GridapROMs.RBSteady

abstract type AbstractTTCore{T,N} <: AbstractArray{T,N} end

Type for nonstandard representations of tensor train cores.

Subtypes:

• DofMapCore
• SparseCore

struct AdaptiveReduction{A,B,R<:DirectReduction{A,B}} <: GreedyReduction{A,B}
  reduction::R
  adaptive_nparams::Int
  adaptive_tol::Float64
  adaptive_maxiter::Int
end

Not implemented yet. Will serve as a parameter-adaptivity greedy reduction algorithm

const AffineContribution{V<:Projection} = Contribution{V}

Contribution whose field values are Projections

struct AffineReduction{A,B} <: DirectReduction{A,B}
  red_style::A
  norm_style::B
end

Reduction employed when the input data is independent with respect to the considered realization. Therefore, simply considering a number of parameters equal to 1 suffices for this type of reduction

struct BlockHRProjection{A,I,N} <: HRProjection{A,I}
  array::Array{HRProjection{A,I},N}
  touched::Array{Bool,N}
end

Block container for HRProjection of type A in a MultiField setting. This type is conceived similarly to ArrayBlock in Gridap

struct BlockProjection{A<:HRProjection,N} <: Projection
  array::Array{A,N}
  touched::Array{Bool,N}
end

Block container for Projection of type A in a MultiField setting. This type is conceived similarly to ArrayBlock in Gridap

abstract type DirectReduction{A,B} <: Reduction{A,B} end

Type representing direct reduction methods, e.g. truncated POD, TTSVD, etc.

Subtypes:

• AffineReduction
• PODReduction
• TTSVDReduction

struct DofMapCore{T,A<:AbstractArray{T,3},B<:AbstractArray} <: AbstractTTCore{T,3}
  core::A
  dof_map::B
end

Represents a tensor train core core reindexed by means of an index mapping dof_map

struct EnergyNorm <: NormStyle
  norm_op::Function
end

Trait indicating that the reduction algorithm will produce a basis orthogonal in the norm specified by norm_op. Note: norm_op should represent a symmetric, positive definite bilinear form (matrix)

struct EuclideanNorm <: NormStyle end

Trait indicating that the reduction algorithm will produce a basis orthogonal in the euclidean norm

struct FixedSVDRank <: ReductionStyle
  rank::Int
end

Struct employed when the chosen reduction algorithm is a truncated POD at a rank rank. Check this reference for more details on the truncated POD algorithm

struct GenericDomain{T,A} <: IntegrationDomain
  cells::Vector{Int32}
  cell_idofs::Table{T,Vector{T},Vector{Int32}}
  metadata::A
end

Integration domain for a projection operator in a steady problem

struct GenericHRProjection{A,B} <: HRProjection{A,B}
  basis::A
  style::B
  interpolation::Interpolation
end

Generic implementation of an HRProjection object

struct GenericRBOperator{O,T,A,B} <: RBOperator{O,T}
  op::ParamOperator{O,T}
  trial::RBSpace
  test::RBSpace
  lhs::A
  rhs::B
end

Fields:

• op: underlying high dimensional FE operator
• trial: reduced trial space
• test: reduced test space
• lhs: hyper-reduced left hand side
• rhs: hyper-reduced right hand side

struct HRParamArray{T,N,A,B,C<:ParamArray{T,N}} <: ParamArray{T,N}
  fecache::A
  coeff::B
  hypred::C
end

Parametric vector returned after the online phase of a hyper-reduction strategy. Fields:

• fecache: represents a parametric residual/Jacobian computed via integration on an IntegrationDomain

• coeff: parameter-dependent coefficient computed during the online phase according to the formula

  coeff = Φi⁻¹ fecache[i,:]

  where (Φi,i) are the interpolation and the reduced integration domain stored in a HyperReduction object.

• hypred: the ouptut of the online phase of a hyper-reduction strategy, acoording to the formula

  hypred = Φrb * coeff

  where Φrb is the basis stored in a HyperReduction object

abstract type HRProjection{A<:ReducedProjection,B<:HyperReduction} <: Projection end

Subtype of a Projection dedicated to the output of a hyper-reduction procedure applied on residual/jacobians of a differential problem. This procedure can be summarized in the following steps:

1. compute a snapshots tensor T
2. construct a Projection Φ by running the function reduction on T
3. implement an interpolation strategy

There are two types of interpolation strategies:

1. Empirical interpolation of the Projection Φ
2. Radial basis interpolation over the parameter space

We recall that a RB method requires the (Petrov-)Galerkin projection of the operators (residuals/Jacobians) on the reduced subspace spanned by Φ:

• for residuals: Φrb = test_basisᵀ Φ
• for Jacobians: Φrb = test_basisᵀ Φ trial_basis

The output of this operation is a ReducedProjection. Therefore, a HRProjection is completely characterized by the couple (Φrb,i), where i indicates the chosen interpolation strategy.

Subtypes:

• GenericHRProjection
• BlockHRProjection

abstract type HyperReduction{A} <: Reduction{A,EuclideanNorm} end

Type representing a hyper-reduction approximation of residuals/Jacobians of a differential problem. Orthogonality with respect to a norm other than the euclidean is not required for this reduction type.

Subtypes:

• MDEIMHyperReduction
• RBFHyperReduction
• HighDimHyperReduction

abstract type IntegrationDomain end

Type representing the set of interpolation rows of a Projection subjected to a EIM approximation. Subtypes:

• GenericDomain
• TransientIntegrationDomain

struct InvProjection <: Projection
  projection::Projection
end

Represents the inverse map of a Projection projection

struct LRApproxRank <: ReductionStyle
  opts::LRAOptions
end

Struct employed when the chosen reduction algorithm is a randomized POD that leverages the package LowRankApprox. The field opts specifies the options needed to run the randomized POD

struct LinearNonlinearRBOperator{A<:RBOperator,B<:RBOperator,T} <: RBOperator{LinearNonlinearParamEq,T}
  op_linear::A
  op_nonlinear::B
end

Extends the concept of GenericRBOperator to accommodate the linear/nonlinear splitting of terms in nonlinear applications

struct LocalRBOperator{O,T,A,B} <: RBOperator{O,T}
  op::ParamOperator{O,T}
  trial::RBSpace
  test::RBSpace
  lhs::A
  rhs::B
end

Fields:

• op: underlying high dimensional FE operator
• trial: local reduced trial spaces
• test: local reduced test spaces
• lhs: local hyper-reduced left hand sides
• rhs: local hyper-reduced right hand sides

struct LocalReduction{A,B,R<:Reduction{A,B}} <: Reduction{A,B}
  reduction::R
  ncentroids::Int
end

struct MDEIMHyperReduction{A} <: HyperReduction{A}
  reduction::Reduction{A,EuclideanNorm}
end

MDEIM struct employed in steady problems

struct MultiFieldRBSpace{S<:MultiFieldFESpace} <: RBSpace{S}
  space::S
  subspace::BlockProjection
end

Reduced basis subspace of a MultiFieldFESpace in Gridap

abstract type NormStyle end

Subtypes:

• EuclideanNorm
• EnergyNorm

struct NormedProjection <: Projection
  projection::Projection
  norm_matrix::MatrixOrTensor
end

Represents a Projection projection spanning a space equipped with an inner product represented by the quantity norm_matrix

struct PODProjection <: Projection
  basis::AbstractMatrix
end

Projection stemming from a truncated proper orthogonal decomposition tpod

struct PODReduction{A,B} <: DirectReduction{A,B}
  red_style::A
  norm_style::B
  nparams::Int
end

Reduction by means of a truncated POD. The field nparams indicates the number of parameters selected for the computation of the snapshots

abstract type Projection <: Map end

Represents a basis for a n-dimensional vector subspace of a N-dimensional vector space (where N >> n), to be used as a Galerkin projection operator. The kernel of a Projection is n-dimensional, whereas its image is N-dimensional.

Subtypes:

• PODProjection
• TTSVDProjection
• NormedProjection
• BlockProjection
• InvProjection
• ReducedProjection
• HRProjection

struct RBFHyperReduction{A} <: HyperReduction{A}
  reduction::Reduction{A,EuclideanNorm}
  strategy::AbstractRadialBasis
end

abstract type RBOperator{O,T} <: ParamOperator{O,T} end

Type representing reduced algebraic operators used within a reduced order modelling framework in steady applications. A RBOperator should contain the following information:

• a reduced test and trial space, computed according to reduced_spaces
• a hyper-reduced residual and jacobian, computed according to reduced_weak_form

Subtypes:

• GenericRBOperator
• LinearNonlinearRBOperator

struct RBParamVector{T,A<:ParamVector{T},B} <: ParamArray{T,1}
  data::A
  fe_data::B
end

Parametric vector obtained by applying a Projection on a high-dimensional parametric FE vector fe_data, which is stored (but mostly unused) for conveniency

struct RBSolver{A<:GridapType,B} <: GridapType
  fesolver::A
  state_reduction::Reduction
  residual_reduction::Reduction
  jacobian_reduction::B
end

Wrapper around a FE solver (e.g. NonlinearSolver or ODESolver in Gridap) with additional information on the reduced basis (RB) method employed to solve a given problem dependent on a set of parameters. A RB method is a projection-based reduced order model where

1. a suitable subspace of a FESpace is sought, of dimension n << Nₕ
2. a matrix-based discrete empirical interpolation method (e.g. MDEIM) is performed

to approximate the manifold of the parametric residuals and jacobians

1. the EIM approximations are compressed with (Petrov-)Galerkin projections

onto the subspace

1. for every desired choice of parameters, numerical integration is performed, and

the resulting n × n system of equations is cheaply solved

In particular:

• tol: tolerance used in the projection-based truncated proper orthogonal decomposition (TPOD) or in the tensor train singular value decomposition (TT-SVD), where a basis spanning the reduced subspace is computed; the value of tol is responsible for selecting the dimension of the subspace, i.e. n = n(tol)
• nparams_state: number of snapshots considered when running TPOD or TT-SVD
• nparams_res: number of snapshots considered when running hyper-reduction for the residual
• nparams_jac: number of snapshots considered when running hyper-reduction for the jacobian
• nparams_test: number of snapshots considered when computing the error the RB method commits with respect to the FE procedure

abstract type RBSpace <: FESpace end

Represents a vector subspace of a FESpace.

Subtypes:

• SingleFieldRBSpace
• MultiFieldRBSpace

struct RBVector{T,A<:AbstractVector{T},B} <: AbstractVector{T}
  data::A
  fe_data::B
end

Vector obtained by applying a Projection on a high-dimensional FE vector fe_data, which is stored (but mostly unused) for conveniency

struct ROMPerformance
  error
  speedup
end

Allows to compute errors and computational speedups to compare the properties of the algorithm with the FE performance.

abstract type ReducedProjection{A<:AbstractArray} <: Projection end

Type representing a Galerkin projection of a Projection onto a reduced subspace represented by another Projection.

Subtypes:

• ReducedAlgebraicProjection

abstract type Reduction{A<:ReductionStyle,B<:NormStyle} end

Type indicating the reduction strategy to employ, and the information regarding the norm with respect to which the reduction should occur.

Subtypes:

• DirectReduction
• GreedyReduction
• SupremizerReduction
• HyperReduction
• HighDimReduction

abstract type ReductionStyle end

Type indicating the reduction strategy to employ.

Subtypes:

• SearchSVDRank
• FixedSVDRank
• LRApproxRank
• TTSVDRanks

struct SOPTHyperReduction{A} <: HyperReduction{A}
  reduction::Reduction{A,EuclideanNorm}
end

S-OPT struct employed in steady problems

struct SearchSVDRank <: ReductionStyle
  tol::Float64
end

Struct employed when the chosen reduction algorithm is a truncated POD at a tolerance tol. Check this reference for more details on the truncated POD algorithm

struct SingleFieldRBSpace{S<:SingleFieldFESpace} <: RBSpace{S}
  space::S
  subspace::Projection
end

Reduced basis subspace of a SingleFieldFESpace in Gridap

abstract type SparseCore{T,N} <: AbstractTTCore{T,N} end

Tensor train cores for sparse matrices.

Subtypes:

• SparseCoreCSC
• SparseCoreCSC4D

struct SparseCoreCSC{T,Ti} <: SparseCore{T,3}
  array::Array{T,3}
  sparsity::SparsityCSC{T,Ti}
end

Tensor train cores for sparse matrices in CSC format

struct SparseCoreCSC4D{T,Ti} <: SparseCore{T,4}
  core::SparseCoreCSC{T,Ti}
  sparse_indexes::Vector{CartesianIndex{2}}
end

Tensor train cores for sparse matrices in CSC format, reshaped as 4D arrays

struct SupremizerReduction{A,R<:Reduction{A,EnergyNorm}} <: Reduction{A,EnergyNorm}
  reduction::R
  supr_op::Function
  supr_tol::Float64
end

Wrapper for reduction methods reduction that require an additional step of stabilization, by means of a supremizer enrichment. Check this for more details in a steady setting, and this for more details in a transient setting. The fields supr_op and supr_tol (which is only needed only in transient applications) are respectively the supremizing operator and the tolerance involved in the enrichment. For a saddle point problem with a Jacobian of the form

[ A Bᵀ B 0 ]

this operator is given by the bilinear form representing the matrix Bᵀ.

struct TTSVDProjection <: Projection
  cores::AbstractVector{<:AbstractArray{T,3} where T}
  dof_map::AbstractDofMap
end

Projection stemming from a tensor train SVD ttsvd. For reindexing purposes a field dof_map is provided along with the tensor train cores cores

struct TTSVDRanks <: ReductionStyle
  style::Vector{<:ReductionStyle}
end

Struct employed when the chosen reduction algorithm is a TTSVD, with reduction algorithm at each step specified in the vector of reduction styles style. Check this reference for more details on the TTSVD algorithm

struct TTSVDReduction{B} <: DirectReduction{TTSVDRanks,B}
  red_style::TTSVDRanks
  norm_style::B
  nparams::Int
end

Reduction by means of a TTSVD. The field nparams indicates the number of parameters selected for the computation of the snapshots

struct TrivialSparseCore{T,A<:SparsityPattern} <: SparseCore{T,3}
  array::Array{T,3}
  sparsity::A
end

Trivial tensor train core for sparse matrices format

contraction(Φₗₖ::AbstractArray{T,3},Aₖ::AbstractArray{T,3}) -> AbstractArray{T,4}
contraction(Φₗₖ::AbstractArray{T,3},Aₖ::AbstractArray{T,3},Φᵣₖ::AbstractArray{T,3}) -> AbstractArray{T,6}

Contraction of tensor train cores, as a result of a (Petrov-)Galerkin projection. The contraction of Aₖ by Φₗₖ is the left-contraction of a TT core Aₖ by a (left, test) TT core Φₗₖ, whereas the contraction of Aₖ by Φᵣₖ is the right-contraction of a TT core Aₖ by a (right, trial) TT core Φᵣₖ. The dimension of the output of a contraction involving N factors is: 3N - N = 2N.

contraction(basis::AbstractArray,coefficient::AbstractArray) -> AbstractArray

Multiplies a reduced basis basis by a set of reduced coeffiecients coefficient. It acts as a generalized linear combination, since basis might have a dimension higher than 2.

cores2basis(cores::AbstractArray{T,3}...) -> AbstractMatrix

Returns a basis in a matrix format from a list of tensor train cores cores. When also supplying the indexing strategy dof_map, the result is reindexed accordingly

create_dir(dir::String) -> Nothing

Recursive creation of a directory dir; does not do anything if dir exists

empirical_interpolation(a::Projection) -> (AbstractVector,AbstractMatrix)

Computes the EIM of a. The outputs are:

• a vector of integers i, corresponding to a list of interpolation row indices
• a matrix Φi = view(Φ,i), where Φ = get_basis(a). This quantity represents the restricted basis on the set of interpolation rows i

enrich!(
  red::SupremizerReduction,
  a::BlockProjection,
  norm_matrix::MatrixOrTensor,
  supr_matrix::MatrixOrTensor) -> Nothing

In-place augmentation of the primal block of a BlockProjection a. This function has the purpose of stabilizing the reduced equations stemming from a saddle point problem

eval_performance(
  solver::RBSolver,
  feop::ParamOperator,
  fesnaps::AbstractSnapshots,
  rbsnaps::AbstractSnapshots,
  festats::CostTracker,
  rbstats::CostTracker
  ) -> ROMPerformance

Arguments:

• solver: solver for the reduced problem
• feop: FE operator representing the PDE
• fesnaps: online snapshots of the FE solution
• rbsnaps: reduced approximation of fesnaps
• festats: time and memory consumption needed to compute fesnaps
• rbstats: time and memory consumption needed to compute rbsnaps

Returns the performance of the reduced algorithm, in terms of the (relative) error between rbsnaps and fesnaps, and the computational speedup between rbstats and festats

galerkin_projection(Φₗ,A) -> Any
galerkin_projection(Φₗ,A,Φᵣ) -> Any

Galerkin projection of A on the subspaces specified by a (left, test) subspace Φₗ (row projection) and a (right, trial) subspace Φᵣ (column projection)

galerkin_projection(a::Projection,b::Projection) -> ReducedProjection
galerkin_projection(a::Projection,b::Projection,c::Projection,args...) -> ReducedProjection

(Petrov) Galerkin projection of a projection map b onto the subspace a (row projection) and, if applicable, onto the subspace c (column projection)

get_basis(a::Projection) -> AbstractMatrix

Returns the basis spanning the reduced subspace represented by the projection a

get_fe_solver(s::RBSolver) -> NonlinearSolver

Returns the underlying NonlinearSolver from a RBSolver s

get_interpolation(a::HRProjection) -> Interpolation

For a HRProjection a represented by the couple (Φrb,i), returns i

get_reduced_subspace(r::RBSpace) -> Projection

Returns the Projection spanning the reduced subspace contained in r

get_rowcols_to_cells(
  cell_row_ids::AbstractArray{<:AbstractArray},
  cell_col_ids::AbstractArray{<:AbstractArray},
  rows::AbstractVector,cols::AbstractVector) -> AbstractVector

Returns the list of cells containing the row ids rows and the col ids cols

get_rows_to_cells(
  cell_row_ids::AbstractArray{<:AbstractArray},
  rows::AbstractVector
  ) -> AbstractVector

Returns the list of cells containing the row ids rows

gram_schmidt(A::AbstractMatrix,args...) -> AbstractMatrix
gram_schmidt(A::AbstractMatrix,X::AbstractSparseMatrix,args...) -> AbstractMatrix

Gram-Schmidt orthogonalization for a matrix A under a Euclidean norm. A (positive definite) sparse matrix X representing an inner product on the row space of A can be provided to make the result orthogonal under a different norm

inv_project!(x::AbstractArray,a::Projection,x̂::AbstractArray) -> Nothing

In-place recasting of a low-dimensional object x̂ the high-dimensional space in which a is immersed

inv_project(a::Projection,x::AbstractArray) -> AbstractArray

Recasts a low-dimensional object x onto the high-dimensional space in which a is immersed

jacobian_snapshots(solver::RBSolver,op::ParamOperator,s::AbstractSnapshots) -> Contribution
jacobian_snapshots(solver::RBSolver,op::ODEParamOperator,s::AbstractSnapshots) -> Tuple{Vararg{Contribution}}

Returns a Jacobian Contribution relative to the FE operator op. The quantity s denotes the solution snapshots in which we evaluate the jacobian. In transient settings, the output is a tuple whose nth element is the Jacobian relative to the nth temporal derivative

load_operator(dir,feop::ParamOperator;kwargs...) -> RBOperator

Given a FE operator feop, load its reduced counterpart stored in the directory dir. Throws an error if the reduced operator has not been previously saved to file

load_snapshots(dir;label="") -> AbstractSnapshots

Load the snapshots at the directory dir. Throws an error if the snapshots have not been previously saved to file

num_fe_dofs(a::Projection) -> Int

For a projection map a from a low dimensional space n to a high dimensional one N, returns N

num_reduced_dofs(a::Projection) -> Int

For a projection map a from a low dimensional space n to a high dimensional one N, returns n

orth_complement!(v::AbstractVector,basis::AbstractMatrix,args...) -> Nothing

In-place orthogonal complement of v on the column space of basis. When a symmetric, positive definite matrix X is provided as an argument, the output is X-orthogonal, otherwise it is ℓ²-orthogonal

orth_projection(v::AbstractVector, basis::AbstractMatrix, args...) -> AbstractVector

Orthogonal projection of v on the column space of basis. When a symmetric, positive definite matrix X is provided as an argument, the output is X-orthogonal, otherwise it is ℓ²-orthogonal

plot_a_solution(
  dir::String,
  feop::ParamOperator,
  sol::AbstractSnapshots,
  sol_approx::AbstractSnapshots,
  args...;
  kwargs...
  ) -> Nothing

Plots a single FE solution, RB solution, and the point-wise error between the two, by selecting the first FE snapshot in sol and the first reduced snapshot in sol_approx

project!(x̂::AbstractArray,a::Projection,x::AbstractArray,args...) -> Nothing

In-place projection of a high-dimensional object x onto the subspace represented by a

project(a::Projection,x::AbstractArray,args...) -> AbstractArray

Projects a high-dimensional object x onto the subspace represented by a

projection_eltype(a::Projection) -> DataType

Returns the eltype of the projection a

reduced_basis(red::Reduction,s::AbstractSnapshots,args...) -> Projection

Computes the basis by compressing the snapshots s

reduced_jacobian(
  solver::RBSolver,
  op::ParamOperator,
  red_trial::RBSpace,
  red_test::RBSpace,
  s::AbstractSnapshots
  ) -> Union{AffineContribution,TupOfAffineContribution}

Reduces the Jacobian contained in op via hyper-reduction. This function first builds the Jacobian snapshots, which are then reduced according to the strategy reduced_jacobian specified in the reduced solver solver. In transient applications, the output is a tuple of length equal to the number of Jacobians(i.e., equal to the order of the ODE plus one)

reduced_operator(solver::RBSolver,feop::ParamOperator,args...;kwargs...) -> RBOperator
reduced_operator(solver::RBSolver,feop::TransientParamOperator,args...;kwargs...) -> TransientRBOperator

Computes a RB operator from the FE operator feop

reduced_residual(
  solver::RBSolver,
  op::ParamOperator,
  red_test::RBSpace,
  s::AbstractSnapshots
  ) -> AffineContribution

Reduces the residual contained in op via hyper-reduction. This function first builds the residual snapshots, which are then reduced according to the strategy residual_reduction specified in the reduced solver solver

reduced_spaces(solver::RBSolver,feop::ParamOperator,s::AbstractSnapshots
  ) -> (RBSpace, RBSpace)

Computes the subspace of the test, trial FESpaces contained in the FE operator feop by compressing the snapshots s

reduced_subspace(space::FESpace,basis::Projection) -> RBSpace

Generic constructor of a RBSpace from a FESpace space and a projection basis

reduced_triangulation(trian::Triangulation,a::HRProjection)

Returns the triangulation view of trian on the integration cells contained in a

reduced_weak_form(
  solver::RBSolver,
  op::ParamOperator,
  red_trial::RBSpace,
  red_test::RBSpace,
  s::AbstractSnapshots
  ) -> (AffineContribution,Union{AffineContribution,TupOfAffineContribution})

Reduces the residual/Jacobian contained in op via hyper-reduction. Check the functions reduced_residual and reduced_jacobian for more details

reduction(red::Reduction,A::AbstractArray,args...) -> AbstractArray
reduction(red::Reduction,A::AbstractArray,X::AbstractSparseMatrix) -> AbstractArray

Given an array (of snapshots) A, returns a reduced basis obtained by means of the reduction strategy red

residual_snapshots(solver::RBSolver,op::ParamOperator,s::AbstractSnapshots) -> Contribution
residual_snapshots(solver::RBSolver,op::ODEParamOperator,s::AbstractSnapshots) -> Contribution

Returns a residual Contribution relative to the FE operator op. The quantity s denotes the solution snapshots in which we evaluate the residual

s_opt(a::Projection) -> (AbstractVector,AbstractMatrix)

Computes the S-OPT hyper-reduction of a. Check this reference for more information

sequential_product(a::AbstractArray,b::AbstractArray...) -> AbstractArray

This function sequentially multiplies the results of several (sequential as well) calls to contraction

solution_snapshots(solver::NonlinearSolver,feop::ParamOperator,r::Realization) -> SteadySnapshots
solution_snapshots(solver::ODESolver,feop::TransientParamOperator,r::TransientRealization,u0) -> TransientSnapshots

The problem encoded in the FE operator feop is solved several times, and the solution snapshots are returned along with the information related to the computational cost of the FE method. In transient settings, an initial condition u0 should be provided.

tpod(red_style::ReductionStyle,A::AbstractMatrix) -> AbstractMatrix
tpod(red_style::ReductionStyle,A::AbstractMatrix,X::AbstractSparseMatrix) -> AbstractMatrix

Truncated proper orthogonal decomposition of A. When provided, X is a (symmetric, positive definite) norm matrix with respect to which the output is made orthogonal. If X is not provided, the output is orthogonal with respect to the euclidean norm

ttsvd(red_style::TTSVDRanks,A::AbstractArray) -> AbstractVector{<:AbstractArray}
ttsvd(red_style::TTSVDRanks,A::AbstractArray,X::AbstractRankTensor) -> AbstractVector{<:AbstractArray}

Tensor train SVD of A. When provided, X is a norm tensor (representing a symmetric, positive definite matrix) with respect to which the output is made orthogonal. Note: if ndims(A) = N, the length of the ouptput is N-1, since we are not interested in reducing the axis of the parameters. Check this reference for more details

union_bases(a::Projection,b::Projection,args...) -> Projection

Computes the projection corresponding to the union of a and b. In essence this operation performs as

gram_schmidt(union(get_basis(a),get_basis(b)))