| 31. |
- Burdakov, Oleg, 1953-, et al.
(författare)
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Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions
- 2011
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- We suggest here a least-change correction to available finite element (FE) solution.This postprocessing procedure is aimed at recoveringthe monotonicity and some other important properties that may not beexhibited by the FE solution. It is based on solvinga monotonic regression problem with some extra constraints.One of them is a linear equality-type constraint which models the conservativityrequirement. The other ones are box-type constraints, andthey originate from the discrete maximum principle.The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessedFE solution preserves the accuracy of the discrete FE approximation.We introduce an algorithm for solving the postprocessingproblem. It can be viewed as a dual ascent method basedon the Lagrangian relaxation of the equality constraint.We justify theoretically its correctness.Its efficiency is demonstrated by the presented results of numerical experiments.
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| 32. |
- Burdakov, Oleg, 1953-, et al.
(författare)
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On Efficiently Combining Limited Memory and Trust-Region Techniques
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Limited memory quasi-Newton methods and trust-region methods represent two efficient approaches used for solving unconstrained optimization problems. A straightforward combination of them deteriorates the efficiency of the former approach, especially in the case of large-scale problems. For this reason, the limited memory methods are usually combined with a line search. We show how to efficiently combine limited memory and trust-region techniques. One of our approaches is based on the eigenvalue decomposition of the limited memory quasi-Newton approximation of the Hessian matrix. The decomposition allows for finding a nearly-exact solution to the trust-region subproblem defined by the Euclidean norm with an insignificant computational overhead compared with the cost of computing the quasi-Newton direction in line-search limited memory methods. The other approach is based on two new eigenvalue-based norms. The advantage of the new norms is that the trust-region subproblem is separable and each of the smaller subproblems is easy to solve. We show that our eigenvalue-based limited-memory trust-region methods are globally convergent. Moreover, we propose improved versions of the existing limited-memory trust-region algorithms. The presented results of numerical experiments demonstrate the efficiency of our approach which is competitive with line-search versions of the L-BFGS method.
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| 33. |
- Burdakov, Oleg, et al.
(författare)
-
Optimal placement of communications relay nodes
- 2009
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Rapport (populärvet., debatt m.m.)abstract
- We consider a constrained optimization problem with mixed integer and real variables. It models optimal placement of communications relay nodes in the presence of obstacles. This problem is widely encountered, for instance, in robotics, where it is required to survey some target located in one point and convey the gathered information back to a base station located in another point. One or more unmanned aerial or ground vehicles (UAVs or UGVs) can be used for this purpose as communications relays. The decision variables are the number of unmanned vehicles (UVs) and the UV positions. The objective function is assumed to access the placement quality. We suggest one instance of such a function which is more suitable for accessing UAV placement. The constraints are determined by, firstly, a free line of sight requirement for every consecutive pair in the chain and, secondly, a limited communication range. Because of these requirements, our constrained optimization problem is a difficult multi-extremal problem for any fixed number of UVs. Moreover, the feasible set of real variables is typically disjoint. We present an approach that allows us to efficiently find a practically acceptable approximation to a global minimum in the problem of optimal placement of communications relay nodes. It is based on a spatial discretization with a subsequent reduction to a shortest path problem. The case of a restricted number of available UVs is also considered here. We introduce two label correcting algorithms which are able to take advantage of using some peculiarities of the resulting restricted shortest path problem. The algorithms produce a Pareto solution to the two-objective problem of minimizing the path cost and the number of hops. We justify their correctness. The presented results of numerical 3D experiments show that our algorithms are superior to the conventional Bellman-Ford algorithm tailored to solving this problem.
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| 34. |
- Burdakov, Oleg, et al.
(författare)
-
Regularized monotonic regression
- 2016
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Monotonic (isotonic) Regression (MR) is a powerful tool used for solving a wide range of important applied problems. One of its features, which poses a limitation on its use in some areas, is that it produces a piecewise constant fitted response. For smoothing the fitted response, we introduce a regularization term in the MR formulated as a least distance problem with monotonicity constraints. The resulting Smoothed Monotonic Regrassion (SMR) is a convex quadratic optimization problem. We focus on the SMR, where the set of observations is completely (linearly) ordered. Our Smoothed Pool-Adjacent-Violators (SPAV) algorithm is designed for solving the SMR. It belongs to the class of dual activeset algorithms. We proved its finite convergence to the optimal solution in, at most, n iterations, where n is the problem size. One of its advantages is that the active set is progressively enlarging by including one or, typically, more constraints per iteration. This resulted in solving large-scale SMR test problems in a few iterations, whereas the size of that problems was prohibitively too large for the conventional quadratic optimization solvers. Although the complexity of the SPAV algorithm is O(n2), its running time was growing in our computational experiments in proportion to n1:16.
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| 35. |
- Cengiz, Cigdem, et al.
(författare)
-
High-dimensional profile analysis
- 2020
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The three tests of profile analysis: test of parallelism, test of level and test of flatness have been studied. Likelihood ratio tests have been derived. Firstly, a traditional setting, where the sample size is greater than the dimension of the parameter space, is considered. Then, all tests have been derived in a high-dimensional setting. In high-dimensional data analysis, it is required to use some techniques to tackle the problems which arise with the dimensionality. We propose a dimension reduction method using scores which was first proposed by Läuter et al. (1996).
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| 36. |
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| 37. |
- Daneva, Maria, et al.
(författare)
-
A Conjugate Direction Frank-Wolfe Method for Nonconvex Problems
- 2003
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Rapport (refereegranskat)abstract
- In this paper we propose an algorithm for solving problems with nonconvex objective function and linear constraints. We extend the previously suggested Conjugate direction Frank–Wolfe algorithm to nonconvex problems. We apply our method to multi-class user equilibria under social marginal cost pricing. Results of numerical experiments on Sioux Falls and Winnipeg are reported.
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| 38. |
- de Dieu Niyigena, Jean, et al.
(författare)
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Approximation of misclassification probabilities using quadratic classifier for repeated measurements with known covariance matrices
- 2024
-
Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Quadratic discriminant analysis is a well-established supervised classification method, which extends the linear the linear discriminant analysis by relaxing the assumption of equal variances across classes. In this study, quadratic discriminant analysis is used to develop a quadratic classification rule based on repeated measurements. We employ a bilinear regression model to assign new observations to predefined populations and approximate the misclassification probability. Through weighted estimators, we estimate unknown mean parameters and derive moments of the quadratic classifier. We then conduct numerical simulations to compare misclassification probabilities using true and estimated mean parameters, as well as probabilities computed through simulation. Our findings suggest that as the distance between groups widens, the misclassification probability curve decreases, indicating that classifying observations is easier in widely separated groups compared to closely clustered ones.
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| 39. |
- Erickson, Brittany. A., et al.
(författare)
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Stable, High Order Accurate Adaptive Schemes for Long Time, Highly Intermittent Geophysics Problems
- 2013
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Many geophysical phenomena are characterized by properties that evolve over a wide range of scales which introduce difficulties when attempting to model these features in one computational method. We have developed a high-order finite difference method for the elastic wave equation that is able to efficiently handle varying temporal scales in a single, stand-alone framework. We apply this method to earthquake cycle models characterized by extremely long interseismic periods interspersed with abrupt, short periods of dynamic rupture. Through the use of summation-by-parts operators and weak enforcement of boundary conditions we derive a provably stable discretization. Time stepping is achieved through the implicit θ-method which allows us to take large time steps during the intermittent period between earthquakes and adapts appropriately to fully resolve rupture.
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| 40. |
- Eriksson, Sofia, et al.
(författare)
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Finite difference schemes with transferable interfaces for parabolic problems
- 2018
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Ingår i: Journal of Computational Physics. - Linköping : Elsevier. - 0021-9991 .- 1090-2716. ; 375, s. 935-949
-
Tidskriftsartikel (refereegranskat)abstract
- We derive a method to locally change the order of accuracy of finite difference schemes that approximate the second derivative. The derivation is based on summation-by-parts operators, which are connected at interfaces using penalty terms. At such interfaces, the numerical solution has a double representation, with one representation in each domain. We merge this double representation into a single one, yielding a new scheme with unique solution values in all grid points. The resulting scheme is proven to be stable, accurate and dual consistent. (C) 2018 Elsevier Inc. All rights reserved.
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