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Solving monotone inclusions via compositions of nonexpansive averaged operators
 Optimization
, 2004
"... A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analys ..."
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Cited by 145 (31 self)
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A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analysis covers proximal methods for common zero problems as well as various splitting methods for finding a zero of the sum of monotone operators.
Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows
 SIAM J. Sci. Comput
"... this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is ..."
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Cited by 120 (17 self)
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this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is devoted to the derivation of weighted a posteriori error bounds for linear functionals of the solution. Finally, in Section 4 we present some numerical examples to demonstrate the performance of the resulting adaptive finite element algorithm. 2 Model problem and discretisation Given an open bounded polyhedral domain fl in lI n, n _> 1, with boundary 0fl, we consider the following problem: find u: fl > lI m, m _> 1, such that div(u) = 0 in , (2.1) where, ,: m __> mxn is continuously differentiable. We assume that the system of conservation laws (2.1) may be supplemented by appropriate initial/boundary conditions. For example, assuming that B(u, y) := EiL1 biVu'(u) has m real eigenvalues and a complete set of linearly independent eigenvectors for all y = (yl,, Yn) C n; then at inflow/outflow boundaries, we require that B(u, n)(u g) = 0, where n denotes the unit outward normal vector to 0fl, B(u, n) is the negative part of B(u, n) and g is a (given) realvalued function. To formulate the discontinuous Galerkin finite element method (DGFEM, for short) for (2.1), we first introduce some notation. Let 7 = {n} be an admissible subdivision of fl into open element domains n; here h is a piecewise constant mesh function with h(x) = diam(n) 2 Houston e al. when x is in element n. For p Iq0, we define the following finite element space n,  {v [L()]": vl [%(n)] " Vn }, where Pp(n) denotes the set of polynomials of degree at most p over n. Given that v [Hi(n)] m for each n...
A Theoretical Framework for Convex Regularizers in PDEBased Computation of Image Motion
, 2000
"... Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness consta ..."
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Cited by 99 (25 self)
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Many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy. Typically, these energy functionals consist of two terms: a data term, which requires e.g. that a brightness constancy assumption holds, and a regularizer that encourages global or piecewise smoothness of the flow field. In this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images. This taxonomy provides a unifying framework for datadriven and flowdriven, isotropic and anisotropic, as well as spatial and spatiotemporal regularizers. While some of these techniques are classic methods from the literature, others are derived here for the first time. We prove that all these methods are wellposed: they posses a unique solution that depends in a continuous way on the initial data. An interesting structural relation between isotropic and anisotropic flowdriven regularizers is identified, and a design criterion is proposed for constructing anisotropic flowdriven regularizers in a simple and direct way from isotropic ones. Its use is illustrated by several examples.
A douglasRachford splitting approach to nonsmooth convex variational signal recovery
 IEEE Journal of Selected Topics in Signal Processing
, 2007
"... Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is propo ..."
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Cited by 89 (23 self)
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Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the DouglasRachford algorithm for monotone operatorsplitting, is obtained under general conditions. Applications to nonGaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, DouglasRachford, frame, nondifferentiable optimization, Poisson noise,
Multiple Boundary Peak Solutions For Some Singularly Perturbed Neumann Problems
"... We consider the problem ae " 2 \Deltau \Gamma u + f(u) = 0 in\Omega u ? 0 in\Omega ; @u @ = 0 on @\Omega ; where\Omega is a bounded smooth domain in R N , " ? 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses bound ..."
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Cited by 76 (51 self)
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We consider the problem ae " 2 \Deltau \Gamma u + f(u) = 0 in\Omega u ? 0 in\Omega ; @u @ = 0 on @\Omega ; where\Omega is a bounded smooth domain in R N , " ? 0 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses boundary spike solutions such that the spike concentrates, as " approaches zero, at a critical point of the mean curvature function H(P ); P 2 @ It is also known that this equation has multiple boundary spike solutions at multiple nondegenerate critical points of H(P ) or multiple local maximum points of H(P ). In this paper, we prove that for any fixed positive integer K there exist boundary K \Gamma peak solutions at a local minimum point of H(P ). This implies that for any smooth and bounded domain there always exist boundary K \Gamma peak solutions. We first use the LiapunovSchmidt method to reduce the problem to finite dimensions. Then we use a maximizing procedure to obtain multiple boundary spikes. 1.
Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin
 Dyn. Syst
"... Abstract. The existence of a global attractor in the natural energy space is proved for the semilinear wave equation utt + βut − ∆u + f(u) = 0 on a bounded domain Ω ⊂ Rn with Dirichlet boundary conditions. The nonlinear term f is supposed to satisfy an exponential growth condition for n =2,andforn≥ ..."
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Cited by 71 (1 self)
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Abstract. The existence of a global attractor in the natural energy space is proved for the semilinear wave equation utt + βut − ∆u + f(u) = 0 on a bounded domain Ω ⊂ Rn with Dirichlet boundary conditions. The nonlinear term f is supposed to satisfy an exponential growth condition for n =2,andforn≥3thegrowth condition f(u)  ≤c0(u  γ +1), where1≤γ≤ n. No Lipschitz condition on f n−2 is assumed, leading to presumed nonuniqueness of solutions with given initial data. The asymptotic compactness of the corresponding generalized semiflow is proved using an auxiliary functional. The system is shown to possess Kneser’s property, which implies the connectedness of the attractor. In the case n ≥ 3andγ> n the existence of a global attractor is proved under n−2 the (unproved) assumption that every weak solution satisfies the energy equation. Dedicated to M.I. Vishik on the occasion of his 80 th birthday
2002): “Perturbation Methods for General Dynamic Stochastic Models,”unpublished
"... Abstract. We describe a general Taylor series method for computing asymptotically valid approximations to deterministic and stochastic rational expectations models near the deterministic steady state. Contrary to conventional wisdom, the higherorder terms are conceptually no more difficult to comp ..."
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Cited by 69 (5 self)
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Abstract. We describe a general Taylor series method for computing asymptotically valid approximations to deterministic and stochastic rational expectations models near the deterministic steady state. Contrary to conventional wisdom, the higherorder terms are conceptually no more difficult to compute than the conventional deterministic linear approximations. We display the solvability conditions for second and thirdorder approximations and show how to compute the solvability conditions in general. We use an implicit function theorem to prove a local existence theorem for the general stochastic model given existence of the degenerate deterministic model. We describe an algorithm which takes as input the equilibrium equations and an integer k, and computes the order k Taylor series expansion along with diagnostic indices indicating the quality of the approximation. We apply this algorithm to some multidimensional problems and show that the resulting nonlinear approximations are far superior to linear approximations. 1 Perturbation methods for general dynamic stochastic models 2
On Besov Regularity of Solutions to Nonlinear Elliptic Partial Differential Equations
, 2011
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
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Cited by 66 (12 self)
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The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors.