Quadrature

Index

• TransportMaps.ClenshawCurtisKnots
• TransportMaps.GaussHermiteKnots
• TransportMaps.GaussLegendreKnots
• TransportMaps.LatinHypercubeWeights
• TransportMaps.MonteCarloWeights
• TransportMaps.SparseSmolyakWeights
• TransportMaps.TensorProductWeights
• TransportMaps.GaussHermiteWeights
• TransportMaps.GaussLegendreWeights
• TransportMaps.gaussquadrature

Quadrature Knots

TransportMaps.GaussHermiteKnots Type

GaussHermiteKnots

One-dimensional Gauss-Hermite quadrature weights for computing integrals for the form:

 

where is the standard normal density.

source

TransportMaps.GaussLegendreKnots Type

GaussLegendreKnots

One-dimensional Gauss-Legendre quadrature weights for computing integrals for the form:

 

where is the uniform density.

source

TransportMaps.ClenshawCurtisKnots Type

ClenshawCurtisKnots

One-dimensional Clenshaw-Curtis quadrature weights for computing integrals for the form:

 

where is the uniform density.

source

Types

TransportMaps.TensorProductWeights Type

TensorProductWeights{T<:AbstractQuadratureKnots}

Multi-dimensional quadrature rule constructed from a tensor-product of one-dimensional quadrature knots. The level parameter controls accuracy.

Fields

• points::Matrix{Float64}: Quadrature points (tensor grid)

• weights::Vector{Float64}: Quadrature weights

• knots::AbstractQuadratureKnots: The 1D quadrature knots, e.g., GaussHermiteKnots or GaussLegendreKnots

Constructors

• TensorProductWeights(level::Int64, dim::Int64, knots::AbstractQuadratureKnots)

• TensorProductWeights(level::Int64, map::AbstractTransportMap)

See also GaussHermiteWeights and GaussLegendreWeights, and SparseSmolyakWeights for spare grids.

source

TransportMaps.GaussHermiteWeights Function

GaussHermiteWeights(level::Int64, dim::Int64)

Tensor-product Gauss-Hermite weights for integration with respect to standard Gaussian. Returns a TensorProductWeights object.

source

GaussHermiteWeights(level::Int64, map::AbstractTransportMap)

Tensor-product Gauss-Hermite weights for integration with respect to standard Gaussian constructed from the transport map map with normal reference density. Returns a TensorProductWeights object.

source

TransportMaps.GaussLegendreWeights Function

GaussLegendreWeights(level::Int64, dim::Int64, domain=[0, 1])

Tensor-product Gauss-Legendre weights for integration with respect to Uniform[0,1]. Returns a TensorProductWeights object.

source

GaussLegendreWeights(level::Int64, map::AbstractTransportMap)

Tensor-product Gauss-Legendre weights for integration with respect to Uniform[a,b] constructed from the transport map map with uniform reference density. Returns a TensorProductWeights object.

source

TransportMaps.SparseSmolyakWeights Type

SparseSmolyakWeights{T<:AbstractQuadratureKnots}

Multi-dimensional quadrature rule constructed using a sparse Smolyak grid of one-dimensional quadrature knots. The level parameter controls accuracy.

Fields

• points::Matrix{Float64}: Quadrature points (sparse grid)

• weights::Vector{Float64}: Quadrature weights

• knots::AbstractQuadratureKnots: The 1D quadrature knots, e.g., GaussHermiteKnots or GaussLegendreKnots

Constructors

• SparseSmolyakWeights(level::Int64, dim::Int64, knots=GaussHermiteKnots()): Construct sparse Smolyak grid with specified level, dim and knots.

• SparseSmolyakWeights(level::Int64, map::AbstractTransportMap): Construct sparse Smolyak grid with the help of the transport map. The knots are chosen based on the reference distribution (GaussHermiteKnots for a Normal reference and GaussLegendreKnots for a Uniform reference).

See also TensorProductWeights.

source

TransportMaps.MonteCarloWeights Type

MonteCarloWeights

Monte Carlo quadrature using random samples from a reference distribution. All points receive uniform weights 1/numberpoints.

Fields

• points::Matrix{Float64}: Quadrature points (random samples)

• weights::Vector{Float64}: Quadrature weights (uniform)

• distribution::Distributions.UnivariateDistribution: Reference distribution

Constructors

• MonteCarloWeights(numberpoints::Int64, dimension::Int64, distr::Distributions.UnivariateDistribution=Normal()): Construct weights with random sampling from distr (defaults to Normal()).

• MonteCarloWeights(numberpoints::Int64, map::AbstractTransportMap): Get number of dimensions and sample from the map's reference density.

• MonteCarloWeights(points::Matrix{Float64}, weights::Vector{Float64}=Float64[]): Construct from custom points and weights.

source

TransportMaps.LatinHypercubeWeights Type

LatinHypercubeWeights

Latin Hypercube sampling for quasi-Monte Carlo integration. Provides better space-filling properties than pure Monte Carlo. All points receive uniform weights 1/n.

Fields

• points::Matrix{Float64}: Quadrature points (Latin Hypercube samples)

• weights::Vector{Float64}: Quadrature weights (uniform)

• distribution::Distributions.UnivariateDistribution: Reference distribution

Constructors

• LatinHypercubeWeights(n::Int64, d::Int64, dist::Distributions.UnivariateDistribution=Normal()): Construct Latin Hypercube samples for d dimensions using dist (defaults to Normal()).

• LatinHypercubeWeights(n::Int64, map::AbstractTransportMap): Get number of dimensions and sample according to the map's reference density.

source

Functions

TransportMaps.gaussquadrature Function

gaussquadrature(fun::Function, n::Int, a::Real, b::Real)

Numerically integrates the function fun over the interval [a, b] using the Gauss-Legendre quadrature rule with n points.

Arguments

• fun::Function: The function to integrate. Should accept a single Float64 argument.

• n::Int: The number of quadrature points to use.

• a::Real: The lower bound of the integration interval.

• b::Real: The upper bound of the integration interval.

Returns

• Float64: The approximate value of the integral of fun over [a, b].

source