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We propose in this paper a new minimization algorithm based on a slightly modified version of the scalar auxialiary variable (SAV) approach coupled with a relaxation step and an adaptive strategy. It enjoys several distinct advantages over popular gradient based methods: (i) it is unconditionally energy diminishing with a modified energy which is intrinsically related to the original energy, and thus no parameter tuning is needed for stability; (ii) it allows the use of large step-sizes which can effectively accelerate the convergence rate. We also present a convergence analysis for some SAV based algorithms, which include our new algorithm without the relaxation step as a special case. We apply our new algorithm to several illustrative and benchmark problems and compare its performance with several popular gradient based methods. The numerical results indicate that the new algorithm is very robust, and its adaptive version usually converges significantly faster than those popular gradient decent based methods. [ABSTRACT FROM AUTHOR]

Applying smoothed aggregation (SA) multigrid to solve a nonsymmetric linear system, Ax = b, is often impeded by the lack of a minimization principle that can be used as a basis for the coarse-grid correction process. This paper proposes a Petrov-Galerkin (PG) approach based on applying SA to either of two symmetric positive definite (SPD) matrices, √AtA or √AAt. These matrices, however, are typically full and difficult to compute, so it is not computationally efficient to use them directly to form a coarse-grid correction. The proposed approach approximates these coarse-grid corrections by using SA to accurately approximate the right and left singular vectors of A that correspond to the lowest singular value. These left and right singular vectors are used to construct the restriction and interpolation operators, respectively. A preliminary two-level convergence theory is presented, suggesting that more relaxation should be applied than for an SPD problem. Additionally, a nonsymmetric version of adaptive SA (αSA) is given that automatically constructs SA multi grid hierarchies using a stationary relaxation process on all levels. Numerical results are reported for convection-diffusion problems in two dimensions with varying amounts of convection for constant, variable, and recirculating convection fields. The results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar PDEs (with the exception of recirculating flow). This paper serves as a first step for nonsymmetric aSA. The long-term goal of this effort is to develop nonsymmetric aSA for systems of PDEs, where the SA framework has proven to be well suited for adaptivity in SPD problems. [ABSTRACT FROM AUTHOR]

When solving partial differential equations using boundary integral equation methods, accurate evaluation of singular and nearly singular integrals in layer potentials is crucial. A recent scheme for this is quadrature by expansion (QBX), which solves the problem by locally approximating the potential using a local expansion centered at some distance from the source boundary. In this paper we introduce an extension of the QBX scheme in two dimensions (2D) denoted AQBX--adaptive quadrature by expansion--which combines QBX with an algorithm for automated selection of parameters, based on a target error tolerance. A key component in this algorithm is the ability to accurately estimate the numerical errors in the coefficients of the expansion. Combining previous results for flat panels with a procedure for taking the panel shape into account, we derive such error estimates for arbitrarily shaped boundaries in 2D that are discretized using panel-based Gauss-Legendre quadrature. Applying our scheme to numerical solutions of Dirichlet problems for the Laplace and Helmholtz equations, and also for solving these equations, we find that the scheme is able to satisfy a given target tolerance to within an order of magnitude, making it useful for practical applications. This represents a significant simplification over the original QBX algorithm, in which choosing a good set of parameters can be hard. [ABSTRACT FROM AUTHOR]

Three-dimensional (3D) image registration is a computationally intensive problem which is commonly solved in medical imaging. The complexity of the problem stems from its size and nonlinearity. In this paper we present an approach that drastically reduces the problem size by using adaptive mesh refinement. Our approach requires special and careful discretization of the variational form on adaptive quad/octree grids. It further requires an appropriate refinement criterion. We show that in some cases this approach can reduce the computational time by a factor of approximately 10 or so in two dimensions and 5 in three dimensions compared to the nonadaptive approach. [ABSTRACT FROM AUTHOR]

The Uintah software framework for the solution of a broad class of fluid-structure interaction problems has been developed by using a problem-driven approach that dates back to its inception. Uintah uses a layered task-graph approach that decouples the problem specification as a set of tasks from the adaptive runtime system that executes these tasks. Using this approach, it is possible to improve the performance of the software components to enable the solution of broad classes of problems as well as the driving problem itself. This process is illustrated by a motivating problem: the computational modeling of the hazards posed by thousands of explosive devices during a deflagration-to-detonation transition that occurred on Highway 6 in Utah. In order to solve this complex fluid-structure interaction problem at the required scale, substantial algorithmic and data structure improvements were needed to Uintah. These improvements enabled scalable runs for the target problem and provided the capability to model the transition to detonation. The solution to the target problem from these runs provided insight as to why the detonation happened, as well as demonstrating a possible remediation strategy that may have avoided detonation. [ABSTRACT FROM AUTHOR]

The inverse scattering problem under consideration is to determine the shape of an obstacle in R3 from a knowledge of the time harmonic incident acoustic wave and the far field pattern of the scattered wave with frequency in the resonance region. A method for solving this nonlinear and improperly posed problem is presented which is based on solving a linear integral equation of the first kind and avoids the use of nonlinear optimization methods. Numerical examples are given showing the practicality of this new approach. [ABSTRACT FROM AUTHOR]

An adaptive coarse space approach including a condition number bound for dual primal finite element tearing and interconnecting (FETI-DP) methods applied to three dimensional problems with coefficient jumps inside subdomains and across subdomain boundaries is presented. The approach is based on a known adaptive coarse space approach enriched by a small number of additional local edge eigenvalue problems. These edge eigenvalue problems serve to make the method robust and permit a condition number bound which depends only on the tolerance of the local eigenvalue problems and some properties of the domain decomposition. The introduction of the edge eigenvalue problems thus turns a well-known condition number indicator for FETI-DP and balancing domain decomposition by constraints (BDDC) methods into a condition number estimate. Numerical results are presented for linear elasticity and heterogeneous materials supporting our theoretical findings. The problems considered include those with random coefficients and almost incompressible material components. [ABSTRACT FROM AUTHOR]