Showing papers in "Mathematics of Computation in 2016"
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TL;DR: Photographs courtesy of Universidad de Buenos Aires and Departamento de Matematica.
Abstract: Fil: Guarnieri, Leandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matematica; Argentina
270 citations
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TL;DR: In this paper, a hybrid high-order (HHO) method for steady non-linear Leray-Lions problems is proposed, which is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high order stabilization term.
Abstract: In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady non-linear Leray–Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, $L^p$-stability and $W^{s,p}$-approximation properties for $L^2$-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.
129 citations
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TL;DR: A superconvergence property of the velocity is proved which allows us to obtain an elementwise postprocessed approximate velocity, H(div)-conforming and divergence-free, which converges with order k + 2 for k ≥ 1.
Abstract: We present the first a priori error analysis of the hybridizable discontinuous Galerkin method for the approximation of the Navier-Stokes equations proposed in J. Comput. Phys. vol. 230 (2011), pp. 1147-1170. The method is defined on conforming meshes made of simplexes and provides piecewise polynomial approximations of fixed degree k to each of the components of the velocity gradient, velocity and pressure. For the stationary case, and under the usual smallness condition for the source term, we prove that the method is well defined and that the global L2-norm of the error in each of the above-mentioned variables converges with the optimal order of k+1 for k ≥ 0. We also prove a superconvergence property of the velocity which allows us to obtain an elementwise postprocessed approximate velocity, H(div)-conforming and divergence-free, which converges with order k + 2 for k ≥ 1. In addition, we show that these results only depend on the inverse of the stabilization parameter of the jump of the normal component of the velocity. Thus, if we superpenalize those jumps, these converegence results do hold by assuming that the pressure lies in H1(Ω) only. Moreover, by letting such stabilization parameters go to infinity, we obtain new H(div)-conforming methods with the above-mentioned convergence properties.
97 citations
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TL;DR: This work proposes an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work, and presents theoretical analysis to motivate the algorithm, and numerical results that show the method is superior to standard Monte Carlo methods in many situations of interest.
Abstract: We propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation frame- work. Our method is motivated by generalized Polynomial Chaos approximation in uncertainty quantification where a polynomial approximation is formed from a combination of orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density of orthogonality. Our proposed algorithm samples with respect to the equilibrium measure of the parametric domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.
76 citations
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TL;DR: A novel discretization of the Monge-Ampere operator is introduced, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications, and achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid.
Abstract: We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.
66 citations
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TL;DR: A new multiscale method for the wave equation that does not require any assumptions on space regularity or scale-separation and is formulated in the framework of the Localized Orthogonal Decomposition (LOD).
Abstract: In this paper we propose and analyze a new multiscale method for the wave equation. The proposed method does not require any assumptions on space regularity or scale-separation and it is formulated in the framework of the Localized Orthogonal Decomposition (LOD). We derive rigorous a priori error estimates for the L2-approximation properties of the method, finding that convergence rates of up to third order can be achieved. The theoretical results are confirrmed by various numerical experiments.
65 citations
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62 citations
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TL;DR: This paper considers the local discontinuous Galerkin method based on the generalized alternating numerical fluxes, for solving the linear convection-diffusion equations in one dimension and two dimensions and obtains directly the optimal L2-norm error estimate in a uniform framework.
Abstract: In this paper we consider the local discontinuous Galerkin method based on the generalized alternating numerical fluxes, for solving the linear convection-diffusion equations in one dimension and two dimensions. As an application of generalized Gauss–Radau projections, we get rid of the dual argument and obtain directly the optimal L2-norm error estimate in a uniform framework. The sharpness of the theoretical results is demonstrated by numerical experiments.
53 citations
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TL;DR: In this paper, a Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number in bounded domains in the sense that the discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale corrections in the spirit of numerical homogenization.
Abstract: We introduce a new Petrov-Galerkin multiscale method for the numerical approximation of the Helmholtz equation with large wave number $\kappa$ in bounded domains in $\mathbb{R}^d$. The discrete trial and test spaces are generated from standard mesh-based finite elements by local subscale corrections in the spirit of numerical homogenization. The precomputation of the corrections involves the solution of coercive cell problems on localized subdomains of size $\ell H$; $H$ being the mesh size and $\ell$ being the oversampling parameter. If the mesh size and the oversampling parameter are such that $H\kappa$ and $\log(\kappa)/\ell$ fall below some generic constants and if the cell problems are solved sufficiently accurate on some finer scale of discretization, then the method is stable and its error is proportional to $H$; pollution effects are eliminated in this regime.
53 citations
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TL;DR: In this article, the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters is analyzed.
Abstract: We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called "non-intrusive" sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a functional of the parametric solution is then computed via compressive sensing methods from samples of functionals of the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sufficient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coefficients of the gpc expansion are contained in certain weighted $\ell_p$-spaces for $0
52 citations
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TL;DR: An approximate Riemann solver is exhibited that satisfies all the needed properties (robustness and well-balancing) and is proved to be positive preserving, entropy satisfying and to exactly capture the nonlinear steady states at rest.
Abstract: The present paper concerns the derivation of numerical schemes to approximate the weak solutions of the Ripa model, which is an extension of the shallow-water model where a gradient of temperature is considered. Here, the main motivation lies in the exact capture of the steady states involved in the model. Because of the temperature gradient, the steady states at rest, of prime importance from the physical point of view, turn out to be very nonlinear and their exact capture by a numerical scheme is very challenging. We propose a relaxation technique to derive the required scheme. In fact, we exhibit an approximate Riemann solver that satisfies all the needed properties (robustness and well-balancing). We show three relaxation strategies to get a suitable interpretation of this adopted approximate Riemann solver. The resulting relaxation scheme is proved to be positive preserving, entropy satisfying and to exactly capture the nonlinear steady states at rest. Several numerical experiments illustrate the relevance of the method.
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TL;DR: This work establishes estimates which generalize existing estimates formerly obtained for homogeneous Helmholtz problems and provides numerical results which illustrate the good level of accuracy of the solution methodology.
Abstract: The numerical simulation of time-harmonic waves in heterogeneous media is a tricky task which consists in reproducing oscillations. These oscillations become stronger as the frequency increases, and high-order finite element methods have demonstrated their capability to reproduce the oscilla-tory behavior. However, they keep coping with limitations in capturing fine scale heterogeneities. We propose a new approach which can be applied in highly heterogeneous propagation media. It consists in constructing an approximate medium in which we can perform computations for a large variety of frequencies. The construction of the approximate medium can be understood as applying a quadrature formula locally. We establish estimates which generalize existing estimates formerly obtained for homogeneous Helmholtz problems. We then provide numerical results which illustrate the good level of accuracy of our solution methodology.
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TL;DR: Two new algorithms for the efficient and rigorous computation of Dirichlet L-functions are described and their use to verify the Generalised Riemann Hypothesis for all such L-Functions associated with primitive characters of modulus q<=400,000 is described.
Abstract: We describe two new algorithms for the efficient and rigorous computation of Dirichlet L-functions and their use to verify the Generalised Riemann Hypothesis for all such L-functions associated with primitive characters of modulus q<=400,000. For even q, we check to height t_0=max(1e8/q,7.5e7/q+200) and for odd q to height t_0=max(1e8/q,3.75e7/q+200).
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TL;DR: A priori error estimates of optimal order in the energy norm and the L-2-norm are derived and a reliable and efficient a posteriori error estimator is derived with the help of an auxiliary problem.
Abstract: In this article, an alternative energy-space based approach is proposed for the Dirichlet boundary control problem and then a finite-element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the L-2-norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help of an auxiliary problem. The theoretical results are illustrated by the numerical experiments.
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TL;DR: In this paper, the decay of the n-widths can be controlled by that of the error achieved by best n-term approximations using polynomials in the parametric variable.
Abstract: Kolmogorov n-widths and low-rank approximations are studied for families of ellip-tic diffusion PDEs parametrized by the diffusion coefficients. The decay of the n-widths can be controlled by that of the error achieved by best n-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the n-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the n-widths.
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TL;DR: A multilevel method with line smoothers and a nearly uniform convergence result on anisotropic meshes is presented and the so-called Xu-Zikatanov (XZ) identity is derived under the assumption that the underlying mesh is quasi-uniform.
Abstract: We develop and analyze multilevel methods for nonuniformly elliptic operators whose ellipticity holds in a weighted Sobolev space with an A2–Muckenhoupt weight. Using the so-called Xu-Zikatanov (XZ) identity, we derive a nearly uniform convergence result under the assumption that the underlying mesh is quasi-uniform. As an application we also consider the socalled α-harmonic extension to localize fractional powers of elliptic operators. Motivated by the scheme proposed by the second, third and fourth authors, we present a multilevel method with line smoothers and obtain a nearly uniform convergence result on anisotropic meshes. Numerical experiments illustrate the performance of our method.
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TL;DR: A novel algorithm is provided that produces much simpler linkages, but works only for parametric curves, and shows how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.
Abstract: Designing mechanical devices, called linkages, that draw a given plane curve has been a topic that interested engineers and mathematicians for hundreds of years, and recently also computer scientists. Already in 1876, Kempe proposed a procedure for solving the problem in full generality, but his constructions tend to be extremely complicated. We provide a novel algorithm that produces much simpler linkages, but works only for parametric curves. Our approach is to transform the problem into a factorization task over some noncommutative algebra. We show how to compute such a factorization, and how to use it to construct a linkage tracing a given curve.
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TL;DR: The exponential convergence of the perfectly matched layer (PML) method in terms of the thickness of the PML layer and the strength of PML medium property is proved.
Abstract: In this paper we study the convergence of the perfectly matched layer (PML) method for solving the time harmonic elastic wave scattering problems. We introduce a simple condition on the PML complex coordinate stretching function to guarantee the ellipticity of the PML operator. We also introduce a new boundary condition at the outer boundary of the PML layer which allows us to extend the reflection argument of Bramble and Pasciak to prove the stability of the PML problem in the truncated domain. The exponential convergence of the PML method in terms of the thickness of the PML layer and the strength of PML medium property is proved. Numerical results are included.
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TL;DR: The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor e, provided that the time stepsize is chosen as an integer fraction of the period.
Abstract: In this work, the error behavior of operator splitting methods is analyzed for highly-oscillatory differential equations. The scope of applications includes time-dependent nonlinear Schrodinger equations, where the evolution operator associated with the principal linear part is highly-oscillatory and periodic in time. In a first step, a known convergence result for the second-order Strang splitting method applied to the cubic Schrodinger equation is adapted to a wider class of nonlinearities. In a second step, the dependence of the global error on the decisive parameter 0 < e < < 1, defining the length of the period, is examined. The main result states that, compared to established error estimates, the Strang splitting method is more accurate by a factor e, provided that the time stepsize is chosen as an integer fraction of the period. This improved error behavior over a time interval of fixed length, which is independent of the period, is due to an averaging effect. The extension of the convergence result to higher-order splitting methods and numerical illustrations complement the investigations.
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TL;DR: An arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law and the local maximum principle is developed.
Abstract: In this paper, we develop and analyze an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method with a time-dependent approximation space for one dimensional conservation laws, which satisfies the geometric conservation law. For the semi-discrete ALE-DG method, when applied to nonlinear scalar conservation laws, a cell entropy inequality, L2 stability and error estimates are proven. More precisely, we prove the sub-optimal (k + 12) convergence for monotone fluxes, and optimal (k + 1) convergence for an upwind flux, when a piecewise P k polynomial approximation space is used. For the fully-discrete ALE-DG method, the geometric conservation law and the local maximum principle are proven. Moreover we state conditions for slope limiters, which ensure total variation stability of the method. Numerical examples show the capability of the method.
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TL;DR: The method is shown to be an absolutely stable HDG method for the indefinite time-harmonic Maxwell equations with high wave number and the dependence of convergence on the wave number k, the mesh size h and the polynomial order p is obtained.
Abstract: We present and analyze a hybridizable discontinuous Galerkin (HDG) method for the time-harmonic Maxwell equations. The divergence-free condition is enforced on the electric field, then a Lagrange multiplier is introduced, and the problem becomes the solution of a mixed curl-curl formulation of the Maxwell's problem. The method is shown to be an absolutely stable HDG method for the indefinite time-harmonic Maxwell equations with high wave number. By exploiting the duality argument, the dependence of convergence of the HDG method on the wave number k, the mesh size h and the polynomial order p is obtained. Numerical results are given to verify the theoretical analysis.
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TL;DR: This paper presents dynamic algorithms for the factorization set, length set, delta set, and $\omega$-primality in numerical monoids and demonstrates that these algorithms give significant improvements in runtime and memory usage.
Abstract: Studying the factorization theory of numerical monoids relies on understanding several important factorization invariants, including length sets, delta sets, and $\omega$-primality. While progress in this field has been accelerated by the use of computer algebra systems, many existing algorithms are computationally infeasible for numerical monoids with several irreducible elements. In this paper, we present dynamic algorithms for the factorization set, length set, delta set, and $\omega$-primality in numerical monoids and demonstrate that these algorithms give significant improvements in runtime and memory usage. In describing our dynamic approach to computing $\omega$-primality, we extend the usual definition of this invariant to the quotient group of the monoid and show that several useful results naturally extend to this broader setting.
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TL;DR: The classical Banach's fixed point Theorem and Lax-Milgram's Lemma are applied to prove well-posedness of the continuous problem and establish wellposedness and the corresponding Cea's estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown.
Abstract: In this paper we propose and analyze a new augmented mixed finite element method for the Navier-Stokes problem. Our approach is based on the introduction of a “nonlinear-pseudostress” tensor linking the pseudostress tensor with the convective term, which leads to a mixed formulation with the nonlinearpseudostress tensor and the velocity as the main unknowns of the system. Further variables of interest, such as the fluid pressure, the fluid vorticity and the fluid velocity gradient, can be easily approximated as a simple postprocess of the finite element solutions with the same rate of convergence. The resulting mixed formulation is augmented by introducing Galerkin least-squares type terms arising from the constitutive and equilibrium equations of the Navier-Stokes equation and from the Dirichlet boundary condition, which are multiplied by stabilization parameters that are chosen in such a way that the resulting continuous formulation becomes well-posed. Then, the classical Banach's fixed point Theorem and Lax-Milgram's Lemma are applied to prove well-posedness of the continuous problem. Similarly, we establish wellposedness and the corresponding Cea's estimate of the associated Galerkin scheme considering any conforming finite element subspace for each unknown. In particular, the associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the nonlinear-pseudostress tensor, and continuous piecewise polynomial elements of degree k+1 for the velocity, which leads to an optimal convergent scheme. In addition, we provide two iterative methods to solve the corresponding nonlinear system of equations and analyze their convergence. Finally, several numerical results illustrating the good performance of the method are provided. This contribution is based on joint work with Ricardo Oyarzua (Universidad del Bio-Bio) and Giordano Tierra (Charles University).
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TL;DR: A random particle blob method for the Keller-Segel equation (with dimension d 2) is introduced and a rigorous convergence analysis is established.
Abstract: In this paper, we introduce a random particle blob method for the Keller-Segel equation (with dimension d 2) and establish a rigorous convergence analysis.
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TL;DR: This paper investigates a family of L stable high order discontinuous Galerkin methods defined through a general form of numerical fluxes, and identifies a sub-family, termed αβ-fluxes, which is proven to have optimal L error estimates and superconvergence properties.
Abstract: Simulating wave propagation is one of the fundamental problems in scientific computing. In this paper, we consider one-dimensional two-way wave equations, and investigate a family of L stable high order discontinuous Galerkin methods defined through a general form of numerical fluxes. For these L stable methods, we systematically establish stability (hence energy conservation), error estimates (in both L and negative-order norms), and dispersion analysis. One novelty of this work is to identify a sub-family of the numerical fluxes, termed αβ-fluxes. Discontinuous Galerkin methods with αβ-fluxes are proven to have optimal L error estimates and superconvergence properties. Moreover, both the upwind and alternating fluxes belong to this sub-family. Dispersion analysis, which examines both the physical and spurious modes, provides insights into the sub-optimal accuracy of the methods using the central flux and the odd degree polynomials, and demonstrates the importance of numerical initialization for the proposed non-dissipative schemes. Numerical examples are presented to illustrate the accuracy and the long-term behavior of the methods under consideration.
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TL;DR: The main technique is a damping adding-removing procedure, which establishes the well-posedness of linear (or linearized) half-space equations with general boundary conditions and quasi-optimality of the numerical scheme.
Abstract: We study half-space linear kinetic equations with general boundary conditions that consist of both given incoming data and various type of reflections, extending our previous work [LLS14] on half-space equations with incoming boundary conditions. As in [LLS14], the main technique is a damping adding-removing procedure. We establish the well-posedness of linear (or linearized) half-space equations with general boundary conditions and quasi-optimality of the numerical scheme. The numerical method is validated by examples including a two-species transport equation, a multi-frequency transport equation, and the linearized BGK equation in 2D velocity space.
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TL;DR: The main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters.
Abstract: This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings. This article is dedicated to the computation of the moments of the solution to elliptic partial differential equations with random, log-normally distributed diffusion coefficients by the quasi-Monte Carlo method. Our main result is that the convergence rate of the quasi-Monte Carlo method based on the Halton sequence for the moment computation depends only linearly on the dimensionality of the stochastic input parameters. Especially, we attain this rather mild dependence on the stochastic dimensionality without any randomization of the quasi-Monte Carlo method under consideration. For the proof of the main result, we require related regularity estimates for the solution and its powers. These estimates are also provided here. Numerical experiments are given to validate the theoretical findings.
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TL;DR: An inexact version of this scheme is investigated which allows the decomposed subproblems to be solved approximately subject to certain inexactness criteria.
Abstract: The augmented Lagrangian method (ALM) is a benchmark for solving convex minimization problems with linear constraints. Solving the augmented subproblems over the primal variables can be regarded as sequentially providing inputs for updating the Lagrange multiplier (i.e., the dual variable). We consider the separable case of a convex minimization problem where its objective function is the sum of more than two functions without coupled variables. When applying the ALM to this case, at each iteration we can (sometimes must) split the resulting augmented subproblem in order to generate decomposed subproblems which are often easy enough to have closedform solutions. But the decomposition of primal variables only provides less accurate inputs for updating the Lagrange multiplier, and it points out the lack of convergence for such a decomposition scheme. To remedy this difficulty, we propose to update the Lagrange multiplier sequentially once each decomposed subproblem over the primal variables is solved. This scheme updates both the primal and dual variables in Gauss-Seidel fashion. In addition to the exact version which is useful enough for the case where the functions in the objective are all simple such that the decomposed subproblems all have closed-form solutions, we investigate an inexact version of this scheme which allows the decomposed subproblems to be solved approximately subject to certain inexactness criteria. Some preliminary numerical results when the proposed scheme is respectively applied to an image decomposition problem and an allocation problem are reported.