Technical Reports from UMIACS
http://hdl.handle.net/1903/7
2015-10-30T03:30:59ZAccurate computation of Galerkin double surface integrals in the 3-D boundary element method
http://hdl.handle.net/1903/16394
Accurate computation of Galerkin double surface integrals in the 3-D boundary element method
Adelman, Ross; Gumerov, Nail A.; Duraiswami, Ramani
Many boundary element integral equation kernels are based on the Green’s functions of the Laplace and Helmholtz equations in three dimensions. These include, for example, the Laplace, Helmholtz, elasticity, Stokes, and Maxwell equations. Integral equation formulations lead to more compact, but dense linear systems. These dense systems are often solved iteratively via Krylov subspace methods, which may be accelerated via the fast multipole method. There are advantages to Galerkin formulations for such integral equations, as they treat problems associated with kernel singularity, and lead to symmetric and better conditioned matrices. However, the Galerkin method requires each entry in the system matrix to be created via the computation of a double surface integral over one or more pairs of triangles. There are a number of semi-analytical methods to treat these integrals, which all have some issues, and are discussed in this paper. We present novel methods to compute all the integrals that arise in Galerkin formulations involving kernels based on the Laplace and Helmholtz Green’s functions to any specified accuracy. Integrals involving completely geometrically separated triangles are non-singular and are computed using a technique based on spherical harmonics and multipole expansions and translations, which results in the integration of polynomial functions over the triangles.
Integrals involving cases where the triangles have common vertices, edges, or are coincident are treated via scaling and symmetry arguments, combined with automatic recursive geometric decomposition of the integrals. Example results are presented, and the developed software is available as open source.
2015-05-29T00:00:00ZA Stochastic Approach to Uncertainty in the Equations of MHD Kinematics
http://hdl.handle.net/1903/15523
A Stochastic Approach to Uncertainty in the Equations of MHD Kinematics
Phillips, Edward G.; Elman, Howard C.
The magnetohydodynamic (MHD) kinematics model describes the
electromagnetic behavior of an electrically conducting fluid when its
hydrodynamic properties are assumed to be known. In particular, the MHD
kinematics equations can be used to simulate the magnetic field induced
by a given velocity field. While prescribing the velocity field leads to
a simpler model than the fully coupled MHD system, this may introduce
some epistemic uncertainty into the model. If the velocity of a physical
system is not known with certainty, the magnetic field obtained from the
model may not be reflective of the magnetic field seen in experiments.
Additionally, uncertainty in physical parameters such as the magnetic
resistivity may affect the reliability of predictions obtained from this
model. By modeling the velocity and the resistivity as random variables
in the MHD kinematics model, we seek to quantify the effects of
uncertainty in these fields on the induced magnetic field. We develop
stochastic expressions for these quantities and investigate their impact
within a finite element discretization of the kinematics equations. We
obtain mean and variance data through Monte-Carlo simulation for several
test problems. Toward this end, we develop and test an efficient block
preconditioner for the linear systems arising from the discretized
equations.
2014-07-10T00:00:00ZPreconditioning Techniques for Reduced Basis Methods for Parameterized Partial Differential Equations
http://hdl.handle.net/1903/15078
Preconditioning Techniques for Reduced Basis Methods for Parameterized Partial Differential Equations
Elman, Howard C.; Forstall, Virginia
The reduced basis methodology is an efficient approach to solve
parameterized discrete partial differential equations when the solution
is needed at many parameter values. An offline step approximates the
solution space and an online step utilizes this approximation, the
reduced basis, to solve a smaller reduced problem, which provides an
accurate estimate of the solution. Traditionally, the reduced problem is
solved using direct methods. However, the size of the reduced system
needed to produce solutions of a given accuracy depends on the
characteristics of the problem, and it may happen that the size is
significantly smaller than that of the original discrete problem but
large enough to make direct solution costly. In this scenario, it may be
more effective to use iterative methods to solve the reduced problem. We
construct preconditioners for reduced iterative methods which are
derived from preconditioners for the full problem. This approach permits
reduced basis methods to be practical for larger bases than direct
methods allow. We illustrate the effectiveness of iterative methods for
solving reduced problems by considering two examples, the steady-state
diffusion and convection-diffusion-reaction equations.
2014-05-27T00:00:00ZAnomaly Detection for Symbolic Representations
http://hdl.handle.net/1903/15073
Anomaly Detection for Symbolic Representations
Cox, Michael T.; Paisner, Matt; Oates, Tim; Perlis, Don
A fully autonomous agent recognizes new problems, explains what causes
such problems, and generates its own goals to solve these problems. Our
approach to this goal-driven model of autonomy uses a methodology called
the Note-Assess-Guide procedure. It instantiates a monitoring process in
which an agent notes an anomaly in the world, assesses the nature and
cause of that anomaly, and guides appropriate modifications to behavior.
This report describes a novel approach to the note phase of that
procedure. A-distance, a sliding-window statistical distance metric, is
applied to numerical vector representations of intermediate states from
plans generated for two symbolic domains. Using these representations,
the metric is able to detect anomalous world states caused by
restricting the actions available to the planner.
2014-03-25T00:00:00Z