Many problems arising in computational science and engineering can be described in terms of approximating a smooth function of d variables, defined over an unknown domain of interest Ω ⊂ R d , from sample data. Here both the underlying dimensionality of the problem (in the case d 1) as well as the lack of domain knowledge-with Ω potentially irregular and/or disconnected-are confounding factors for sampling-based methods. Naïve approaches for such problems often lead to wasted samples and inefficient approximation schemes. For example, uniform sampling can result in upwards of 20% wasted samples in some problems considered herein. In applications such as surrogate model construction in computational uncertainty quantification (UQ), the high cost of computing samples necessitates a more efficient sampling procedure. Over the last several years methods for computing such approximations from sample data have been studied in the case of irregular domains, and the advantages of computing sampling measures depending on an approximation space P of dim(P ) = N have been shown. More specifically, such approaches confer advantages such as stability and well-conditioning, with an asymptotically optimal sample complexity scaling O(N log(N )). The recently-proposed adaptive sampling for general domains (ASGD) strategy is one such technique to construct these sampling measures. The main contribution of this paper is a procedure to improve upon the ASGD approach by adaptively updating the sampling measure in the case of unknown domains. We achieve this by first introducing a general domain adaptivity strategy (GDAS), which computes an approximation of the function and domain of interest from the sample points. Second, we propose an adaptive sampling strategy, termed adaptive sampling for unknown domains (ASUD), which generates sampling measures over a domain that may not be known in advance, based on the ideas introduced in the ASGD approach. We then derive (weighted) least squares and augmented least squares techniques for polynomial approximation on unknown domains. We present numerical experiments demonstrating the efficacy of the adaptive sampling techniques with least squares-based polynomial approximation schemes. Our results show that the ASUD approach can reduce the computational cost by as much as 50% when compared with uniform sampling for such problems.

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New oral fXa inhibitors have been increasingly adopted for VTE or AF treatment in the outpatient setting instead of warfarin. Andexanet alfa (AnXa) is a modified, recombinant human fXa molecule developed as a specific antidote to reverse anticoagulant activity of fXa inhibitors during episodes of serious bleeding or before urgent surgery. PER977, a small molecule under development by Perosphere, Inc., is reported to reverse the effect of a broad range of anticoagulants (fXa and thrombin inhibitors, LMWH). In order to compare AnXa and PER977 mechanisms of action, we studied the in vitro activity of both agents in the presence or absence of fXa inhibitors rivaroxaban, apixaban, edoxaban or enoxaparin. In a buffer system containing purified human fXa with physiologic Ca 2+ (5 mM), AnXa dose-dependently reversed oral fXa inhibitor activity. Reversal activity was not observed for any fXa inhibitor with PER977 over a wide concentration range (≤2 mM). In human plasma, AnXa reversed both direct and ATIII-dependent fXa inhibitors, whereas no reversal effect by PER977 was detected. In the absence of a fXa inhibitor, PER977 potentiated fX activation by fIXa in a buffer system (purified human fX, fIXa and 5mM Ca 2+ ). Similar cofactor activity was observed with polylysine, indicating that PER977 may have properties similar to poly-cationic molecules. Procoagulant activity was observed in human plasma with PER977 at low concentrations (≤100 μM) as measured by clotting (aPTT) or thrombin generation, but inhibition in these assays was seen at higher concentrations ( ~1 mM). PER977 also potentiated human platelet activation as determined by P-selectin expression induced by 10 μM ADP. Platelet aggregation, whole blood hemolysis, or complement activation testing showed no effect with either AnXa or PER977. These data indicate that PER977 does not reverse the anticoagulant activity of a direct or indirect fXa inhibitor in vitro. Its observed in vivo effect on blood loss in animals may not be mediated by direct interaction of PER977 with the inhibitor, but may be in part attributed to an off-target effect, suggested by its potential procoagulant activity. In contrast, AnXa acts as a specific antidote to fXa inhibitors by sequestering them with a defined stoichiometry (1:1 molar ratio).

In this paper, we address the problem of approximating a multivariate function defined on a general domain in d dimensions from sample points. We consider weighted least-squares approximation in an arbitrary finite-dimensional space P from independent random samples taken according to a suitable measure. In general, least-squares approximations can be inaccurate and ill conditioned when the number of sample points M is close to N = dim(P ). To counteract this, we introduce a novel method for sampling in general domains which leads to provably accurate and well-conditioned weighted least-squares approximations. The resulting sampling measure is discrete, and therefore straightforward to sample from. Our main result shows near optimal sample complexity for this procedure; specifically, M = O(N log(N )) samples suffice for a well conditioned and accurate approximation. Numerical experiments on polynomial approximation in general domains confirm the benefits of this method over standard sampling.