The Penalized Lebesgue Constant for Surface Spline Interpolation

Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity, and is far less well understood. In this article we consider the stability of surface spline interpolation (a popular type of RBF interpolation) for data with nonuniform arrangements. Using techniques similar to those recently employed by Hangelbroek, Narcowich and Ward to demonstrate the stability of interpolation from quasi-uniform data on manifolds, we show that surface spline interpolation on R^d is stable, but in a stronger, local sense. We also obtain pointwise estimates showing that the Lagrange function decays very rapidly, and at a rate determined by the local spacing of datasites. These results, in conjunction with a Lebesgue lemma, show that surface spline interpolation enjoys the same rates of convergence as those of the local approximation schemes recently developed by DeVore and Ron.Comment: 20 pages; corrected typos; to appear in Proc. Amer. Math. So

Nonlinear Approximation Using Gaussian Kernels

It is well-known that non-linear approximation has an advantage over linear schemes in the sense that it provides comparable approximation rates to those of the linear schemes, but to a larger class of approximands. This was established for spline approximations and for wavelet approximations, and more recently by DeVore and Ron for homogeneous radial basis function (surface spline) approximations. However, no such results are known for the Gaussian function, the preferred kernel in machine learning and several engineering problems. We introduce and analyze in this paper a new algorithm for approximating functions using translates of Gaussian functions with varying tension parameters. At heart it employs the strategy for nonlinear approximation of DeVore and Ron, but it selects kernels by a method that is not straightforward. The crux of the difficulty lies in the necessity to vary the tension parameter in the Gaussian function spatially according to local information about the approximand: error analysis of Gaussian approximation schemes with varying tension are, by and large, an elusive target for approximators. We show that our algorithm is suitably optimal in the sense that it provides approximation rates similar to other established nonlinear methodologies like spline and wavelet approximations. As expected and desired, the approximation rates can be as high as needed and are essentially saturated only by the smoothness of the approximand.Comment: 15 Pages; corrected typos; to appear in J. Funct. Ana

Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds

This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given by the authors for restricted surface splines on $\mathbb{R}^d$. The kernels for which the theory applies includes the Sobolev-Mat\'ern kernels for closed, compact, connected, $C^\infty$ Riemannian manifolds.Comment: 29 pages. To appear in Festschrift for the 80th Birthday of Ian Sloa

Kernel Approximation on Manifolds I: Bounding the Lebesgue Constant

The purpose of this paper is to establish that for any compact, connected C^{\infty} Riemannian manifold there exists a robust family of kernels of increasing smoothness that are well suited for interpolation. They generate Lagrange functions that are uniformly bounded and decay away from their center at an exponential rate. An immediate corollary is that the corresponding Lebesgue constant will be uniformly bounded with a constant whose only dependence on the set of data sites is reflected in the mesh ratio, which measures the uniformity of the data. The analysis needed for these results was inspired by some fundamental work of Matveev where the Sobolev decay of Lagrange functions associated with certain kernels on \Omega \subset R^d was obtained. With a bit more work, one establishes the following: Lebesgue constants associated with surface splines and Sobolev splines are uniformly bounded on R^d provided the data sites \Xi are quasi-uniformly distributed. The non-Euclidean case is more involved as the geometry of the underlying surface comes into play. In addition to establishing bounded Lebesgue constants in this setting, a "zeros lemma" for compact Riemannian manifolds is established.Comment: 33 pages, 2 figures, new title, accepted for publication in SIAM J. on Math. Ana

Molecular assessment of muscle health and function : The effect of age, nutrition and physical activity on the human muscle transcriptome and metabolom

Prolonged lifespan and decreased fertility will lead to an increased proportion of older adults in the world population (population aging). An important strategy to deal with population aging has been to promote healthy aging; not only to prevent mounting health care costs, but also to maintain independence and quality of life of older populations for as long as possible. Close to the opposite of the healthy aging is frailty. A major component of (physical) frailty is sarcopenia: age-related loss of muscle mass. Decreased muscle size and strength has been associated with a wide variety of negative health outcomes, including increased risk of hospitalization, physical disability and even death. Therefore, maintaining muscle size and strength is very important for healthy aging. Nutrition and physical activity are possible strategies to maintain or even improve muscle function with age. The effect of nutrition, age, frailty and physical activity on the function of skeletal muscle is complex. A better understanding of the molecular mechanisms involved can provide new insights in potential strategies to maintain muscle function over the life course. This thesis aims to investigate these mechanisms and processes that underlie the effects of age, frailty and physical activity by leveraging the sensitivity and comprehensiveness of transcriptomics and metabolomics. Chapter 2 and 3 describe the effects of age, frailty and resistance-type exercise training on the skeletal muscle transcriptome and metabolome. Both the transcriptome and metabolome show significant differences between frail and healthy older adults. These differences are similar to the differneces between healthy young men and healthy older adults, suggesting that frailty presents itself as a more pronounced form of aging, somewhat independent of chronological age. These age and frailty related differences in the transcriptome are partially reversed by resistance-type exercise training, in accordance with the observed improvement in muscle strength. Regression analysis revealed that the protocadherin gamma gene cluster may be important to skeletal muscle function. Protocadherin gamma is involved in axon guidance and may be upregulated due to the denervation-reinnervation cycles observed in skeletal muscle of older individuals. The metabolome suggested that resistance-type exercise training led to a decrease in branched-chain amino acid oxidation, as shown by a decrease in amino acid derived carnitines. Lastly, the blood metabolome showed little agreement with the metabolome in skeletal muscle, indicating that blood is a poor read-out of muscle metabolism. We assessed the effect of knee immobilization with creatine supplementation or placebo on the skeletal muscle transcriptome and metabolome in chapter 4. Knee immobilization caused muscle mass loss and strength loss in all participants, with no differences between creatine and placebo groups. Knee immobilization appeared to induce the HDAC4-myogenin axis, which is primarily associated with denervation and motor neuron diseases. The metabolome showed changes consistent with the decreased expression of energy metabolism genes. While acyl-carnitine levels tended to decrease with knee immobilization, one branched-chain amino acid-derived acyl carnitine was increased after knee immobilization, suggesting increased amino acid oxidation. Vitamin D deficiency is common among older adults and has been linked to muscle weakness. Vitamin D supplementation has been proposed as a strategy to improve muscle function among older populations. In chapter 5, supplementation with vitamin D (calcifediol, 25(OH)D) is investigated as nutritional strategy to improve muscle function among frail older adults. However, we observed no effect of vitamin D on the muscle transcriptome. These findings indicate the effects of vitamin D supplementation on skeletal muscle may be either absent, weak, or limited to a small subset of muscle cells. Transcriptomic changes due to different forms of muscle disuse are compared in chapter 6 (primarily knee immobilization and bed rest). The goal was to determine the similarities and differences among various causes of muscle atrophy in humans (primarily muscle disuse). Both knee immobilization and bed rest led to significant changes in the muscle transcriptome. However, the overlap in significantly changed genes was relatively small. Knee immobilization was characterized by ubiquitin-mediated proteolysis and induction of the HDAC4/Myogenin axis, whereas bed rest revealed increased expression of genes of the immune system and increased expression of lysosomal genes. Knee immobilization showed the highest similarity with age and frailty-related transcriptomic changes. This finding suggests that knee immobilization may be the most suitable form of disuse atrophy to assess the effectiveness of strategies to prevent age-related muscle loss in humans. The transcriptome and metabolome are incredibly useful tools in describing the wide array of biological systems within skeletal muscle. These systems can be modulated using physical activity (or lack thereof) as well as nutrition. This thesis describes some of these processes and highlights several unexplored genes and metabolites that may be important for maintaining or even optimizing muscle function. In the future, it may be possible to optimize both exercise and nutrition for each individual using these techniques; or even better, cheaper and less invasive alternatives.</p