The idea of 1 -minimization is the basis of the widely adopted compressive sensing method for function approximation. In this paper, we extend its application to high-dimensional stochastic collocation methods. To facilitate practical implementation, we employ orthogonal polynomials, particularly Legendre polynomials, as basis functions, and focus on the cases where the dimensionality is high such that one can not afford to construct high-degree polynomial approximations. We provide theoretical analysis on the validity of the approach. The analysis also suggests that using the Chebyshev measure to precondition the 1 -minimization, which has been shown to be numerically advantageous in one dimension in the literature, may in fact become less efficient in high dimensions. Numerical tests are provided to examine the performance of the methods and validate the theoretical findings.

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The regeneration of large bone defects is an osteoinductive, osteoconductive, and osteogenic process that often requires a bone graft for support. Limitations associated with naturally autogenic or allogenic bone grafts have demonstrated the need for synthetic substitutes. The present study investigates the feasibility of using novel hollow hydroxyapatite microspheres as an osteoconductive matrix and a carrier for controlled local delivery of bone morphogenetic protein 2 (BMP2), a potent osteogenic inducer of bone regeneration. Hollow hydroxyapatite microspheres (100±25 μm) with a core (60±18 μm) and a mesoporous shell (180±42 m 2 /g surface area) were prepared by a glass conversion technique and loaded with recombinant human BMP2 (1 μg/mg). There was a gentle burst release of BMP2 from microspheres into the surrounding phosphate-buffered saline in vitro within the initial 48 hours, and continued at a low rate for over 40 days. In comparison with hollow hydroxyapatite microspheres without BMP2 or soluble BMP2 without a carrier, BMP2-loaded hollow hydroxyapatite microspheres had a significantly enhanced capacity to reconstitute radial bone defects in rabbit, as shown by increased serum alkaline phosphatase; quick and complete new bone formation within 12 weeks; and great biomechanical flexural strength. These results indicate that BMP2-loaded hollow hydroxyapatite microspheres could be a potential new option for bone graft substitutes in bone regeneration.