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An Average-case Analysis of the Gaussian Algorithm for Lattice Reduction Hervé Daudé, Philippe Flajolet, Brigitte Vallée.. To cite this version: Hervé Daudé, Philippe Flajolet, Brigitte [r]

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Theorem 4. When OP T LP (G) = (1 + )n and G is subquartic, there is a feasible circulation **for**
C(G, T ) with cost at most n/8 + 2n.
Consider a vertex v ∈ T exp . If both incoming back edges had f -value 1/2, then this vertex
would not contribute anything to **the** cost **of** **the** circulation. Thus, on a high level, our goal is to find f -values that are as close to half as possible, while at **the** same time not creating any additional unsatisfied vertices. **The** f -value therefore corresponds to a decreased x-value if **the** x-value is high, and **an** increased x-value if **the** x-value is low. A set **of** f -values corresponding to decreased x-values may pose a problem if they correspond to **the** set **of** back edges that cover **an** LP-unsatisfied vertex. However, we note that in Section 4.1 , we only used (j)/2 to satisfy **an** LP-unsatisfied vertex j. We can actually use at least (j). This observation allows us to decrease **the** x-values. We use **the** rules depicted in Figure 1 to determine **the** values f : B(T ) → [0, 1].

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1. Introduction
[ 2 ] U ‐Pb geochronology by isotope dilution ther-
mal ionization mass spectrometry (ID‐TIMS) has become **the** gold standard **for** calibrating geologic time due to precisely determined uranium decay constants, high‐precision measurement methods, and **an** internal check **for** open ‐system behavior provided by **the** dual decay **of** 235 U and 238 U. Pre- cise, accurate ID ‐TIMS dates have been used to test and calibrate detailed tectonic models [e.g., Schoene et al., 2008], determine **the** timing and tempo **of** mass extinctions and ecological recovery [Bowring et al., 1998], calibrate a global geologic timescale [Davydov et al., 2010], and establish a precise chronology **for** **the** early solar system [Amelin et al., 2002]. These results rely on **analysis** and interpre- tation **of** precisely measured data, **for** which correct and transparent data **reduction** and error propagation are imperative.

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We wish to find **the** global minimum **of** a function f , min x∈D f (x), where **the** search space
D = [LB, U B] d
is a compact subset **of** R d . We assume that f is **an** expensive-to-compute
black-box function. Subsequently, optimization can only be attempted **for** a low number **of** function evaluations. **The** Efficient Global Optimization (EGO) **algorithm** [3, 4] has become standard **for** optimizing such expensive unconstrained continuous problems. Its efficiency stems from **an** embedded conditional **Gaussian** process (GP), also known as kriging, which acts as a surrogate **for** **the** objective function. Certainly, other surrogate techniques can be employed instead **of** GPs. **For** example, [9] proposes a variant **of** EGO in which a quadratic regression model serves as a surrogate. However, it is shown by some **of** their examples that **the** standard EGO performs better than this variant.

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perturbed reduced basis remains reduced, possibly with respect to weaker reduc- tion parameters), we introduce a new notion **of** LLL-**reduction** (Definition 5.3). Matrices reduced in this new sense satisfy essentially **the** same properties as those satisfied by matrices reduced in **the** classical sense. But **the** new notion **of** **reduction** is more natural with respect to column-wise perturbations, as **the** perturbation **of** a reduced basis remains reduced (this is not **the** case with **the** classical notion **of** **reduction**). Another important ingredient **of** **the** main result, that may be **of** inde- pendent interest, is **the** improvement **of** **the** perturbation analyses **of** [1] and [28] **for** general full column rank matrices (section 2). More precisely, all our bounds are fully rigorous, in **the** sense that no higher order error term is neglicted, and explicit constant factors are provided. Explicit and rigorous bounds are invaluable **for** guaranteeing computational accuracy: one can choose a precision that will be known in advance to provide a certain degree **of** accuracy in **the** result. In [1, §6], a rigorous error bound was proved. A (much) smaller bound was given in [1, §8], but it is a first-order bound, i.e., high-order terms were neglected. Our rigorous bound is close to this improved bound. Our approach to deriving this rigorous bound is new and has been extended to **the** perturbation **analysis** **of** some other important matrix factorizations [3]. Finally, we give explicit constants in **the** back- ward stability **analysis** **of** Householder’s **algorithm** from [8, §19], which, along with **the** perturbation **analysis**, provides fully rigorous and explicit error bounds **for** **the** computed R-factor **of** a LLL-reduced matrix.

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two events were not known, it was assumed that **the** two possibilities (glide versus non-glide) were equally likely. Normally, this may seem unusual, but in **the** anal- ysis given in **the** following chapters, it is noted that **the** test **for** a glide occurs by taking data starting thirty milliseconds from **the** point **of** **the** onset **of** voicing. In cases where **the** distance between **the** landmark and **the** following vowel was less than thirty milliseconds, **the** region in question was automatically labeled as a non-glide section. This is reasonable because **the** duration from a glide into a vowel is more than thirty milliseconds, with minimums being only as small as fifty milliseconds in con- tinuous speech. Thus, only regions where **the** duration from landmark to vowel was more than thirty milliseconds were considered in **the** hypothesis test. **The** remaining speech segments in question were modeled to have a 50% chance **of** being a glide, and 50% chance **of** being a non-glide. Albeit a rough estimate, setting **the** two apriori probabilities equal served as a reasonable assumption which would at **the** same time simplify computation. Next, **the** probability distribution had to be modeled. **The** variability **of** **the** parameters selected was modeled to be distributed in a **Gaussian** probability density function around its mean. This distribution was used as a simpli- fication **for** detection without specification **of** context, and can likely be improved if specific contexts are considered separately since **the** measurements will be localized around different means **for** different vowels which follow **the** glides. However, in this thesis, a general hypothesis test is performed on each new set **of** measurements with- out regard to specific context. **The** covariances **of** this distribution were determined by taking **the** covariance **of** **the** sampled data from **the** training set. Likewise, **the** means were **the** sample means **of** **the** training data.

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In Table 5 , we summarize **the** peak ﬂux ratios and size ratio data shown in Figures 7 and 8 so that accurate completeness can be estimated **for** future core-population studies. We emphasize that even **for** analyses **of** relatively compact sources using **the** DR2 external-mask **reduction**, extra attention should be paid to three factors. First, **the** population **of** sources near **the** completeness limit (peak ﬂuxes **of** 3–5 times **the** noise) likely have contributions from even fainter sources (peak ﬂuxes **of** 1–2 times **the** noise) that have been boosted to higher ﬂuxes through noise spikes, etc. If **the** true underlying source population is expected to increase with decreasing peak ﬂux, then this contribution **of** fainter sources could be signiﬁcant. Second, faint compact sources could be either intrinsically faint and compact or brighter and larger sources that are not fully recovered. Examination **of** **the** size distribution **of** **the** brighter sources in **the** map should help determine what **the** expected properties **of** **the** fainter sources are. Third, **for** analyses where **the** source-detection rate is important (e.g., applying correc- tions to **an** observed core mass function), **the** source recovery rates presented in Section 4.1 should not be blindly applied, as they do not include factors such as crowding or **the** limitations **of** core-ﬁnding algorithms running without prior knowledge on a map, both **of** which are expected to decrease **the** real observational detection rate. Furthermore, while **the** results presented here are uncontaminated by false-positive detections, such complications will need to be carefully considered when running source-identiﬁcation algorithms on real observations.

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Suppose a design goal is to obtain **an** MSE which is less than a speciÞed value . **The** following procedure could be applied. 1) Set a coherence threshold and deÞne a set **of** kernel bandwidths to be tested. **For** **the** following design exam- ples, a set **of** values **for** was chosen as equally spaced points in (0, 1). Then, was chosen to yield reasonable values **of** **for** **the** chosen set **of** values. **The** value **of** is determined **for** each pair by training **the** dic- tionary with **the** input signal until its size stabilizes. **The** training is repeated several times. A value is de- termined **for** **the** th realization. **The** value **of** associated with **the** pair is **the** average **of** **the** dictionary sizes **for** all realizations, rounded to **the** nearest integer. This is **the** value **of** to be used in **the** theoretical model. 2) Using **the** system input , determine **the** desired output

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Figure 2 shows **the** confidence ellipse **of** **the** scores
obtained by each **algorithm** on each pair **of** scales. It shows how **the** EM and MM perform slightly better than KAMIR in both **of** its fashions. As in Figure 3, we see **the** benefits **of** **the** sparsity penalty as improving background suppression at **the** cost **of** introducing some artifacts. **An** interesting observation is that EM + S and MM + S appear closer to K and ˜ K than EM and MM. Regardless **of** **the** amount **of** noise that may affect **the** evaluation results, **the** EM method presented in this paper leads to slightly better results than state **of** **the** art. Close investigation reveals that its main difference with KAMIR lies in handling **the** uncertainty **of** **the** model through **the** posterior variance in (7). Then, **the** W-disjoint orthogonality penalty γ in (19) is seen as controlling **the** trade-off between isolation and distor- tion. **The** MM approach does not seem to perform sig- nificantly better than KAMIR algorithms, especially **for** **the** suppression **of** background. Still, adding a penalty γ brings it closer to EM, while having a significantly smaller computational complexity.

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ence equation like x n = X(t n ), **the** behavior can be highly irregular and extremely complex. In some cases **the** behavior is estimated like chaotic. In **the** first approximation, we can determine **the** chaocity by **the** property **of** **the** system to construct it’s trajectories in a bounded domain **of** **the** phase space. Properties **of** dynamical systems which generate chaotic solutions, have been widely discussed by **the** authors (results and references in **the** monographs [13, 14]). **The** simplest example is **an** one-dimensional dynami- cal system x n+1 = f (x n , µ) which generates chaotic solution **for** some func- tions f and values **of** parameter µ . In particular, **for** logistic function f such as x n+1 = µx n (1 − x n ), **the** plot **of** solution looks like white noise with some values µ > 3.6. So, **the** problem statement **the** nature **of** time series **analysis** nature is do **the** observed data have stochastic nature, or deterministic. Let B (t), 0 ≤ t ≤ 1 be a fractional Brownian motion with Hurst exponent H. Let’s consider **the** normalized increments

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We notice that **lattice**-based quantization gives results which are optimal (or **the** best known ones) in eight cases out **of** fifteen and **the** other seven cases are very close to **the** optimal ones. It outperforms telescoping rounding in twelve cases out **of** fifteen and yields **the** same (optimal) result in a thirteenth case: **the** use **of** **the** LLL option gives a better result in twelve cases and **an** identical (optimal) result in a thirteenth case, **the** use **of** **the** BKZ option gives a better result in eleven cases and **the** use **of** **the** HKZ option in nine cases. We remark, in particular, **the** good behavior **of** **the** LLL **algorithm** in all 15 test cases, further emphasizing **the** idea that **the** **lattice** bases we use are close to being reduced. Our approach seems to work particularly well when **the** gap between **the** minimax error and **the** naive rounding error is significant, which is **the** case where one is most interested in improvements over **the** naive rounding filter. Eventually, note that, in **the** C125/21 and D125/22 cases, our approach returns (in less than 8 seconds, see Subsection VI-D ) results that are better than **the** ones provided by (time-limited) MILP tools.

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in most cases **the** bases need not to be chosen a priori, but much work should be done, and in particular we need **an** **algorithm** which generates uniformly convex **lattice** sets **of** a given size at random.
Secondly what can we say about **the** reconstruction problem **for** any set **of** **lattice** directions not uniquely determining convex **lattice** sets ? Does there exist a polynomial **algorithm** in this case ? We could apply our **algorithm** until **the** **reduction** to a 2-SAT formula, but then we do not see any way to express **the** convexity by a formula whose satisfiability could be checked in polynomial time.

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ODD L ( x ) = f a j a k x; 9 b;c 2 Inf ( x ) with a = b _ L c g will denote **the** set **of** LUB **of** elements in Inf ( x ), called lower odds. **The** duality principle, allows us to consider only ODD U ( x ).
Odds give rise to auxiliary elements so they may be compared to generators in [8] and to canonical representatives in [1]. **The** existing strategies **for** **lattice** insertion dier as to **the** way odds are detected and **the** number **of** auxiliaries per odd. Thus, **the** **algorithm** in [2] checks **the** GLB **of** all couples **of** elements in Sup ( x ) and inserts **an** auxiliary each time this GLB is incomparable to x . In this way, **the** same odd may provoke **the** generation **of** a set **of** auxiliaries. **For** example, **the** element g on Fig.1 is **an** odd since g = V

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Suppose a design goal is to obtain **an** MSE which is less than a specified value . **The** following procedure could be applied. 1) Set a coherence threshold and define a set **of** kernel bandwidths to be tested. **For** **the** following design exam- ples, a set **of** values **for** was chosen as equally spaced points in (0, 1). Then, was chosen to yield reasonable values **of** **for** **the** chosen set **of** values. **The** value **of** is determined **for** each pair by training **the** dic- tionary with **the** input signal until its size stabilizes. **The** training is repeated several times. A value is de- termined **for** **the** th realization. **The** value **of** associated with **the** pair is **the** average **of** **the** dictionary sizes **for** all realizations, rounded to **the** nearest integer. This is **the** value **of** to be used in **the** theoretical model. 2) Using **the** system input , determine **the** desired output

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Keywords: A posteriori error estimation, Navier-Stokes problem, iterative method.
1. Introduction
**The** a posteriori **analysis** controls **the** overall discretization error **of** a problem by providing error indicators easy to compute. Once these error indicators are constructed, we prove their efficiency by bounding each indicator by **the** local error. This **analysis** was first introduced by I. Babuˇ ska [2], and developed by R. Verf¨ urth [12]. **The** present work investigates a posteriori error estimates **of** **the** finite element discretization **of** **the** Navier-Stokes equations in polygonal domains. In fact, many works have been carried out in this field. In [3], C. Bernardi, F. Hecht and R. Verfürth considered a variational formulation **of** **the** three-dimensional Navier-Stokes equations with mixed boundary conditions and they proved that it admits a solution if **the** domain satisfies a suitable regularity assumption. In addition, they established **the** a priori and **the** a posteriori error estimates. As well, in [8], V. Ervin, W. Layton and J. Maubach present locally calculable a posteriori error estimators **for** **the** basic two-level discretization **of** **the** Navier- Stokes equations. In this work, we propose a finite element discretization **of** **the** Navier-Stokes equations relying on **the** Galerkin method. In order to solve **the** discrete problem we propose **an** iterative method. Therefore two sources **of** error appear, due to **the** discretization and **the** **algorithm**. Balancing these two errors leads to important computational savings. We apply this strategy on **the** following Navier-Stokes equations:

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4. Conclusion and Perspectives
In **the** framework **of** model **reduction** techniques, a strategy has been developed that reduces **the** compu- tational cost **of** post-buckling simulations. This approach makes **the** most **of** **an** on **the** fly adaptive procedure and little knowledge on post-buckling phenomenon. More precisely, as stated in **the** semi-analytical meth- ods, **the** post-buckling equilibrium state **of** structures is taken to be a combination **of** **the** pre-buckling state (known) and a variation, which is decomposed into a buckling mode component and a higher order variation arising from **an** automatic completion procedure. This results in a fast **algorithm** **for** solving post-buckling problems, which does not require expensive pre-calculations and is positioned both as **an** alternative to POD-based model **reduction** and as a way to build POD snapshots **for** post-buckling **analysis** at lower cost. In this paper, **the** relevance **of** this approach has been demonstrated in **the** case **of** post-buckling **analysis** **of** plates. In spite **of** **the** limits **of** **the** home-made finite element research code, insight was gained into **the** computational performances **of** **the** strategy **for** **an** in-plane load reaching more than twice **the** buckling load

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However, before exploring potential departures from Gaus- sianity, we need to adopt a robust enough numerical strategy in order to numerically evaluate integrals such as Eq. ( 4 ), a task which is notoriously di fficult even with Maxwell–Boltzmann VDFs. It is very easy to verify that, **for** instance, a standard Gauss–Hermite (GH) quadrature, even at high rank k, fails at properly computing a somewhat simpler expression like **the** Voigt 3 function given in Eq. ( 13 ). We display, in Fig. 1 , **the** com- parison between a GH integration and **the** new numerical scheme that is presented hereafter.

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Applying POLRED to P(X) we obtain
thus showing that **the** fields generated by **the** roots **of** **the** polynomials given in [PMD] are isomorphic, and also that is a subfield. **The** fact that
**the** same polynomial is obtained several times gives also some information on **the** Galois group **of** **the** Galois closure **of** **the** number field I, since it