. We extend the norms from the vector to the matrix setting and we introduce the spectral k-support norm. We study its properties and show that it is closely related to the multitask learning cluster norm. We apply the norms to real and synthetic matrix completion datasets. Our findings indicate that spectral k-support norm regularization gives state of the art performance, consistently improving over trace norm regularization and the matrix elastic net.
39 citations
TL;DR: In this letter, after showing that the DWSL1 is indeed a norm, two key tools for its use as a regularizer are derived: the dual norm and the Moreau proximity operator.
Abstract: —We consider a new family of regularizers, termedweighted sorted ‘ 1 norms (WSL1), which generalizes the recentlyintroduced octagonal shrinkage and clustering algorithm for re-gression (OSCAR) and also contains the ‘ 1 and ‘ 1 norms asparticular instances. We focus on a special case of the WSL1, thedecreasing WSL1 (DWSL1), where the elements of the argumentvector are sorted in non-increasing order and the weights are alsonon-increasing. In this paper, after showing that the DWSL1 isindeed a norm, we derive two key tools for its use as a regularizer:the dual norm and the Moreau proximity operator.Index Terms—Structured sparsity, sorted ‘ 1 norm, proximalsplitting algorithms. I. I NTRODUCTION In recent years, much research has been devoted not only tosparsity, but also to structured/group sparsity [1]. The OSCAR[2] is a convex regularizer, which was proposed to promotevariable/feature grouping; unlike other methods, it does notrequire previous knowledge of the group structure and it isnot tied to any particular order of the variables. The OSCARcriterion for linear regression with a quadratic loss functionhas the formmin
36 citations
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TL;DR: This paper derives the atomic formulation of the OWL and exploits this formulation to show how Tikhonov regularization schemes can be handled using state-of-the-art proximal splitting algorithms, while IvanovRegularization can be efficiently implemented via the Frank-Wolfe algorithm.
Abstract: The ordered weighted $\ell_1$ norm (OWL) was recently proposed, with two different motivations: because of its good statistical properties as a sparsity promoting regularizer, and as generalization of the so-called {\it octagonal shrinkage and clustering algorithm for regression} (OSCAR). The OSCAR is a convex group-sparsity inducing regularizer, which does not require the prior specification of the group structure. Also recently, much interest has been raised by the atomic norm formulation of several regularizers, not only because it provides an new avenue for their theoretical characterization, but also because it is particularly well suited to a type of method known as {\it conditional gradient} (CG), or Frank-Wolfe, algorithm. In this paper, we derive the atomic formulation of the OWL and exploit this formulation to show how Tikhonov regularization schemes can be handled using state-of-the-art proximal splitting algorithms, while Ivanov regularization can be efficiently implemented via the Frank-Wolfe algorithm.
26 citations
TL;DR: A key ingredient in the method is the proper choice of dual norm used to measure the residual, and two choices are observed to be robust in both convection and diffusion-dominated regimes, as well as a proof of stability for quasi-uniform meshes and a method for the weak imposition of boundary conditions.
Abstract: We present a minimum-residual finite element method (based on a dual PetrovGalerkin formulation) for convectiondiffusion problems in a higher order, adaptive, continuous Galerkin setting. The method borrows concepts from both the Discontinuous PetrovGalerkin (DPG) method by Demkowicz and Gopalakrishnan (2011) and the method of variational stabilization by Cohen, Dahmen, and Welper (2012), and it can also be interpreted as a variational multiscale method in which the fine-scales are defined through a dual-orthogonality condition. A key ingredient in the method is the proper choice of dual norm used to measure the residual, and we present two choices which are observed to be robust in both convection and diffusion-dominated regimes, as well as a proof of stability for quasi-uniform meshes and a method for the weak imposition of boundary conditions. Numerically obtained convergence rates in 2D are reported, and benchmark numerical examples are given to illustrate the behavior of the method.
26 citations
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TL;DR: Using the NP-hardness of the approximation of the spectral norm of tensors, the main result of this paper shows that the weak membership problem in the unit ball of a given norm is NP- hard if and only if the weak membership problem of the unitBall of the dual norm isNP-hard.
Abstract: The main result of this paper shows that the weak membership problem in the unit ball of a given norm is NP-hard if and only if the weak membership problem in the unit ball of the dual norm is NP-hard. Equivalently, the approximation of a given norm is polynomial time if and only if the approximation of the dual norm is polynomial time. Using the NP-hardness of the approximation of the spectral norm of tensors we show that the approximation of the nuclear norm is NP-hard. We relate our results to bipartite separable states in quantum mechanics.
26 citations
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TL;DR: The residual-based a posteriori error estimators are computed by an adaptive dual wavelet expansion, which allows us to compute a surrogate of the dual norm of the residual, which underline the potential of this approach.
Abstract: We use asymptotically optimal \emph{adaptive} numerical methods (here specifically a wavelet scheme) for snapshot computations within the offline phase of the Reduced Basis Method (RBM). The resulting discretizations for each snapshot (i.e., parameter-dependent) do not permit the standard RB `truth space', but allow for error estimation of the RB approximation with respect to the exact solution of the considered parameterized partial differential equation.
The residual-based a posteriori error estimators are computed by an adaptive dual wavelet expansion, which allows us to compute a surrogate of the dual norm of the residual. The resulting adaptive RBM is analyzed. We show the convergence of the resulting adaptive Greedy method. Numerical experiments for stationary and instationary problems underline the potential of this approach.
17 citations
Proceedings Article•
21 Jun 2014TL;DR: This paper forms the problem of learning robust structural SVMs when there are perturbations in the sample space, and shows that robustness to perturbation in the feature space is equivalent to additional regularization, the first theoretical analysis of robust optimization of structural support vector machines.
Abstract: Previous analysis of binary support vector machines (SVMs) has demonstrated a deep connection between robustness to perturbations over uncertainty sets and regularization of the weights. In this paper, we explore the problem of learning robust models for structured prediction problems. We first formulate the problem of learning robust structural SVMs when there are perturbations in the sample space, and show how we can construct corresponding bounds on the perturbations in the feature space. We then show that robustness to perturbations in the feature space is equivalent to additional regularization. For an ellipsoidal uncertainty set, the additional regularizer is based on the dual norm of the norm that constrains the ellipsoidal uncertainty. For a polyhedral uncertainty set, the robust optimization problem is equivalent to adding a linear regularizer in a transformed weight space related to the linear constraints of the polyhedron. We also show that these constraint sets can be combined and demonstrate a number of interesting special cases. This represents the first theoretical analysis of robust optimization of structural support vector machines. Our experimental results show that our method outperforms the nonrobust structural SVMs on real world data when the test data distribution has drifted from the training data distribution.
16 citations
TL;DR: In this article, two exact descriptions for the dual norm in the Kothe dual of an Orlicz-Lorentz function space and a sequence space equipped with either Luxemburg or Amemiya norms are presented.
Abstract: For an Orlicz function $\varphi$ and a decreasing weight $w$, two intrinsic exact descriptions are presented for the norm in the Kothe dual of an Orlicz-Lorentz function space $\Lambda_{\varphi,w}$ or a sequence space $\lambda_{\varphi,w}$, equipped with either Luxemburg or Amemiya norms. The first description of the dual norm is given via the modular $\inf\{\int\varphi_*(f^*/|g|)|g|: g\prec w\}$, where $f^*$ is the decreasing rearrangement of $f$, $g\prec w$ denotes the submajorization of $g$ by $w$ and $\varphi_*$ is the complementary function to $\varphi$. The second one is stated in terms of the modular $\int_I \varphi_*((f^*)^0/w)w$, where $(f^*)^0$ is Halperin's level function of $f^*$ with respect to $w$. That these two descriptions are equivalent results from the identity $\inf\{\int\psi(f^*/|g|)|g|: g\prec w\}=\int_I \psi((f^*)^0/w)w$ valid for any measurable function $f$ and Orlicz function $\psi$. Analogous identity and dual representations are also presented for sequence spaces.
TL;DR: In this paper, it was shown that the resolvent norm of an operator with a compact resolver on a Banach space cannot be constant on an open set if the underlying space or its dual is complex strictly convex.
Abstract: It is proved that the resolvent norm of an operator with a compact resolvent on a Banach space $X$ cannot be constant on an open set if the underlying space or its dual is complex strictly convex. It is also shown that this is not the case for an arbitrary Banach space: there exists a separable, reflexive space $X$ and an unbounded, densely defined operator acting in $X$ with a compact resolvent whose norm is constant in a neighbourhood of zero; moreover $X$ is isometric to a Hilbert space on a subspace of co-dimension $2$. There is also a bounded linear operator acting on the same space whose resolvent norm is constant in a neighbourhood of zero. It is shown that similar examples cannot exist in the co-dimension $1$ case.
01 Feb 2014
TL;DR: Numerical results demonstrate that the space-time formulation provides improved stability constants compared to classical L2-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T , unlike the exponentially growing estimate of the classical formulation for high PEClet number cases.
Abstract: We present a space-time certified reduced basis method for Burgers’ equation over the spatial interval (0, 1) and the temporal interval (0, T ] parametrized with respect to the Peclet number. We first introduce a Petrov-Galerkin space-time finite element discretization, which enjoys a favorable inf-sup constant that decreases slowly with Peclet number and final time T . We then consider an hp interpolation-based space-time reduced basis approximation and associated Brezzi-Rappaz-Raviart a posteriori error bounds. We detail computational procedures that permit offline-online decomposition for the three key ingredients of the error bounds: the dual norm of the residual, a lower bound for the inf-sup constant, and the space-time Sobolev embedding constant. Numerical results demonstrate that our space-time formulation provides improved stability constants compared to classical L2-error estimates; the error bounds remain sharp over a wide range of Peclet numbers and long integration times T , unlike the exponentially growing estimate of the classical formulation for high Peclet number cases.
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TL;DR: An analogue of Banach's theorem for tensor spectral norm and Comon's conjecture for Tensor rank is established --- for a symmetric tensor, its symmetric nuclear norm always equals its nuclear norm.
Abstract: We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the nuclear norm of a real 3-tensor depends on whether we regard it as a real 3-tensor or a complex 3-tensor with real entries. We show that every tensor has a nuclear norm attaining decomposition and every symmetric tensor has a symmetric nuclear norm attaining decomposition. There is a corresponding notion of nuclear rank that, unlike tensor rank, is upper semicontinuous. We establish an analogue of Banach's theorem for tensor spectral norm and Comon's conjecture for tensor rank --- for a symmetric tensor, its symmetric nuclear norm always equals its nuclear norm. We show that computing tensor nuclear norm is NP-hard in several sense. Deciding weak membership in the nuclear norm unit ball of 3-tensors is NP-hard, as is finding an $\varepsilon$-approximation of nuclear norm for 3-tensors. In addition, the problem of computing spectral or nuclear norm of a 4-tensor is NP-hard, even if we restrict the 4-tensor to be bi-Hermitian, bisymmetric, positive semidefinite, nonnegative valued, or all of the above. We discuss some simple polynomial-time approximation bounds. As an aside, we show that the nuclear $(p,q)$-norm of a matrix is NP-hard in general but can be computed in polynomial-time if $p=1$, $q = 1$, or $p=q=2$, with closed-form expressions for the nuclear $(1,q)$- and $(p,1)$-norms.
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TL;DR: In this paper, two exact descriptions for the dual norm in the Kothe dual of an Orlicz-Lorentz function space and a sequence space equipped with either Luxemburg or Amemiya norms are presented.
Abstract: For an Orlicz function $\varphi$ and a decreasing weight $w$, two intrinsic exact descriptions are presented for the norm in the Kothe dual of an Orlicz-Lorentz function space $\Lambda_{\varphi,w}$ or a sequence space $\lambda_{\varphi,w}$, equipped with either Luxemburg or Amemiya norms. The first description of the dual norm is given via the modular $\inf\{\int\varphi_*(f^*/|g|)|g|: g\prec w\}$, where $f^*$ is the decreasing rearrangement of $f$, $g\prec w$ denotes the submajorization of $g$ by $w$ and $\varphi_*$ is the complementary function to $\varphi$. The second one is stated in terms of the modular $\int_I \varphi_*((f^*)^0/w)w$, where $(f^*)^0$ is Halperin's level function of $f^*$ with respect to $w$. That these two descriptions are equivalent results from the identity $\inf\{\int\psi(f^*/|g|)|g|: g\prec w\}=\int_I \psi((f^*)^0/w)w$ valid for any measurable function $f$ and Orlicz function $\psi$. Analogous identity and dual representations are also presented for sequence spaces.
01 Jan 2014
TL;DR: In this article, the authors generalize the Eidelheit-Yood method of proof to Calkin algebras and show that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm of the same algebra.
Abstract: A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.
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TL;DR: This work defines notions like Minkowskian curvature, evolutes and involutes for polygons of constant U -width, and proves that many properties of the smooth case, which is already completely studied, are preserved.
Abstract: Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We define notions like Minkowskian curvature, evolutes and involutes for polygons of constant U-width, and we prove that many properties of the smooth case, which is already completely studied, are preserved. The iteration of involutes generates a pair of sequences of polygons of constant width with respect to the Minkowski norm and its dual norm, respectively. We prove that these sequences are converging to symmetric polygons with the same center, which can be regarded as a central point of the polygon P.
TL;DR: In this article, the authors investigated the inverse problem to the minimal branching geodesic searching problem in a normed space, where the form of the minimal geodesics is determined by finding all possible normed spaces with the same form as the one in the considered norm.
Abstract: In this paper, we investigate the inverse problem to the minimal branching geodesic searching problem in a normed space. Let us consider a normed space. Then the form of the minimal branching geodesic is determined. We must find all possible normed spaces with the same form of the minimal branching geodesics as the one in the considered normed space. The case of Euclidean norms is analyzed in detail.
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TL;DR: In this article, the diameter two properties of the Banach spaces were studied and the dual norm of these spaces was shown to be octahedral, and a closed hyperplane of the spaces was found to have a strong diameter two property.
Abstract: We study the diameter two properties in the spaces $JH$, $JT_\infty$ and $JH_\infty$. We show that the topological dual space of the previous Banach spaces fails every diameter two property. However, we prove that $JH$ and $JH_{\infty}$ satisfy the strong diameter two property, and so the dual norm of these spaces is octahedral. Also we find a closed hyperplane $M$ of $JH_\infty$ whose topological dual space enjoys the $w^*$-strong diameter two property and also $M$ and $M^*$ have an octahedral norm.
TL;DR: The spectral norm of the matrix A x = ( x i x j ) i, j = 1 n , where x i are ith components of any integer sequence ( x n ) .
01 Jan 2014
TL;DR: The concept of rotundity is not far from differentiability as mentioned in this paper, and there are several papers devoted to the study of relations between rotundness and smoothness in Banach spaces.
Abstract: The concept of rotundity is not far from differentiability. There are several papers in the literature devoted to the study of relations between rotundity and smoothness in Banach spaces. In this paper, we study new relations between some kinds of rotundity and smoothness in Banach spaces. In particular, we investigate relations between one kind of rotundity, which is called strongly very rotund, and very smoothness, in Banach spaces. A Banach space is rotund if the midpoint of every two distinct points of unit sphere is in the open unit ball of the Banach space. A Banach space is smooth if its norm is Gateaux differentiable at every non- zero point of the space and it is very smooth if the norm is very Gateaux differentiable, that is, the second dual norm in the second dual of is Gateaux differentiable at every non zero point of .
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TL;DR: In this paper, the dual norm and the Moreau proximity operator were derived for the DWSL1, which is a special case of the Weighted Sorted Weighted Weighted Normalization (WSL1) regularizer.
Abstract: We consider a new family of regularizers, termed {\it weighted sorted $\ell_1$ norms} (WSL1), which generalizes the recently introduced {\it octagonal shrinkage and clustering algorithm for regression} (OSCAR) and also contains the $\ell_1$ and $\ell_{\infty}$ norms as particular instances. We focus on a special case of the WSL1, the {\sl decreasing WSL1} (DWSL1), where the elements of the argument vector are sorted in non-increasing order and the weights are also non-increasing. In this paper, after showing that the DWSL1 is indeed a norm, we derive two key tools for its use as a regularizer: the dual norm and the Moreau proximity operator.
01 Jun 2014
TL;DR: This work derives some a posteriori error estimates for the Richards equation, based on the dual norm of the residual, and proposes a strategy to decrease the computational cost relying on a splitting of the error terms in three parts: linearization, time discretization, and space discretized.
Abstract: We derive some a posteriori error estimates for the Richards equation, based on the dual norm of the residual This equation is nonlinear in space and in time, thus its resolution requires fixed-point iterations within each time step We propose a strategy to decrease the computational cost relying on a splitting of the error terms in three parts: linearization, time discretization, and space discretization In practice, we stop the fixed-point iterations after the linearization error becomes negligible, and choose the time step in order to balance the time and space errors
01 Jan 2014
TL;DR: In this paper, a class of generalised best approximation problems is formulated in a fuzzy normed linear space in order to solve inequality-based optimization problems.
Abstract: In this paper , a class of generalised best approximation problems is formulated in a fuzzy normed linear space.
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TL;DR: In this article, different estimates for the norm of the self-commutator of a Hilbert space operator are proposed, and an isoperimetric-type inequality is proved.
Abstract: Different estimates for the norm of the self-commutator of a Hilbert space operator are proposed. Particularly, this norm is bounded from above by twice of the area of the numerical range of the operator. An isoperimetric-type inequality is proved.
01 Jan 2014
TL;DR: In this paper, the best approximation on probabilistic normed spaces is shown to be equivalent with p-best approximation in the induced normed space, which is the first countable approximation.
Abstract: In this paper, there was investigated the best approximation on probabilistic normed spaces. Firstly, there were defined the best approximation, approximatively compact set, proximal set and the best approximation, and then we found that Serstnev probabilistic normed space with strong probabilistic norm is the first countable and we obtained an interesting proposition about the mentioned definitions. Then we showed that best approximation on probabilistic normed spaces is equivalent with p-best approximation in the in the induced normed space.
Journal Article•
TL;DR: In this paper, almost linear space is introduced and the relations between these two concepts are obtained in the context of almost linear spaces, and the relation between the two concepts is analyzed.
Abstract: Almost linear space is also introduced and the relations between these two concepts are obtained in this paper.
TL;DR: In this paper, the boundedness and compactness of the weighted composition operator from mixed norm space to Bloch-type space on the unit ball of a unit ball was discussed.
Abstract: We discuss the boundedness and compactness of the weighted composition operator from mixed norm space to Bloch-type space on the unit ball of .