Computation of Electromagnetic Fields Scattered From Objects With Uncertain Shapes Using Multilevel Monte Carlo Method

Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization the number of MC samples has to be large. In this work, to address this challenge, the continuation multilevel Monte Carlo (CMLMC) method is used together with a surface integral equation solver. The CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational cost. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.Comment: 25 pages, 10 Figure

General Impossibility of Group Homomorphic Encryption in the Quantum World

Group homomorphic encryption represents one of the most important building blocks in modern cryptography. It forms the basis of widely-used, more sophisticated primitives, such as CCA2-secure encryption or secure multiparty computation. Unfortunately, recent advances in quantum computation show that many of the existing schemes completely break down once quantum computers reach maturity (mainly due to Shor's algorithm). This leads to the challenge of constructing quantum-resistant group homomorphic cryptosystems. In this work, we prove the general impossibility of (abelian) group homomorphic encryption in the presence of quantum adversaries, when assuming the IND-CPA security notion as the minimal security requirement. To this end, we prove a new result on the probability of sampling generating sets of finite (sub-)groups if sampling is done with respect to an arbitrary, unknown distribution. Finally, we provide a sufficient condition on homomorphic encryption schemes for our quantum attack to work and discuss its satisfiability in non-group homomorphic cases. The impact of our results on recent fully homomorphic encryption schemes poses itself as an open question.Comment: 20 pages, 2 figures, conferenc

Inductive machine learning of optimal modular structures: Estimating solutions using support vector machines

Structural optimization is usually handled by iterative methods requiring repeated samples of a physics-based model, but this process can be computationally demanding. Given a set of previously optimized structures of the same topology, this paper uses inductive learning to replace this optimization process entirely by deriving a function that directly maps any given load to an optimal geometry. A support vector machine is trained to determine the optimal geometry of individual modules of a space frame structure given a specified load condition. Structures produced by learning are compared against those found by a standard gradient descent optimization, both as individual modules and then as a composite structure. The primary motivation for this is speed, and results show the process is highly efficient for cases in which similar optimizations must be performed repeatedly. The function learned by the algorithm can approximate the result of optimization very closely after sufficient training, and has also been found effective at generalizing the underlying optima to produce structures that perform better than those found by standard iterative methods

Consistent Weighted Sampling Made Fast, Small, and Easy

Document sketching using Jaccard similarity has been a workable effective technique in reducing near-duplicates in Web page and image search results, and has also proven useful in file system synchronization, compression and learning applications. Min-wise sampling can be used to derive an unbiased estimator for Jaccard similarity and taking a few hundred independent consistent samples leads to compact sketches which provide good estimates of pairwise-similarity. Subsequent works extended this technique to weighted sets and show how to produce samples with only a constant number of hash evaluations for any element, independent of its weight. Another improvement by Li et al. shows how to speedup sketch computations by computing many (near-)independent samples in one shot. Unfortunately this latter improvement works only for the unweighted case. In this paper we give a simple, fast and accurate procedure which reduces weighted sets to unweighted sets with small impact on the Jaccard similarity. This leads to compact sketches consisting of many (near-)independent weighted samples which can be computed with just a small constant number of hash function evaluations per weighted element. The size of the produced unweighted set is furthermore a tunable parameter which enables us to run the unweighted scheme of Li et al. in the regime where it is most efficient. Even when the sets involved are unweighted, our approach gives a simple solution to the densification problem that other works attempted to address. Unlike previously known schemes, ours does not result in an unbiased estimator. However, we prove that the bias introduced by our reduction is negligible and that the standard deviation is comparable to the unweighted case. We also empirically evaluate our scheme and show that it gives significant gains in computational efficiency, without any measurable loss in accuracy

Analyzing Boltzmann Samplers for Bose-Einstein Condensates with Dirichlet Generating Functions

Boltzmann sampling is commonly used to uniformly sample objects of a particular size from large combinatorial sets. For this technique to be effective, one needs to prove that (1) the sampling procedure is efficient and (2) objects of the desired size are generated with sufficiently high probability. We use this approach to give a provably efficient sampling algorithm for a class of weighted integer partitions related to Bose-Einstein condensation from statistical physics. Our sampling algorithm is a probabilistic interpretation of the ordinary generating function for these objects, derived from the symbolic method of analytic combinatorics. Using the Khintchine-Meinardus probabilistic method to bound the rejection rate of our Boltzmann sampler through singularity analysis of Dirichlet generating functions, we offer an alternative approach to analyze Boltzmann samplers for objects with multiplicative structure.Comment: 20 pages, 1 figur