Chapter 9
High-Dimension Model Representation
via Sparse Grid Techniques
9.1 Introduction
The question of high-dimension model representation (HDMR) is of increasing
importance in computational mathematics and science. We introduced this subject
in Chaps. 7 and 8, where we invoked the Karhunen-Loève expansion to reduce the
number of random parameters that are required to define the stochastic model. We
continue the discussion of HDMR in this chapter by turning our attention to the
question of determining a suitable surrogate model for computing the response of
the forward problem via VIC-3D ® . This surrogate takes the form of an interpolation
table, which is then transformed into the conventional table used in NLSE for
solving inverse problems. The surrogate model that we seek falls under the rubric
sparse grids, and has been the subject of intensive research in a number of areas in
recent years [15, 22, 43, 44, 46, 57, 58, 73, 121, 123, 146]. We will apply it to solving
problems of model-based inversion as was developed in [111]. Sparse grids can also
be used to effectively calculate high-dimensional integrals of the form (7.2).
9.2 Mathematical Structure of the Problem
The problems in this set are based on Fig. 6.3, and required 81 VIC-3D ® runs to
establish the interpolating grid. Thus, we say that the grid has 81 nodes in fourdimensional space, speaking abstractly. Each variable (the slab depth in Fig. 6.3)
defines a dimension of the grid. The fifth problem introduced another variable,
the width, with three possible values, making the overall grid a hypercube of 243
nodes in five-dimensional space. As we add dimensions (variables), we will soon
encounter the ‘curse of dimensionality,’ because each node requires a VIC-3D ®
run to produce the corresponding blending function. The question arises as to
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_9
215
High-Dimension Model Representation
via Sparse Grid Techniques
9.1 Introduction
The question of high-dimension model representation (HDMR) is of increasing
importance in computational mathematics and science. We introduced this subject
in Chaps. 7 and 8, where we invoked the Karhunen-Loève expansion to reduce the
number of random parameters that are required to define the stochastic model. We
continue the discussion of HDMR in this chapter by turning our attention to the
question of determining a suitable surrogate model for computing the response of
the forward problem via VIC-3D ® . This surrogate takes the form of an interpolation
table, which is then transformed into the conventional table used in NLSE for
solving inverse problems. The surrogate model that we seek falls under the rubric
sparse grids, and has been the subject of intensive research in a number of areas in
recent years [15, 22, 43, 44, 46, 57, 58, 73, 121, 123, 146]. We will apply it to solving
problems of model-based inversion as was developed in [111]. Sparse grids can also
be used to effectively calculate high-dimensional integrals of the form (7.2).
9.2 Mathematical Structure of the Problem
The problems in this set are based on Fig. 6.3, and required 81 VIC-3D ® runs to
establish the interpolating grid. Thus, we say that the grid has 81 nodes in fourdimensional space, speaking abstractly. Each variable (the slab depth in Fig. 6.3)
defines a dimension of the grid. The fifth problem introduced another variable,
the width, with three possible values, making the overall grid a hypercube of 243
nodes in five-dimensional space. As we add dimensions (variables), we will soon
encounter the ‘curse of dimensionality,’ because each node requires a VIC-3D ®
run to produce the corresponding blending function. The question arises as to
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
H. A. Sabbagh et al., Advanced Electromagnetic Models for Materials
Characterization and Nondestructive Evaluation, Scientific Computation,
https://doi.org/10.1007/978-3-030-67956-9_9
215