| 1. |
- Dmytryshyn, Andrii, 1986-, et al.
(författare)
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Miniversal deformations of matrices of bilinear forms
- 2012
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 436:7, s. 2670-2700
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Tidskriftsartikel (refereegranskat)abstract
- Arnold [V.I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26 (2) (1971) 29–43] constructed miniversal deformations of square complex matrices under similarity; that is, a simple normal form to which not only a given square matrix A but all matrices B close to it can be reduced by similarity transformations that smoothly depend on the entries of B. We construct miniversal deformations of matrices under congruence.
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| 2. |
- Zhang, Ye, 1984-, et al.
(författare)
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Reconstructing gas distribution maps via an adaptive sparse regularization algorithm
- 2016
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Ingår i: Inverse Problems in Science and Engineering. - : Taylor & Francis. - 1741-5977 .- 1741-5985. ; 24:7, s. 1186-1204
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we present an algorithm to be used by an inspectionrobot to produce a gas distribution map and localize gas sources ina large complex environment. The robot, equipped with a remotegas sensor, measures the total absorption of a tuned laser beam andreturns integral gas concentrations. A mathematical formulation ofsuch measurement facility is a sequence of Radon transforms,which isa typical ill-posed problem. To tackle the ill-posedness, we developa new regularization method based on the sparse representationproperty of gas sources and the adaptive finite-element method. Inpractice, only a discrete model can be applied, and the quality ofthe gas distributionmap depends on a detailed 3-D world model thatallows us to accurately localize the robot and estimate the paths of thelaser beam. In this work, using the positivity ofmeasurements and theprocess of concentration, we estimate the lower and upper boundsof measurements and the exact continuous model (mapping fromgas distribution to measurements), and then create a more accuratediscrete model of the continuous tomography problem. Based onadaptive sparse regularization, we introduce a new algorithm thatgives us not only a solution map but also a mesh map. The solutionmap more accurately locates gas sources, and the mesh map providesthe real gas distribution map. Moreover, the error estimation of theproposed model is discussed. Numerical tests for both the syntheticproblem and practical problem are given to show the efficiency andfeasibility of the proposed algorithm.
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| 3. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
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A separating oscillation method of recovering the G-limit in standard and non-standard homogenization problems
- 2016
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Ingår i: Inverse Problems. - Bristol, United Kingdom : IOP Publishing. - 0266-5611 .- 1361-6420. ; 32:2
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Tidskriftsartikel (refereegranskat)abstract
- Reconstructing the homogenized coefficient, which is also called the G-limit, in elliptic equations involving heterogeneous media is a typical nonlinear ill-posed inverse problem. In this work, we develop a numerical technique to determine G-limit that does not rely on any periodicity assumption. The approach is a technique that separates the computation of the deviation of the G-limit from the weak L-2-limit of the sequence of coefficients from the latter. Moreover, to tackle the ill-posedness, based on the classical Tikhonov regularization scheme we develop several strategies to regularize the introduced method. Various numerical tests for both standard and non-standard homogenization problems are given to show the efficiency and feasibility of the proposed method.
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| 4. |
- Cheng, Xiaoliang, et al.
(författare)
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A modified coupled complex boundary method for an inverse chromatography problem
- 2018
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Ingår i: Journal of Inverse and Ill-Posed Problems. - : Walter de Gruyter. - 0928-0219 .- 1569-3945. ; 26:1, s. 33-49
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Tidskriftsartikel (refereegranskat)abstract
- Adsorption isotherms are the most important parameters in rigorous models of chromatographic processes. In this paper, in order to recover adsorption isotherms, we consider a coupled complex boundary method (CCBM), which was previously proposed for solving an inverse source problem [2]. With CCBM, the original boundary fitting problem is transferred to a domain fitting problem. Thus, this method has advantages regarding robustness and computation in reconstruction. In contrast to the traditional CCBM, for the sake of the reduction of computational complexity and computational cost, the recovered adsorption isotherm only corresponds to the real part of the solution of a forward complex initial boundary value problem. Furthermore, we take into account the position of the profiles and apply the momentum criterion to improve the optimization progress. Using Tikhonov regularization, the well-posedness, convergence properties and regularization parameter selection methods are studied. Based on an adjoint technique, we derive the exact Jacobian of the objective function and give an algorithm to reconstruct the adsorption isotherm. Finally, numerical simulations are given to show the feasibility and efficiency of the proposed regularization method.
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| 5. |
- Fasi, Massimiliano, 1989-, et al.
(författare)
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Generating extreme-scale matrices with specified singular values or condition numbers
- 2021
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics. - 1064-8275 .- 1095-7197. ; 43:1, s. A663-A684
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Tidskriftsartikel (refereegranskat)abstract
- A widely used form of test matrix is the randsvd matrix constructed as the product A = U Sigma V*, where U and V are random orthogonal or unitary matrices from the Haar distribution and Sigma is a diagonal matrix of singular values. Such matrices are random but have a specified singular value distribution. The cost of forming an m x n randsvd matrix is m(3) + n(3) flops, which is prohibitively expensive at extreme scale; moreover, the randsvd construction requires a significant amount of communication, making it unsuitable for distributed memory environments. By dropping the requirement that U and V be Haar distributed and that both be random, we derive new algorithms for forming A that have cost linear in the number of matrix elements and require a low amount of communication and synchronization. We specialize these algorithms to generating matrices with a specified 2-norm condition number. Numerical experiments show that the algorithms have excellent efficiency and scalability.
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| 6. |
- Fasi, Massimiliano, 1989-, et al.
(författare)
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Matrices with tunable infinity-norm condition number and no need for pivoting in LU factorization
- 2021
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Ingår i: SIAM Journal on Matrix Analysis and Applications. - : Society for Industrial and Applied Mathematics. - 0895-4798 .- 1095-7162. ; 42:1, s. 417-435
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Tidskriftsartikel (refereegranskat)abstract
- We propose a two-parameter family of nonsymmetric dense n x n matrices A(alpha, beta) for which LU factorization without pivoting is numerically stable, and we show how to choose alpha and beta to achieve any value of the infinity-norm condition number. The matrix A(alpha, beta) can be formed from a simple formula in O(n(2)) flops. The matrix is suitable for use in the HPL-AI Mixed-Precision Benchmark, which requires an extreme scale test matrix (dimension n > 10(7)) that has a controlled condition number and can be safely used in LU factorization without pivoting. It is also of interest as a general-purpose test matrix.
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| 7. |
- Fasi, Massimiliano, 1989-, et al.
(författare)
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Numerical behavior of NVIDIA tensor cores
- 2021
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Ingår i: PeerJ Computer Science. - : PeerJ, Inc. - 2376-5992. ; 7, s. 1-19
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Tidskriftsartikel (refereegranskat)abstract
- We explore the floating-point arithmetic implemented in the NVIDIA tensor cores, which are hardware accelerators for mixed-precision matrix multiplication available on the Volta, Turing, and Ampere microarchitectures. Using Volta V100, Turing T4, and Ampere A100 graphics cards, we determine what precision is used for the intermediate results, whether subnormal numbers are supported, what rounding mode is used, in which order the operations underlying the matrix multiplication are performed, and whether partial sums are normalized. These aspects are not documented by NVIDIA, and we gain insight by running carefully designed numerical experiments on these hardware units. Knowing the answers to these questions is important if one wishes to: (1) accurately simulate NVIDIA tensor cores on conventional hardware; (2) understand the differences between results produced by code that utilizes tensor cores and code that uses only IEEE 754-compliant arithmetic operations; and (3) build custom hardware whose behavior matches that of NVIDIA tensor cores. As part of this work we provide a test suite that can be easily adapted to test newer versions of the NVIDIA tensor cores as well as similar accelerators from other vendors, as they become available. Moreover, we identify a non-monotonicity issue affecting floating point multi-operand adders if the intermediate results are not normalized after each step.
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| 8. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
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Damped Dynamical Systems for Solving Equations and Optimization Problems
- 2019
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Ingår i: Handbook of the Mathematics of the Arts and Sciences. - Cham : Springer. - 9783319706580
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- We present an approach for solving optimization problems with or without constrains which we call Dynamical Functional Particle Method (DFMP). The method consists of formulating the optimization problem as a second order damped dynamical system and then applying symplectic method to solve it numerically. In the first part of the chapter, we give an overview of the method and provide necessary mathematical background. We show that DFPM is a stable, efficient, and given the optimal choice of parameters, competitive method. Optimal parameters are derived for linear systems of equations, linear least squares, and linear eigenvalue problems. A framework for solving nonlinear problems is developed and numerically tested. In the second part, we adopt the method to several important applications such as image analysis, inverse problems for partial differential equations, and quantum physics. At the end, we present open problems and share some ideas of future work on generalized (nonlinear) eigenvalue problems, handling constraints with reflection, global optimization, and nonlinear ill-posed problems.
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| 9. |
- Gulliksson, Mårten, 1963-, et al.
(författare)
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Dynamical representations of constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions
- 2021
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Ingår i: Journal of Physics A: Mathematical and Theoretical. - : IOP Publishing. - 1751-8113 .- 1751-8121. ; 54:27
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Tidskriftsartikel (refereegranskat)abstract
- We present new approaches for solving constrained multicomponent nonlinear Schrödinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose stationary solution is the solution to the time-independent nonlinear Schrödinger equation. Constraints are often considered by projection onto the constraint set, here we include them explicitly into the dynamical system. We show the applicability and efficiency of the methods on examples of relevance in modern physics applications.
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| 10. |
- Lockby, Andreas, et al.
(författare)
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Finding Stationary Solutions of PDEs with Constraints using Damped Dynamical Systems
- 2016
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Ingår i: Comsol Conference 2016.
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Konferensbidrag (refereegranskat)abstract
- The dynamical functional particle method(DFPM) is a method for solving equations, e.g. PDEs, using a second order damped dynamical system. We show how the method can be extended to include constraints both explicitly as global constraints and adding the constraints as additional damped dynamical equations. These methods are implemented in Comsol and we show numerical tests for finding the stationary solution of a nonlinear heat equation with and without constraints (global and dynamical). The results show that DFPM is a very general and robust way of solving PDEs and it should be of interest to implement the approach more generally in Comsol.
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