14,567 research outputs found
Entropic Wasserstein Gradient Flows
This article details a novel numerical scheme to approximate gradient flows
for optimal transport (i.e. Wasserstein) metrics. These flows have proved
useful to tackle theoretically and numerically non-linear diffusion equations
that model for instance porous media or crowd evolutions. These gradient flows
define a suitable notion of weak solutions for these evolutions and they can be
approximated in a stable way using discrete flows. These discrete flows are
implicit Euler time stepping according to the Wasserstein metric. A bottleneck
of these approaches is the high computational load induced by the resolution of
each step. Indeed, this corresponds to the resolution of a convex optimization
problem involving a Wasserstein distance to the previous iterate. Following
several recent works on the approximation of Wasserstein distances, we consider
a discrete flow induced by an entropic regularization of the transportation
coupling. This entropic regularization allows one to trade the initial
Wasserstein fidelity term for a Kulback-Leibler divergence, which is easier to
deal with numerically. We show how KL proximal schemes, and in particular
Dykstra's algorithm, can be used to compute each step of the regularized flow.
The resulting algorithm is both fast, parallelizable and versatile, because it
only requires multiplications by a Gibbs kernel. On Euclidean domains
discretized on an uniform grid, this corresponds to a linear filtering (for
instance a Gaussian filtering when is the squared Euclidean distance) which
can be computed in nearly linear time. On more general domains, such as
(possibly non-convex) shapes or on manifolds discretized by a triangular mesh,
following a recently proposed numerical scheme for optimal transport, this
Gibbs kernel multiplication is approximated by a short-time heat diffusion
Volume 2: Explicit, multistage upwind schemes for Euler and Navier-Stokes equations
The objective of this study was to develop a high-resolution-explicit-multi-block numerical algorithm, suitable for efficient computation of the three-dimensional, time-dependent Euler and Navier-Stokes equations. The resulting algorithm has employed a finite volume approach, using monotonic upstream schemes for conservation laws (MUSCL)-type differencing to obtain state variables at cell interface. Variable interpolations were written in the k-scheme formulation. Inviscid fluxes were calculated via Roe's flux-difference splitting, and van Leer's flux-vector splitting techniques, which are considered state of the art. The viscous terms were discretized using a second-order, central-difference operator. Two classes of explicit time integration has been investigated for solving the compressible inviscid/viscous flow problems--two-state predictor-corrector schemes, and multistage time-stepping schemes. The coefficients of the multistage time-stepping schemes have been modified successfully to achieve better performance with upwind differencing. A technique was developed to optimize the coefficients for good high-frequency damping at relatively high CFL numbers. Local time-stepping, implicit residual smoothing, and multigrid procedure were added to the explicit time stepping scheme to accelerate convergence to steady-state. The developed algorithm was implemented successfully in a multi-block code, which provides complete topological and geometric flexibility. The only requirement is C degree continuity of the grid across the block interface. The algorithm has been validated on a diverse set of three-dimensional test cases of increasing complexity. The cases studied were: (1) supersonic corner flow; (2) supersonic plume flow; (3) laminar and turbulent flow over a flat plate; (4) transonic flow over an ONERA M6 wing; and (5) unsteady flow of a compressible jet impinging on a ground plane (with and without cross flow). The emphasis of the test cases was validation of code, and assessment of performance, as well as demonstration of flexibility
Monte-Carlo methods for the pricing of American options: a semilinear BSDE point of view
We extend the viscosity solution characterization proved in [5] for call/put
American option prices to the case of a general payoff function in a
multi-dimensional setting: the price satisfies a semilinear re-action/diffusion
type equation. Based on this, we propose two new numerical schemes inspired by
the branching processes based algorithm of [8]. Our numerical experiments show
that approximating the discontinu-ous driver of the associated
reaction/diffusion PDE by local polynomials is not efficient, while a simple
randomization procedure provides very good results
Locally adaptive image denoising by a statistical multiresolution criterion
We demonstrate how one can choose the smoothing parameter in image denoising
by a statistical multiresolution criterion, both globally and locally. Using
inhomogeneous diffusion and total variation regularization as examples for
localized regularization schemes, we present an efficient method for locally
adaptive image denoising. As expected, the smoothing parameter serves as an
edge detector in this framework. Numerical examples illustrate the usefulness
of our approach. We also present an application in confocal microscopy
Inverse airfoil design procedure using a multigrid Navier-Stokes method
The Modified Garabedian McFadden (MGM) design procedure was incorporated into an existing 2-D multigrid Navier-Stokes airfoil analysis method. The resulting design method is an iterative procedure based on a residual correction algorithm and permits the automated design of airfoil sections with prescribed surface pressure distributions. The new design method, Multigrid Modified Garabedian McFadden (MG-MGM), is demonstrated for several different transonic pressure distributions obtained from both symmetric and cambered airfoil shapes. The airfoil profiles generated with the MG-MGM code are compared to the original configurations to assess the capabilities of the inverse design method
Variational Downscaling, Fusion and Assimilation of Hydrometeorological States via Regularized Estimation
Improved estimation of hydrometeorological states from down-sampled
observations and background model forecasts in a noisy environment, has been a
subject of growing research in the past decades. Here, we introduce a unified
framework that ties together the problems of downscaling, data fusion and data
assimilation as ill-posed inverse problems. This framework seeks solutions
beyond the classic least squares estimation paradigms by imposing proper
regularization, which are constraints consistent with the degree of smoothness
and probabilistic structure of the underlying state. We review relevant
regularization methods in derivative space and extend classic formulations of
the aforementioned problems with particular emphasis on hydrologic and
atmospheric applications. Informed by the statistical characteristics of the
state variable of interest, the central results of the paper suggest that
proper regularization can lead to a more accurate and stable recovery of the
true state and hence more skillful forecasts. In particular, using the Tikhonov
and Huber regularization in the derivative space, the promise of the proposed
framework is demonstrated in static downscaling and fusion of synthetic
multi-sensor precipitation data, while a data assimilation numerical experiment
is presented using the heat equation in a variational setting
Consistent thermodynamic derivative estimates for tabular equations of state
Numerical simulations of compressible fluid flows require an equation of
state (EOS) to relate the thermodynamic variables of density, internal energy,
temperature, and pressure. A valid EOS must satisfy the thermodynamic
conditions of consistency (derivation from a free energy) and stability
(positive sound speed squared). When phase transitions are significant, the EOS
is complicated and can only be specified in a table. For tabular EOS's such as
SESAME from Los Alamos National Laboratory, the consistency and stability
conditions take the form of a differential equation relating the derivatives of
pressure and energy as functions of temperature and density, along with
positivity constraints. Typical software interfaces to such tables based on
polynomial or rational interpolants compute derivatives of pressure and energy
and may enforce the stability conditions, but do not enforce the consistency
condition and its derivatives. We describe a new type of table interface based
on a constrained local least squares regression technique. It is applied to
several SESAME EOS's showing how the consistency condition can be satisfied to
round-off while computing first and second derivatives with demonstrated
second-order convergence. An improvement of 14 orders of magnitude over
conventional derivatives is demonstrated, although the new method is apparently
two orders of magnitude slower, due to the fact that every evaluation requires
solving an 11-dimensional nonlinear system.Comment: 29 pages, 9 figures, 16 references, submitted to Phys Rev
Impact of different time series aggregation methods on optimal energy system design
Modelling renewable energy systems is a computationally-demanding task due to
the high fluctuation of supply and demand time series. To reduce the scale of
these, this paper discusses different methods for their aggregation into
typical periods. Each aggregation method is applied to a different type of
energy system model, making the methods fairly incomparable. To overcome this,
the different aggregation methods are first extended so that they can be
applied to all types of multidimensional time series and then compared by
applying them to different energy system configurations and analyzing their
impact on the cost optimal design. It was found that regardless of the method,
time series aggregation allows for significantly reduced computational
resources. Nevertheless, averaged values lead to underestimation of the real
system cost in comparison to the use of representative periods from the
original time series. The aggregation method itself, e.g. k means clustering,
plays a minor role. More significant is the system considered: Energy systems
utilizing centralized resources require fewer typical periods for a feasible
system design in comparison to systems with a higher share of renewable
feed-in. Furthermore, for energy systems based on seasonal storage, currently
existing models integration of typical periods is not suitable
- …