Fundamental Splines on Sparse Grids and Their Application to Gradient-Based Optimization

Valentin, Julian; Pflüger, Dirk

doi:10.1007/978-3-319-75426-0_10

Cited by 9 publications

(20 citation statements)

References 15 publications

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“…However, B-splines are much more flexible than global polynomials (Valentin and Pflüger 2016;Valentin 2019). While global polynomial approximations are bound to certain grid structures to avoid Runge's phenomenon or similar issues, B-spline basis functions can be employed on any nested spatially adaptive grid hierarchy.…”

Section: Introductionmentioning

confidence: 99%

“…Approximations with B-splines of cubic degree (or higher) are twice continuously differentiable, and readily supply smooth and explicit approximations of both, the value function and the gradient. Compared to approximating the derivatives with finite differences, the optimization is not only more accurate but also significantly faster, especially when the number of optimization variables is large (Valentin 2019). B-splines have thus proven useful for computing numerical solutions to numerous dynamic models when finding the root of the gradient is required (Chu et al 2013;Habermann and Kindermann 2007;Judd and Solnick 1994;Philbrick and Kitanidis 2001).…”

Section: Introductionmentioning

confidence: 99%

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Solving High-Dimensional Dynamic Portfolio Choice Models with Hierarchical B-Splines on Sparse Grids

2021

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Discrete time dynamic programming to solve dynamic portfolio choice models has three immanent issues: firstly, the curse of dimensionality prohibits more than a handful of continuous states. Secondly, in higher dimensions, even regular sparse grid discretizations need too many grid points for sufficiently accurate approximations of the value function. Thirdly, the models usually require continuous control variables, and hence gradient-based optimization with smooth approximations of the value function is necessary to obtain accurate solutions to the optimization problem. For the first time, we enable accurate and fast numerical solutions with gradient-based optimization while still allowing for spatial adaptivity using hierarchical B-splines on sparse grids. When compared to the standard linear bases on sparse grids or finite difference approximations of the gradient, our approach saves an order of magnitude in total computational complexity for a representative dynamic portfolio choice model with varying state space dimensionality, stochastic sample space, and choice variables.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Solving High-Dimensional Dynamic Portfolio Choice Models with Hierarchical B-Splines on Sparse Grids

2021

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“…This requirement is equivalent to removing these knots from the knot sequence ξ u l but keeping them in the set of interpolation nodes. Consequently we obtain the uniform not-a-knot sequence of level l and degree p [13], ξ p,nak l := (ξ p,nak l,0 , . .…”

Section: B-splinesmentioning

confidence: 99%

“…This guarantees a basis of the polynomial space for the first levels [13]. Finally, the not-a-knot B-spline basis b p,nak l,i of degree p, level l and index i is given by…”

Section: B-splinesmentioning

confidence: 99%

“…In the context of sparse grids, which we will use later on, the boundary points are usually omitted to reduce the overall effort. In order to keep reasonable approximations, the left-most and right-most B-splines are modified, so that they extrapolate towards the boundary [13]. This results in b p,mod l,i , the modified not-a-knot B-splines of degree p, level l and index i,…”

Section: B-splinesmentioning

confidence: 99%

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Active Subspaces With B-Spline Surrogates on Sparse Grids

Rehme¹,

Pflüger²

2019

Proceedings of the 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECO

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Many complex model functions allow the reduction of their effective dimension through active subspaces. These are computed by an eigenvalue decomposition of the average of the outer product of the function's gradient with itself. The size of the eigenvalues indicates how much the function changes on average along the direction given by the eigenvectors. This motivates to omit directions belonging to small eigenvalues and therefore to effectively reduce the model's dimension without losing much accuracy. The remaining directions form the active subspace, a linear combination of the input parameters. For real-world applications the required gradients are usually not explicitly known and they are thus commonly approximated with finite differences or ridge functions. The average of the outer product is then calculated using Monte Carlo quadrature. This converges slowly, resulting in long runtimes if the evaluation of the model function is expensive. The differentiation and integration of B-splines is numerically fast and analytically exact. Together with adaptive sparse grid discretization, they can be employed in higher-dimensional approximation. We use this to create a surrogate for the objective function, which provides us with better approximations for the gradient and thus better approximations for the active subspace. Furthermore we present a new integration technique for functions with a one-dimensional active subspace, that is based on a geometric interpretation of B-splines.

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Efficiently Transforming from Values of a Function on a Sparse Grid to Basis Coefficients

Wodraszka

Carrington

2021

Lecture Notes in Computational Science and Engineering

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Fundamental Splines on Sparse Grids and Their Application to Gradient-Based Optimization

Cited by 9 publications

References 15 publications

Solving High-Dimensional Dynamic Portfolio Choice Models with Hierarchical B-Splines on Sparse Grids

Solving High-Dimensional Dynamic Portfolio Choice Models with Hierarchical B-Splines on Sparse Grids

Active Subspaces With B-Spline Surrogates on Sparse Grids

Efficiently Transforming from Values of a Function on a Sparse Grid to Basis Coefficients

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