397 research outputs found
Probabilistic Reduced-Order Modeling for Stochastic Partial Differential Equations
We discuss a Bayesian formulation to coarse-graining (CG) of PDEs where the
coefficients (e.g. material parameters) exhibit random, fine scale variability.
The direct solution to such problems requires grids that are small enough to
resolve this fine scale variability which unavoidably requires the repeated
solution of very large systems of algebraic equations. We establish a
physically inspired, data-driven coarse-grained model which learns a low-
dimensional set of microstructural features that are predictive of the
fine-grained model (FG) response. Once learned, those features provide a sharp
distribution over the coarse scale effec- tive coefficients of the PDE that are
most suitable for prediction of the fine scale model output. This ultimately
allows to replace the computationally expensive FG by a generative proba-
bilistic model based on evaluating the much cheaper CG several times. Sparsity
enforcing pri- ors further increase predictive efficiency and reveal
microstructural features that are important in predicting the FG response.
Moreover, the model yields probabilistic rather than single-point predictions,
which enables the quantification of the unavoidable epistemic uncertainty that
is present due to the information loss that occurs during the coarse-graining
process
Bipartite Perfect Matching in Pseudo-Deterministic NC
We present a pseudo-deterministic NC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, on the same graph it returns the same matching for almost all choices of randomness. As an immediate consequence we also find a pseudo-deterministic NC algorithm for constructing a depth first search (DFS) tree. We introduce a method for computing the union of all min-weight perfect matchings of a weighted graph in RNC and a novel set of weight assignments which in combination enable isolating a unique matching in a graph.
We then show a way to use pseudo-deterministic algorithms to reduce the number of random bits used by general randomized algorithms. The main idea is that random bits can be reused by successive invocations of pseudo-deterministic randomized algorithms. We use the technique to show an RNC algorithm for constructing a depth first search (DFS) tree using only O(log^2 n) bits whereas the previous best randomized algorithm used O(log^7 n), and a new sequential randomized algorithm for the set-maxima problem which uses fewer random bits than the previous state of the art.
Furthermore, we prove that resolving the decision question NC = RNC, would imply an NC algorithm for finding a bipartite perfect matching and finding a DFS tree in NC. This is not implied by previous randomized NC search algorithms for finding bipartite perfect matching, but is implied by the existence of a pseudo-deterministic NC search algorithm
Final report on the evaluation of RRM/CRRM algorithms
Deliverable public del projecte EVERESTThis deliverable provides a definition and a complete evaluation of the RRM/CRRM algorithms selected in D11 and D15, and evolved and refined on an iterative process. The evaluation will be carried out by means of simulations using the simulators provided at D07, and D14.Preprin
Comparing Spatial and Scenario Decomposition for Stochastic Hydrothermal Unit Commitment Problems
Solving very-large-scale optimization problems frequently require to decompose them in smaller subproblems, that are iteratively solved to produce useful information. One such approach is the Lagrangian Relaxation (LR), a general technique that leads to many different decomposition schemes. The LR produces a lower bound of the objective function and useful information for heuristics aimed at constructing feasible primal solutions. In this paper, we compare the main LR strategies used so far for Stochastic Hydrothermal Unit Commitment problems, where uncertainty mainly concerns water availability in reservoirs and demand (weather conditions). The problem is customarily modeled as a two-stage mixed-integer optimization problem. We compare different decomposition strategies (unit and scenario schemes) in terms of quality of produced lower bound and running time. The schemes are assessed with various hydrothermal systems, considering different configuration of power plants, in terms of capacity and number of units
On k-Column Sparse Packing Programs
We consider the class of packing integer programs (PIPs) that are column
sparse, i.e. there is a specified upper bound k on the number of constraints
that each variable appears in. We give an (ek+o(k))-approximation algorithm for
k-column sparse PIPs, improving on recent results of and
. We also show that the integrality gap of our linear programming
relaxation is at least 2k-1; it is known that k-column sparse PIPs are
-hard to approximate. We also extend our result (at the loss
of a small constant factor) to the more general case of maximizing a submodular
objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail
Sketching Cuts in Graphs and Hypergraphs
Sketching and streaming algorithms are in the forefront of current research
directions for cut problems in graphs. In the streaming model, we show that
-approximation for Max-Cut must use space;
moreover, beating -approximation requires polynomial space. For the
sketching model, we show that -uniform hypergraphs admit a
-cut-sparsifier (i.e., a weighted subhypergraph that
approximately preserves all the cuts) with
edges. We also make first steps towards sketching general CSPs (Constraint
Satisfaction Problems)
A Two-Scale Complexity Measure for Deep Learning Models
We introduce a novel capacity measure 2sED for statistical models based on
the effective dimension. The new quantity provably bounds the generalization
error under mild assumptions on the model. Furthermore, simulations on standard
data sets and popular model architectures show that 2sED correlates well with
the training error. For Markovian models, we show how to efficiently
approximate 2sED from below through a layerwise iterative approach, which
allows us to tackle deep learning models with a large number of parameters.
Simulation results suggest that the approximation is good for different
prominent models and data sets
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