Data.Manifold.Web

The web data type

data PointsWeb :: * -> * -> *

A PointsWeb is almost, but not quite a mesh. It is a stongly connected† directed graph, backed by a tree for fast nearest-neighbour lookup of points.

†In general, there can be disconnected components, but every connected component is strongly connected.

Instances

Functor (PointsWeb a)

Methods

fmap :: (a -> b) -> PointsWeb a a -> PointsWeb a b

(<$) :: a -> PointsWeb a b -> PointsWeb a a

Foldable (PointsWeb a)

Methods

fold :: Monoid m => PointsWeb a m -> m

foldMap :: Monoid m => (a -> m) -> PointsWeb a a -> m

foldr :: (a -> b -> b) -> b -> PointsWeb a a -> b

foldr' :: (a -> b -> b) -> b -> PointsWeb a a -> b

foldl :: (b -> a -> b) -> b -> PointsWeb a a -> b

foldl' :: (b -> a -> b) -> b -> PointsWeb a a -> b

foldr1 :: (a -> a -> a) -> PointsWeb a a -> a

foldl1 :: (a -> a -> a) -> PointsWeb a a -> a

toList :: PointsWeb a a -> [a]

null :: PointsWeb a a -> Bool

length :: PointsWeb a a -> Int

elem :: Eq a => a -> PointsWeb a a -> Bool

maximum :: Ord a => PointsWeb a a -> a

minimum :: Ord a => PointsWeb a a -> a

sum :: Num a => PointsWeb a a -> a

product :: Num a => PointsWeb a a -> a

Traversable (PointsWeb a)

Methods

traverse :: Applicative f => (a -> f b) -> PointsWeb a a -> f (PointsWeb a b)

sequenceA :: Applicative f => PointsWeb a (f a) -> f (PointsWeb a a)

mapM :: Monad m => (a -> m b) -> PointsWeb a a -> m (PointsWeb a b)

sequence :: Monad m => PointsWeb a (m a) -> m (PointsWeb a a)

Foldable (PointsWeb x) (->) (->)

Methods

ffoldl :: (ObjectPair (->) a b, ObjectPair (->) a (PointsWeb x b)) => ((a, b) -> a) -> (a, PointsWeb x b) -> a

foldMap :: (Object (->) a, Object (->) (PointsWeb x a), Monoid m, Object (->) m, Object (->) m) => (a -> m) -> PointsWeb x a -> m

Generic (PointsWeb a b)

Associated Types

type Rep (PointsWeb a b) :: * -> *

Methods

from :: PointsWeb a b -> Rep (PointsWeb a b) x

to :: Rep (PointsWeb a b) x -> PointsWeb a b

(NFData x, NFData (Metric x), NFData (Needle' x), NFData y) => NFData (PointsWeb x y)

Methods

rnf :: PointsWeb x y -> ()

type Rep (PointsWeb a b)

type Rep (PointsWeb a b) = D1 (MetaData "PointsWeb" "Data.Manifold.Web.Internal" "manifolds-0.4.4.0-4U7mLxDASVE34WPYCdNRgy" True) (C1 (MetaCons "PointsWeb" PrefixI True) (S1 (MetaSel (Just Symbol "webNodeRsc") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (Shaded a (Neighbourhood a b)))))

Construction

fromWebNodes :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => MetricChoice x -> [(x, y)] -> PointsWeb x y

fromShadeTree_auto :: forall x. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => ShadeTree x -> PointsWeb x ()

fromShadeTree :: forall x. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => (Shade x -> Metric x) -> ShadeTree x -> PointsWeb x ()

fromShaded

Arguments

:: (WithField ℝ Manifold x, SimpleSpace (Needle x))

=> MetricChoice x

Local scalar-product generator. You can always use recipMetric . _shadeExpanse (but this may give distortions compared to an actual Riemannian metric).

-> (x `Shaded` y)

Source tree.

-> PointsWeb x y

Lookup

nearestNeighbour :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => PointsWeb x y -> x -> Maybe (x, y)

fmap from the co-Kleisli category of WebLocally.

indexWeb :: PointsWeb x y -> WebNodeId -> Maybe (x, y)

toGraph :: (WithField ℝ Manifold x, SimpleSpace (Needle x)) => PointsWeb x y -> (Graph, Vertex -> (x, y))

webBoundary :: WithField ℝ Manifold x => PointsWeb x y -> [(Cutplane x, y)]

Decomposition

sliceWeb_lin :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x), Geodesic x, Geodesic y) => PointsWeb x y -> Cutplane x -> [(x, y)]

Fetch a point between any two neighbouring web nodes on opposite sides of the plane, and linearly interpolate the values onto the cut plane.

sampleWeb_2Dcartesian_lin :: (x ~ ℝ, y ~ ℝ, Geodesic z) => PointsWeb (x, y) z -> ((x, x), Int) -> ((y, y), Int) -> [(y, [(x, Maybe z)])]

sampleEntireWeb_2Dcartesian_lin :: (x ~ ℝ, y ~ ℝ, Geodesic z) => PointsWeb (x, y) z -> Int -> Int -> [(y, [(x, Maybe z)])]

Local environments

localFocusWeb :: WithField ℝ Manifold x => PointsWeb x y -> PointsWeb x ((x, y), [(Needle x, y)])

Uncertain functions

differentiateUncertainWebFunction :: forall x y. ModellableRelation x y => PointsWeb x (Shade' y) -> PointsWeb x (Shade' (LocalLinear x y))

differentiate²UncertainWebFunction :: forall x y. ModellableRelation x y => PointsWeb x (Shade' y) -> PointsWeb x (Shade' (Needle x `⊗〃+>` Needle y))

localModels_CGrid :: forall x y ㄇ. (ModellableRelation x y, LocalModel ㄇ) => PointsWeb x (Shade' y) -> [(x, ㄇ x y)]

Calculate a quadratic fit with uncertainty margin centered around the connection between any two adjacent nodes. In case of a regular grid (which we by no means require here!) this corresponds to the vector quantities of an Arakawa type C/D grid (cf. A. Arakawa, V.R. Lamb (1977): Computational design of the basic dynamical processes of the UCLA general circulation model)

Differential equations

Fixed resolution

iterateFilterDEqn_static :: (ModellableRelation x y, MonadPlus m, LocalModel ㄇ) => InformationMergeStrategy [] m (x, Shade' y) iy -> Embedding (->) (Shade' y) iy -> DifferentialEqn ㄇ x y -> PointsWeb x (Shade' y) -> Cofree m (PointsWeb x (Shade' y))

iterateFilterDEqn_static_selective :: (ModellableRelation x y, MonadPlus m, badness ~ ℝ, LocalModel ㄇ) => InformationMergeStrategy [] m (x, Shade' y) iy -> Embedding (->) (Shade' y) iy -> (x -> iy -> badness) -> DifferentialEqn ㄇ x y -> PointsWeb x (Shade' y) -> Cofree m (PointsWeb x (Shade' y))

Automatic resolution

filterDEqnSolutions_adaptive

Arguments

:: (ModellableRelation x y, AffineManifold y, badness ~ ℝ, Monad m, LocalModel ㄇ)

=> MetricChoice x

Scalar product on the domain, for regularising the web.

-> InconsistencyStrategy m x (Shade' y)

-> DifferentialEqn ㄇ x y

-> (x -> Shade' y -> badness)

-> PointsWeb x (SolverNodeState x y)

-> m (PointsWeb x (SolverNodeState x y))

iterateFilterDEqn_adaptive

Arguments

:: (ModellableRelation x y, AffineManifold y, LocalModel ㄇ, Monad m)

=> MetricChoice x

Scalar product on the domain, for regularising the web.

-> InconsistencyStrategy m x (Shade' y)

-> DifferentialEqn ㄇ x y

-> (x -> Shade' y -> ℝ)

Badness function for local results.

-> PointsWeb x (Shade' y)

-> [PointsWeb x (Shade' y)]

Configuration

data InconsistencyStrategy m x y where

Constructors

AbortOnInconsistency :: InconsistencyStrategy Maybe x y

IgnoreInconsistencies :: InconsistencyStrategy Identity x y

HighlightInconsistencies :: y -> InconsistencyStrategy Identity x y

Instances

Functor (InconsistencyStrategy m x)

Methods

fmap :: (a -> b) -> InconsistencyStrategy m x a -> InconsistencyStrategy m x b

(<$) :: a -> InconsistencyStrategy m x b -> InconsistencyStrategy m x a

newtype InformationMergeStrategy n m y' y

Constructors

InformationMergeStrategy

Fields

• mergeInformation :: y -> n y' -> m y

naïve :: (NonEmpty y -> y) -> InformationMergeStrategy [] Identity (x, y) y

inconsistencyAware :: (NonEmpty y -> m y) -> InformationMergeStrategy [] m (x, y) y

indicateInconsistencies :: (NonEmpty υ -> Maybe υ) -> InformationMergeStrategy [] (Except (PropagationInconsistency x υ)) (x, υ) υ

postponeInconsistencies :: Monad m => (NonEmpty υ -> Maybe υ) -> InformationMergeStrategy [] (WriterT [PropagationInconsistency x υ] m) (x, υ) υ

data PropagationInconsistency x υ

Constructors

PropagationInconsistency

Fields

• _inconsistentPropagatedData :: [(x, υ)]
• _inconsistentAPrioriData :: υ

PropagationInconsistencies [PropagationInconsistency x υ]

Instances

(Show υ, Show x) => Show (PropagationInconsistency x υ)

Methods

showsPrec :: Int -> PropagationInconsistency x υ -> ShowS

show :: PropagationInconsistency x υ -> String

showList :: [PropagationInconsistency x υ] -> ShowS

Monoid (PropagationInconsistency x υ)

Methods

mempty :: PropagationInconsistency x υ

mappend :: PropagationInconsistency x υ -> PropagationInconsistency x υ -> PropagationInconsistency x υ

mconcat :: [PropagationInconsistency x υ] -> PropagationInconsistency x υ

Misc

data ConvexSet x

Constructors

EmptyConvex

ConvexSet

Fields

• convexSetHull :: Shade' x

  If p is in all intersectors, it must also be in the hull.

• convexSetIntersectors :: [Shade' x]

Instances

LtdErrorShow x => Show (ConvexSet x)

Methods

showsPrec :: Int -> ConvexSet x -> ShowS

show :: ConvexSet x -> String

showList :: [ConvexSet x] -> ShowS

Refinable x => Semigroup (ConvexSet x)

Under intersection.

Methods

(<>) :: ConvexSet x -> ConvexSet x -> ConvexSet x

sconcat :: NonEmpty (ConvexSet x) -> ConvexSet x

stimes :: Integral b => b -> ConvexSet x -> ConvexSet x

ellipsoid :: Shade' x -> ConvexSet x

ellipsoidSet :: Embedding (->) (Maybe (Shade' x)) (ConvexSet x)

coerceWebDomain :: forall a b y. (Manifold a, Manifold b, LocallyCoercible a b, SimpleSpace (Needle b)) => PointsWeb a y -> PointsWeb b y

rescanPDELocally :: forall x y ㄇ. (ModellableRelation x y, LocalModel ㄇ) => DifferentialEqn ㄇ x y -> WebLocally x (Shade' y) -> Maybe (Shade' y)

webOnions :: forall x y. WithField ℝ Manifold x => PointsWeb x y -> PointsWeb x [[(x, y)]]

knitShortcuts :: forall x y. (WithField ℝ Manifold x, SimpleSpace (Needle x)) => MetricChoice x -> PointsWeb x y -> PointsWeb x y

Consider at each node not just the connections to already known neighbours, but also the connections to their neighbours. If these next-neighbours turn out to be actually situated closer, link to them directly.