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benmkent merged 2 commits into from
Sep 22, 2025
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minor fixes in manuscript and bibliography #37

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c4e57ca
fix minor issues in the manuscript
jbytecode Sep 22, 2025
5abd99a
minor fixes in bibtex
jbytecode Sep 22, 2025
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6 changes: 3 additions & 3 deletions paper/paper.bib
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Expand Up @@ -104,7 +104,7 @@ @article{Piazzola2021
}

@article{Li2024,
title = {Characterizing Aquifer Properties through a Sparse Grid-Based Bayesian Framework and InSAR Measurements: A Basin-Scale Application to {A}lto {G}uadalentín, {S}pain},
title = {Characterizing Aquifer Properties through a Sparse Grid-Based {B}ayesian Framework and InSAR Measurements: A Basin-Scale Application to {A}lto {G}uadalentín, {S}pain},
url = {http://dx.doi.org/10.22541/essoar.172373105.53381390/v1},
DOI = {10.22541/essoar.172373105.53381390/v1},
publisher = {Authorea, Inc.},
Expand Down Expand Up @@ -241,7 +241,7 @@ @url{Dakota
}

@article{spinterp,
title = {Algorithm 847: {spinterp}: piecewise multilinear hierarchical sparse grid interpolation in {MATLAB}},
title = {Algorithm 847: {Spinterp}: piecewise multilinear hierarchical sparse grid interpolation in {MATLAB}},
volume = {31},
ISSN = {1557-7295},
url = {http://dx.doi.org/10.1145/1114268.1114275},
Expand Down Expand Up @@ -356,7 +356,7 @@ @article{julia
}

@article{Bittens2023,
title = {DistributedSparseGrids.jl: A Julia library implementing
title = {DistributedSparseGrids.jl: A {J}ulia library implementing
an Adaptive Sparse Grid collocation method},
volume = {8},
ISSN = {2475-9066},
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20 changes: 10 additions & 10 deletions paper/paper.md
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Expand Up @@ -22,19 +22,19 @@ bibliography: paper.bib

# Summary
Approximating functions with high-dimensional domains is crucial in modern scientific and engineering problems.
An example of this is constructing surrogate models for quantities of interest in high-dimensional, parametrized PDE problems.
An example of this is constructing surrogate models for quantities of interest in high-dimensional, parameterized PDE problems.
These surrogate models are constructed to provide computationally inexpensive yet accurate approximations that can be used in applications such as uncertainty quantification, optimization, and parameter estimation [@UQHandbook2017].
For suitably smooth functions, accurate surrogates can be constructed using global polynomial approximation techniques over the parameter domain, and a common approach is the use of sparse grid polynomial approximation.
In particular, sparse grid polynomial interpolation techniques enable practitioners to approximate solutions to parametric problems in a non-intrusive manner using existing numerical solvers.

`SparseGridsKit.jl` provides a Julia toolbox to manually and adaptively construct sparse grid polynomial approximations [@julia].
`SparseGridsKit.jl` provides a Julia [@julia] toolbox to manually and adaptively construct sparse grid polynomial approximations.
Interpolation and quadrature routines allow evaluation and integration of the surrogate models.
Multi-fidelity approximation via the multi-index stochastic collocation algorithm is also possible [@HajiAli2016; @Jakeman2019; @Piazzola2022].
Approximations can be represented either in a basis of global Lagrange interpolation polynomials or in a basis of domain appropriate spectral-type global polynomials (e.g.\ Legendre, Chebyshev, Hermite...).
Approximations can be represented either in a basis of global Lagrange interpolation polynomials or in a basis of domain appropriate spectral-type global polynomials (e.g.\ Legendre, Chebyshev, Hermite, etc.).

# Statement of need
Sparse grid approximation is a well-developed methodology and is featured in many survey articles and textbook chapters, e.g., [@Bungartz2004; @LeMaitre2010; @Schwab2011; @Cohen2015; @Sullivan2015].
The need for sparse grid surrogate modelling is demonstrated by its use in many applications, from simpler elliptic and parabolic PDEs to complex practical engineering problems, e.g., \ [@Piazzola2021; @Piazzola2022; @Li2024].
Sparse grid approximation is a well-developed methodology and is featured in many survey articles and textbook chapters [@Bungartz2004; @LeMaitre2010; @Schwab2011; @Cohen2015; @Sullivan2015].
The need for sparse grid surrogate modelling is demonstrated by its use in many applications, from simpler elliptic and parabolic PDEs to complex practical engineering problems [@Piazzola2021; @Piazzola2022; @Li2024].
The `SparseGridsKit.jl` implementation offers a rich set of features to enable this.

Specifically, `SparseGridsKit.jl` is a Julia implementation of adaptive sparse grid global polynomial approximation methods.
Expand All @@ -46,7 +46,7 @@ The functionality includes:
- Multi-index set construction and manipulation.
- Combination technique sparse grid approximations, and functionality for interpolation, integration and derivatives of the surrogate model.
- Adaptive sparse grid approximation construction based on the ubiquitous Gerstner-Griebel dimensional adaptive algorithm [@Gerstner2003].
This implementation uses profit indicators as described in [@Nobile2016].
This implementation uses profit indicators as described in @Nobile2016.
- Adaptive multi-fidelity approximation via the Multi-Index Stochastic Collocation (MISC) algorithm [@HajiAli2016; @Jakeman2019; @Piazzola2022].
- Conversion to and from Polynomial Chaos / spectral polynomial series representation.
- Limited support for surrogate model differentiation via automatic differentiation.
Expand All @@ -55,9 +55,9 @@ The functionality described above is tested and documented with examples include

Other sparse grid approximation packages in Julia include:

- [`DistributedSparseGrids.jl`](https://github.com/baxmittens/DistributedSparseGrids.jl) [@Bittens2023]: a Julia package providing adaptive sparse grid approximation using a local hierarchical basis and distributed computing functionality.
- [`Tasmanian.jl`](https://github.com/floswald/Tasmanian.jl): a Julia interface to the C++ [Tasmanian library](https://github.com/ORNL/Tasmanian).
- [`AdaptiveSparseGrids.jl`](https://github.com/jacobadenbaum/AdaptiveSparseGrids.jl): a Julia package offering sparse grid approximation using a local hierarchical basis.
- [`DistributedSparseGrids.jl`](https://github.com/baxmittens/DistributedSparseGrids.jl) [@Bittens2023]: A Julia package providing adaptive sparse grid approximation using a local hierarchical basis and distributed computing functionality.
- [`Tasmanian.jl`](https://github.com/floswald/Tasmanian.jl): A Julia interface to the C++ [Tasmanian library](https://github.com/ORNL/Tasmanian).
- [`AdaptiveSparseGrids.jl`](https://github.com/jacobadenbaum/AdaptiveSparseGrids.jl): A Julia package offering sparse grid approximation using a local hierarchical basis.

As described above, `SparseGridsKit.jl` instead offers functionality based upon global polynomial approximation targeting problems in which the function is assumed to be suitably smooth.
The adaptive approximation algorithm is also split cleanly into `SOLVE-ESTIMATE-MARK-REFINE` steps to aid adaptive algorithm development.
Expand All @@ -71,7 +71,7 @@ Other popular software packages implementing sparse grid approximation include:
- `PyApprox`: A Python package for high-dimensional approximation [@PyApprox].
- `Dakota`: A C++ library for optimisation and surrogate modelling [@Dakota].
- `UQTk`: A collection of C++/Python uncertainty quantification tools including sparse grid quadrature [@DebusschereUQTk:2017].
- `Tasmanian`,`SG++`,: C++ sparse grid approximation implementations with wrappers for many popular software languages [@stoyanov2015tasmanian; @pflueger10spatially].
- `Tasmanian`,`SG++`: C++ sparse grid approximation implementations with wrappers for many popular software languages [@stoyanov2015tasmanian; @pflueger10spatially].

`SparseGridsKit.jl` specifically offers a Julia toolkit sharing the ethos of the `Sparse Grids MATLAB Kit`: to be user-friendly and aid fast algorithm prototyping.
Notably, `SparseGridsKit.jl` also provides an implementation of the multi-index stochastic collocation algorithm which is currently only available in `PyApprox`.
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