title: Construction of lattice rules for multiple integration based on a weighted discrepancy

authors: Sinescu, Vasile

subjects: Lattice rules

abstract: High-dimensional integrals arise in a variety of areas, including quantum physics, the physics and chemistry of molecules, statistical mechanics and more recently, in financial applications.
In order to approximate multidimensional integrals, one may use Monte Carlo methods in which the quadrature points are generated randomly or quasi-Monte Carlo methods, in which points are generated deterministically. One particular class of quasi-Monte Carlo methods for multivariate integration is represented by lattice rules. Lattice rules constructed throughout this thesis allow good approximations to integrals of functions belonging to certain
weighted function spaces. These function spaces were proposed as an explanation as to why integrals in many variables appear to be successfully approximated although the standard theory indicates that the number of quadrature
points required for reasonable accuracy would be astronomical because of the large number of variables.
The purpose of this thesis is to contribute to theoretical results regarding the construction of lattice rules for multiple integration. We consider both lattice rules for integrals over the unit cube and lattice rules suitable for integrals over Euclidean space. The research reported throughout the thesis is devoted to finding the generating vector required to produce lattice rules that have what is termed a  low weighted discrepancy . In simple terms, the  discrepancy  is a measure of the uniformity of the distribution of the quadrature points or in other settings, a worst-case error. One of the assumptions used in these weighted function spaces is that variables are arranged in the decreasing order of their importance and the assignment of weights in this situation results in so-called  product weights . In other applications it is rather the importance of group of variables that matters. This situation is modelled by using function spaces in which the weights are  general . In the weighted
settings mentioned above, the quality of the lattice rules is assessed by the  weighted discrepancy  mentioned earlier. Under appropriate conditions on the weights, the lattice rules constructed here produce a convergence rate of the error that ranges 
from O(n−1/2) to the (believed) optimal O(n−1+δ) for any δ gt 0, with the involved constant independent of the dimension.

text: Construction of lattice rules for multiple integration based on a weighted discrepancy High-dimensional integrals arise in a variety of areas, including quantum physics, the physics and chemistry of molecules, statistical mechanics and more recently, in financial applications.
In order to approximate multidimensional integrals, one may use Monte Carlo methods in which the quadrature points are generated randomly or quasi-Monte Carlo methods, in which points are generated deterministically. One particular class of quasi-Monte Carlo methods for multivariate integration is represented by lattice rules. Lattice rules constructed throughout this thesis allow good approximations to integrals of functions belonging to certain
weighted function spaces. These function spaces were proposed as an explanation as to why integrals in many variables appear to be successfully approximated although the standard theory indicates that the number of quadrature
points required for reasonable accuracy would be astronomical because of the large number of variables.
The purpose of this thesis is to contribute to theoretical results regarding the construction of lattice rules for multiple integration. We consider both lattice rules for integrals over the unit cube and lattice rules suitable for integrals over Euclidean space. The research reported throughout the thesis is devoted to finding the generating vector required to produce lattice rules that have what is termed a  low weighted discrepancy . In simple terms, the  discrepancy  is a measure of the uniformity of the distribution of the quadrature points or in other settings, a worst-case error. One of the assumptions used in these weighted function spaces is that variables are arranged in the decreasing order of their importance and the assignment of weights in this situation results in so-called  product weights . In other applications it is rather the importance of group of variables that matters. This situation is modelled by using function spaces in which the weights are  general . In the weighted
settings mentioned above, the quality of the lattice rules is assessed by the  weighted discrepancy  mentioned earlier. Under appropriate conditions on the weights, the lattice rules constructed here produce a convergence rate of the error that ranges 
from O(n−1/2) to the (believed) optimal O(n−1+δ) for any δ gt 0, with the involved constant independent of the dimension.

year: 2008

title: International tephra conference “Crossing New Frontiers: Tephra Hunt in Transylvania”, 24 June - 1 July, 2018, Moieciu de Sus, Romania.

authors: Lowe, David J.

abstract: The International Focus Group on Tephrochronology and Volcanism (INTAV) held an international tephra conference, “Crossing New Frontiers: Tephra Hunt in Transylvania”, at the spectacular mountain resort ‘Cheile Gradistei’ Fundata near Moieciu de Sus (Fig. 1), about 30 km southwest of the regional city of Braşov, in the southern Carpathian Mountains of Transylvania, Romania, from 24 June to 1 July, 2018. INTAV (which has operated previously under various names including COT, SCOTAV, and ICCT) is a global tephra research group active within the Stratigraphy and Chronology Commission (SACCOM) of INQUA since its initiation in 1961 (Kobayshi, 1965; Westgate and Gold, 1974; Lowe et al., 2011). It organises specialist tephra meetings every four years or so on average, although the most recent meeting (before this one in Romania) was held in 2010 in Kirishima, Japan (Holt and Lowe, 2010). The Romanian tephra meeting was convened by Daniel Veres (Romania) and Ulrich Hambach (Germany), together with support from the INTAV executive committee of Britta Jensen (Canada), Peter Abbott (UK/Switzerland), Takehiko Suzuki (Japan), Siwan Davies (UK), and David Lowe (New Zealand).

text: International tephra conference “Crossing New Frontiers: Tephra Hunt in Transylvania”, 24 June - 1 July, 2018, Moieciu de Sus, Romania. The International Focus Group on Tephrochronology and Volcanism (INTAV) held an international tephra conference, “Crossing New Frontiers: Tephra Hunt in Transylvania”, at the spectacular mountain resort ‘Cheile Gradistei’ Fundata near Moieciu de Sus (Fig. 1), about 30 km southwest of the regional city of Braşov, in the southern Carpathian Mountains of Transylvania, Romania, from 24 June to 1 July, 2018. INTAV (which has operated previously under various names including COT, SCOTAV, and ICCT) is a global tephra research group active within the Stratigraphy and Chronology Commission (SACCOM) of INQUA since its initiation in 1961 (Kobayshi, 1965; Westgate and Gold, 1974; Lowe et al., 2011). It organises specialist tephra meetings every four years or so on average, although the most recent meeting (before this one in Romania) was held in 2010 in Kirishima, Japan (Holt and Lowe, 2010). The Romanian tephra meeting was convened by Daniel Veres (Romania) and Ulrich Hambach (Germany), together with support from the INTAV executive committee of Britta Jensen (Canada), Peter Abbott (UK/Switzerland), Takehiko Suzuki (Japan), Siwan Davies (UK), and David Lowe (New Zealand).

year: 2018

title: Biological flora of New Zealand 11. Eleochairs spacelata, kuta, paopao, bamboo spike sedge.

authors: Kapa, Mieke M

subjects: cultural

abstract: Information relevant to the ecology and morphology of Eleocharis sphacelata (Cyperaceae) available through published and unpublished sources is assembled and reviewed. Eleocharis sphacelata is a freshwater wetland sedge found growing at the edge of lake margins, swamps, lagoons, and bogs within New Zealand, Australia, and Papua New Guinea. It is a perennial rhizomatous herb, with tall green cylindrical culms rising from submerged rhizomes generally buried under water in aerobic soils. A tendency to dominate the deepwater margins makes E. sphacelata crucial to the ecology of this habitat as a source of food, shelter, and protection for invertebrates, birds, and fish. Eleocharis sphacelata is commonly named giant spike-rush, kuta, or paopao. Māori have traditionally used the plant for clothing and warm coverings within whare (houses). Recent interest comes from its use in restoration plantings and constructed wetlands.

text: Biological flora of New Zealand 11. Eleochairs spacelata, kuta, paopao, bamboo spike sedge. Information relevant to the ecology and morphology of Eleocharis sphacelata (Cyperaceae) available through published and unpublished sources is assembled and reviewed. Eleocharis sphacelata is a freshwater wetland sedge found growing at the edge of lake margins, swamps, lagoons, and bogs within New Zealand, Australia, and Papua New Guinea. It is a perennial rhizomatous herb, with tall green cylindrical culms rising from submerged rhizomes generally buried under water in aerobic soils. A tendency to dominate the deepwater margins makes E. sphacelata crucial to the ecology of this habitat as a source of food, shelter, and protection for invertebrates, birds, and fish. Eleocharis sphacelata is commonly named giant spike-rush, kuta, or paopao. Māori have traditionally used the plant for clothing and warm coverings within whare (houses). Recent interest comes from its use in restoration plantings and constructed wetlands.

year: 2009