Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak's algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments

Temporal and spatial earthquake clustering revealed through comparison of millennial strain-rates from ³⁶Cl cosmogenic exposure dating and decadal GPS strain-rate

To assess whether continental extension and seismic hazard are spatially-localized on single faults or spread over wide regions containing multiple active faults, we investigated temporal and spatial slip-rate variability over many millennia using in-situ 36Cl cosmogenic exposure dating for active normal faults near Athens, Greece. We study a ~ NNE-SSW transect, sub-parallel to the extensional strain direction, constrained by two permanent GPS stations located at each end of the transect and arranged normal to the fault strikes. We sampled 3 of the 7 seven normal faults that exist between the GPS sites for 36Cl analyses. Results from Bayesian inference of the measured 36Cl data implies that some faults slip relatively-rapidly for a few millennia accompanied by relative quiescence on faults across strike, defining out-of-phase fault activity. Assuming that the decadal strain-rate derived from GPS applies over many millennia, slip on a single fault can accommodate ~ 30–75% of the regional strain-rate for a few millennia. Our results imply that only a fraction of the total number of Holocene active faults slip over timescales of a few millennia, so continental deformation and seismic hazard are localized on specific faults and over a length-scale shorter than the spacing of the present GPS network over this time-scale. Thus, (1) the identification of clustered fault activity is vital for probabilistic seismic hazard assessments, and (2) a combination of dense geodetic observations and palaeoseismology is needed to identify the precise location and width of actively deforming zones over specific time periods

Temporal and spatial earthquake clustering revealed through comparison of millennial strain-rates from 36Cl cosmogenic exposure dating and decadal GPS strain-rate

To assess whether continental extension and seismic hazard are spatially-localized on single faults or spread over wide regions containing multiple active faults, we investigated temporal and spatial slip-rate variability over many millennia using in-situ 36Cl cosmogenic exposure dating for active normal faults near Athens, Greece. We study a ~ NNE-SSW transect, sub-parallel to the extensional strain direction, constrained by two permanent GPS stations located at each end of the transect and arranged normal to the fault strikes. We sampled 3 of the 7 seven normal faults that exist between the GPS sites for 36Cl analyses. Results from Bayesian inference of the measured 36Cl data implies that some faults slip relatively-rapidly for a few millennia accompanied by relative quiescence on faults across strike, defining out-of-phase fault activity. Assuming that the decadal strain-rate derived from GPS applies over many millennia, slip on a single fault can accommodate ~ 30–75% of the regional strain-rate for a few millennia. Our results imply that only a fraction of the total number of Holocene active faults slip over timescales of a few millennia, so continental deformation and seismic hazard are localized on specific faults and over a length-scale shorter than the spacing of the present GPS network over this time-scale. Thus, (1) the identification of clustered fault activity is vital for probabilistic seismic hazard assessments, and (2) a combination of dense geodetic observations and palaeoseismology is needed to identify the precise location and width of actively deforming zones over specific time periods