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Автор: Grafiati

Опубліковано: 6 вересня 2023

Оновлено: 7 вересня 2023

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1

Cipra, Tomáš. "Some problems of exponential smoothing." Applications of Mathematics 34, no. 2 (1989): 161–69. http://dx.doi.org/10.21136/am.1989.104344.

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2

Asmuss, Svetlana, and Natalja Budkina. "ON SOME GENERALIZATION OF SMOOTHING PROBLEMS." Mathematical Modelling and Analysis 20, no. 3 (June 2, 2015): 311–28. http://dx.doi.org/10.3846/13926292.2015.1048756.

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Анотація:

The paper deals with the generalized smoothing problem in abstract Hilbert spaces. This generalized problem involves particular cases such as the interpolating problem, the smoothing problem with weights, the smoothing problem with obstacles, the problem on splines in convex sets and others. The theorem on the existence and characterization of a solution of the generalized problem is proved. It is shown how the theorem gives already known theorems in special cases as well as some new results.

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3

Yin, Hongxia. "An Adaptive Smoothing Method for Continuous Minimax Problems." Asia-Pacific Journal of Operational Research 32, no. 01 (February 2015): 1540001. http://dx.doi.org/10.1142/s0217595915400011.

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Анотація:

A simple and implementable two-loop smoothing method for semi-infinite minimax problem is given with the discretization parameter and the smoothing parameter being updated adaptively. We prove the global convergence of the algorithm when the steepest descent method or a BFGS type quasi-Newton method is applied to the smooth subproblems. The strategy for updating the smoothing parameter can not only guarantee the convergence of the algorithm but also considerably reduce the ill-conditioning caused by increasing the value of the smoothing parameter. Numerical tests show that the algorithm is robust and effective.

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4

Asmuss, Svetlana, and Natalia Budkina. "ON SMOOTHING PROBLEMS WITH ONE ADDITIONAL EQUALITY CONDITION." Mathematical Modelling and Analysis 14, no. 2 (June 30, 2009): 159–68. http://dx.doi.org/10.3846/1392-6292.2009.14.159-168.

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Анотація:

Two problems of approximation in Hilbert spaces are considered with one additional equality condition: the smoothing problem with a weight and the smoothing problem with an obstacle. This condition is a generalization of the equality, which appears in the problem of approximation of a histogram in a natural way. We characterize the solutions of these smoothing problems and investigate the connection between them.

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5

Zhou, Zhengyong, and Qi Yang. "An Active Set Smoothing Method for Solving Unconstrained Minimax Problems." Mathematical Problems in Engineering 2020 (June 24, 2020): 1–25. http://dx.doi.org/10.1155/2020/9108150.

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Анотація:

In this paper, an active set smoothing function based on the plus function is constructed for the maximum function. The active set strategy used in the smoothing function reduces the number of gradients and Hessians evaluations of the component functions in the optimization. Combing the active set smoothing function, a simple adjustment rule for the smoothing parameters, and an unconstrained minimization method, an active set smoothing method is proposed for solving unconstrained minimax problems. The active set smoothing function is continuously differentiable, and its gradient is locally Lipschitz continuous and strongly semismooth. Under the boundedness assumption on the level set of the objective function, the convergence of the proposed method is established. Numerical experiments show that the proposed method is feasible and efficient, particularly for the minimax problems with very many component functions.

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6

Yang, X. Q. "Smoothing approximations to nonsmooth optimization problems." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 36, no. 3 (January 1995): 274–85. http://dx.doi.org/10.1017/s0334270000010444.

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AbstractWe study certain types of composite nonsmooth minimization problems by introducing a general smooth approximation method. Under various conditions we derive bounds on error estimates of the functional values of original objective function at an approximate optimal solution and at the optimal solution. Finally, we obtain second-order necessary optimality conditions for the smooth approximation prob lems using a recently introduced generalized second-order directional derivative.

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7

Tsar'kov, I. G. "Linear methods in some smoothing problems." Mathematical Notes 56, no. 6 (December 1994): 1255–70. http://dx.doi.org/10.1007/bf02266694.

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8

Haddou, Mounir, and Patrick Maheux. "Smoothing Methods for Nonlinear Complementarity Problems." Journal of Optimization Theory and Applications 160, no. 3 (September 12, 2013): 711–29. http://dx.doi.org/10.1007/s10957-013-0398-1.

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9

Zhu, Jianguang, and Binbin Hao. "A new smoothing method for solving nonlinear complementarity problems." Open Mathematics 17, no. 1 (March 10, 2019): 104–19. http://dx.doi.org/10.1515/math-2019-0011.

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Анотація:

Abstract In this paper, a new improved smoothing Newton algorithm for the nonlinear complementarity problem was proposed. This method has two-fold advantages. First, compared with the classical smoothing Newton method, our proposed method needn’t nonsingular of the smoothing approximation function; second, the method also inherits the advantage of the classical smoothing Newton method, it only needs to solve one linear system of equations at each iteration. Without the need of strict complementarity conditions and the assumption of P0 property, we get the global and local quadratic convergence properties of the proposed method. Numerical experiments show that the efficiency of the proposed method.

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10

Wang, Jian, LingLing Shen, LeSheng Jin, and Gang Qian. "Age Sequence Recursive Models for Long Time Evaluation Problems." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 26, no. 02 (April 2018): 299–325. http://dx.doi.org/10.1142/s0218488518500162.

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The evaluation models for long time historical data is important in many applications. In this study, based on Age measure defined by Yager, we propose the definitions of Age Sequence and Age Series. Then, we provide a Generalized Recursive Smoothing method. Some classical smoothing models in evaluation problems can be seen as special cases of Generalized Recursive Smoothing method. In order to obtain more reasonable and effective aggregation results of the historical data, we propose some different Age Sequences, e.g., the Generalized Harmonic Age Sequence and p Age Sequence, which theoretically can provide infinite more recursive smoothing methods satisfying different preferences of decision makers.

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11

Qi, L., and D. Sun. "Smoothing Functions and Smoothing Newton Method for Complementarity and Variational Inequality Problems." Journal of Optimization Theory and Applications 113, no. 1 (April 2002): 121–47. http://dx.doi.org/10.1023/a:1014861331301.

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12

Liu, Xiangjing, and Jianke Zhang. "Strong Convergence of a Two-Step Modified Newton Method for Weighted Complementarity Problems." Axioms 12, no. 8 (July 28, 2023): 742. http://dx.doi.org/10.3390/axioms12080742.

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Анотація:

This paper focuses on the weighted complementarity problem (WCP), which is widely used in the fields of economics, sciences and engineering. Not least because of its local superlinear convergence rate, smoothing Newton methods have widespread application in solving various optimization problems. A two-step smoothing Newton method with strong convergence is proposed. With a smoothing complementary function, the WCP is reformulated as a smoothing set of equations and solved by the proposed two-step smoothing Newton method. In each iteration, the new method computes the Newton equation twice, but using the same Jacobian, which can avoid consuming a lot of time in the calculation. To ensure the global convergence, a derivative-free line search rule is inserted. At the same time, we develop a different term in the solution of the smoothing Newton equation, which guarantees the local strong convergence. Under appropriate conditions, the algorithm has at least quadratic or even cubic local convergence. Numerical experiments indicate the stability and effectiveness of the new method. Moreover, compared to the general smoothing Newton method, the two-step smoothing Newton method can significantly improve the computational efficiency without increasing the computational cost.

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13

Bagirov, A. M., A. Al Nuaimat, and N. Sultanova. "Hyperbolic smoothing function method for minimax problems." Optimization 62, no. 6 (June 2013): 759–82. http://dx.doi.org/10.1080/02331934.2012.675335.

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14

Grimble, M. J. "H∞ inferential filtering, prediction and smoothing problems." Signal Processing 60, no. 3 (August 1997): 289–304. http://dx.doi.org/10.1016/s0165-1684(97)00079-0.

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15

Schneider, Johannes, Markus Dankesreiter, Werner Fettes, Ingo Morgenstern, Martin Schmid, and Johannes Maria Singer. "Search-space smoothing for combinatorial optimization problems." Physica A: Statistical Mechanics and its Applications 243, no. 1-2 (September 1997): 77–112. http://dx.doi.org/10.1016/s0378-4371(97)00207-0.

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16

de Wiljes, Jana, Sahani Pathiraja, and Sebastian Reich. "Ensemble Transform Algorithms for Nonlinear Smoothing Problems." SIAM Journal on Scientific Computing 42, no. 1 (January 2020): A87—A114. http://dx.doi.org/10.1137/19m1239544.

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17

Kanzow, Christian, and Heiko Pieper. "Jacobian Smoothing Methods for Nonlinear Complementarity Problems." SIAM Journal on Optimization 9, no. 2 (January 1999): 342–73. http://dx.doi.org/10.1137/s1052623497328781.

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18

Slaby, John. "Application of smoothing methods to flash problems." AIChE Journal 50, no. 4 (2004): 883–86. http://dx.doi.org/10.1002/aic.10085.

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19

Asmuss, S., N. Budkina, and P. Oja. "ON SMOOTHING PROBLEMS WITH WEIGHTS AND OBSTACLES." Proceedings of the Estonian Academy of Sciences. Physics. Mathematics 46, no. 4 (1997): 262. http://dx.doi.org/10.3176/phys.math.1997.4.04.

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20

Zheng, Xiuyun, and Jiarong Shi. "Smoothing Newton method for generalized complementarity problems based on a new smoothing function." Applied Mathematics and Computation 231 (March 2014): 160–68. http://dx.doi.org/10.1016/j.amc.2013.12.170.

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21

CUI, X. Y., G. Y. LI, and G. R. LIU. "AN EXPLICIT SMOOTHED FINITE ELEMENT METHOD (SFEM) FOR ELASTIC DYNAMIC PROBLEMS." International Journal of Computational Methods 10, no. 01 (February 2013): 1340002. http://dx.doi.org/10.1142/s0219876213400021.

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Анотація:

This paper presents an explicit smoothed finite element method (SFEM) for elastic dynamic problems. The central difference method for time integration will be used in presented formulations. A simple but general contact searching algorithm is used to treat the contact interface and an algorithm for the contact force is presented. In present method, the problem domain is first divided into elements as in the finite element method (FEM), and the elements are further subdivided into several smoothing cells. Cell-wise strain smoothing operations are used to obtain the stresses, which are constants in each smoothing cells. Area integration over the smoothing cell becomes line integration along its edges, and no gradient of shape functions is involved in computing the field gradients nor in forming the internal force. No mapping or coordinate transformation is necessary so that the element can be used effectively for large deformation problems. Through several examples, the simplicity, efficiency and reliability of the smoothed finite element method are demonstrated.

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22

Yang, X. Q. "A comparative study of smoothing approximations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 38, no. 2 (October 1996): 194–200. http://dx.doi.org/10.1017/s0334270000000588.

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AbstractIt is known that many optimization problems can be reformulated as composite optimization problems. In this paper error analyses are provided for two kinds of smoothing approximation methods of a unconstrained composite nondifferentiable optimization problem. Computational results are presented for nondifferentiable optimization problems by using these smoothing approximation methods. Comparisons are made among these methods.

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23

Bello, M., A. Willsky, B. Levy, and D. Castanon. "Smoothing error dynamics and their use in the solution of smoothing and mapping problems." IEEE Transactions on Information Theory 32, no. 4 (July 1986): 483–95. http://dx.doi.org/10.1109/tit.1986.1057207.

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24

Hinrichs, Richard N. "Endpoint problems in smoothing raw kinematic data: An evaluation of four popular smoothing techniques." Journal of Biomechanics 25, no. 6 (June 1992): 682. http://dx.doi.org/10.1016/0021-9290(92)90201-b.

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25

Zhu, Jianguang, and Binbin Hao. "A new class of smoothing functions and a smoothing Newton method for complementarity problems." Optimization Letters 7, no. 3 (January 4, 2012): 481–97. http://dx.doi.org/10.1007/s11590-011-0432-x.

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26

OSMANI, El Hassene, Mounir Haddou, Naceurdine Bensalem, and Lina Abdallah. "A new smoothing method for nonlinear complementarity problems involving P0-function." Statistics, Optimization & Information Computing 10, no. 4 (September 29, 2022): 1267–92. http://dx.doi.org/10.19139/soic-2310-5070-1493.

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Анотація:

In this paper, we present a family of smoothing methods to solve nonlinear complementarity problems (NCPs) involving P0-function. Several regularization or approximation techniques like Fisher-Burmeister’s method, interior-point methods (IPMs) approaches, or smoothing methods already exist. All the corresponding methods solve a sequence of nonlinear systems of equations and depend on parameters that are difficult to drive to zero. The main novelty of our approach is to consider the smoothing parameters as variables that converge by themselves to zero. We do not need any complicated updating strategy, and then obtain nonparametric algorithms. We prove some global and local convergence results and present several numerical experiments, comparisons, and applications that show the efficiency of our approach.

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27

Feng, Ning, Zhi Yuan Tian, and Xin Lei Qu. "A Smoothing Newton Method for Nonlinear Complementarity Problems." Applied Mechanics and Materials 475-476 (December 2013): 1090–93. http://dx.doi.org/10.4028/www.scientific.net/amm.475-476.1090.

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A new FB-function based on the P0 function is given in this paper. The nonlinear complementarity problem is reformulated to solve equivalent equations based on the FB-function. A modified smooth Newton method is proposed for nonlinear complementarity problem. Under mild conditions, the global convergence of the algorithm is proved. The numerical experiment shows that the algorithm is potentially efficient.

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28

Yang, X. Q., Z. Q. Meng, X. X. Huang, and G. T. Y. Pong. "Smoothing Nonlinear Penalty Functions for Constrained Optimization Problems." Numerical Functional Analysis and Optimization 24, no. 3-4 (January 8, 2003): 351–64. http://dx.doi.org/10.1081/nfa-120022928.

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29

Kersey, Scott N. "On the problems of smoothing and near-interpolation." Mathematics of Computation 72, no. 244 (May 1, 2003): 1873–86. http://dx.doi.org/10.1090/s0025-5718-03-01523-0.

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30

Sun, Defeng, and Liqun Qi. "Solving variational inequality problems via smoothing-nonsmooth reformulations." Journal of Computational and Applied Mathematics 129, no. 1-2 (April 2001): 37–62. http://dx.doi.org/10.1016/s0377-0427(00)00541-0.

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31

Leetma, Evely, and Peeter Oja. "Connection Between Smoothing Problems with Obstacles and Weights." Numerical Functional Analysis and Optimization 35, no. 11 (July 31, 2014): 1435–58. http://dx.doi.org/10.1080/01630563.2014.895761.

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32

Hong-wu, Zhang, He Su-yan, and Li Xing-si. "Non-interior smoothing algorithm for frictional contact problems." Applied Mathematics and Mechanics 25, no. 1 (January 2004): 47–58. http://dx.doi.org/10.1007/bf02437293.

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33

Fukushima, Masao, Zhi-Quan Luo, and Paul Tseng. "Smoothing Functions for Second-Order-Cone Complementarity Problems." SIAM Journal on Optimization 12, no. 2 (January 2002): 436–60. http://dx.doi.org/10.1137/s1052623400380365.

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34

Polak, E., J. O. Royset, and R. S. Womersley. "Algorithms with Adaptive Smoothing for Finite Minimax Problems." Journal of Optimization Theory and Applications 119, no. 3 (December 2003): 459–84. http://dx.doi.org/10.1023/b:jota.0000006685.60019.3e.

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35

Tang, Jingyong, Li Dong, Jinchuan Zhou, and Liang Fang. "A smoothing Newton method for nonlinear complementarity problems." Computational and Applied Mathematics 32, no. 1 (March 26, 2013): 107–18. http://dx.doi.org/10.1007/s40314-013-0015-9.

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36

Hackbusch, Wolfgang, and Thomas Probst. "Downwind Gauß-Seidel Smoothing for Convection Dominated Problems." Numerical Linear Algebra with Applications 4, no. 2 (March 1997): 85–102. http://dx.doi.org/10.1002/(sici)1099-1506(199703/04)4:2<85::aid-nla100>3.0.co;2-2.

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37

Huang, Zheng-Hai, and Tie Ni. "Smoothing algorithms for complementarity problems over symmetric cones." Computational Optimization and Applications 45, no. 3 (April 9, 2008): 557–79. http://dx.doi.org/10.1007/s10589-008-9180-y.

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38

Zhang, Liping, Soon-Yi Wu, and Tingran Gao. "Improved smoothing Newton methods for nonlinear complementarity problems." Applied Mathematics and Computation 215, no. 1 (September 2009): 324–32. http://dx.doi.org/10.1016/j.amc.2009.04.088.

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39

Chen, Pin-Bo, Peng Zhang, Xide Zhu, and Gui-Hua Lin. "Modified Jacobian smoothing method for nonsmooth complementarity problems." Computational Optimization and Applications 75, no. 1 (October 10, 2019): 207–35. http://dx.doi.org/10.1007/s10589-019-00136-3.

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40

Shiojima, Takeo, and Yoji Shimazaki. "A pressure-smoothing scheme for incompressible flow problems." International Journal for Numerical Methods in Fluids 9, no. 5 (May 1989): 557–67. http://dx.doi.org/10.1002/fld.1650090506.

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41

Gopal, Vipin, and Lorenz T. Biegler. "Smoothing methods for complementarity problems in process engineering." AIChE Journal 45, no. 7 (July 1999): 1535–47. http://dx.doi.org/10.1002/aic.690450715.

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42

Wolfgang, Hackbusch, and Probst Thomas. "Downwind Gauß-Seidel-Smoothing for Convection-Dominated Problems." ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik 78, S3 (1998): 933–34. http://dx.doi.org/10.1002/zamm.19980781539.

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43

Qi, Hou-Duo, Li-Zhi Liao, and Zheng-Hua Lin. "Regularized Smoothing Approximations to Vertical Nonlinear Complementarity Problems." Journal of Mathematical Analysis and Applications 230, no. 1 (February 1999): 261–76. http://dx.doi.org/10.1006/jmaa.1998.6205.

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44

Marpaung, Kevin Togos Parningotan, Agus Rusgiyono, and Yuciana Wilandari. "PERBANDINGAN METODE HOLT WINTER’S EXPONENTIAL SMOOTHING DAN EXTREME LEARNING MACHINE UNTUK PERAMALAN JUMLAH BARANG YANG DIMUAT PADA PENERBANGAN DOMESTIK DI BANDARA UTAMA SOEKARNO HATTA." Jurnal Gaussian 11, no. 3 (November 18, 2022): 439–46. http://dx.doi.org/10.14710/j.gauss.11.3.439-446.

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Анотація:

The loading of goods carried out at the airport is an essential part of the transporting goods system. In this regard, it is necessary to have a prediction to make the right policy or to solve the problems that occur. Holt Winter's Exponential Smoothing, which one of the classic methods of analyzing time series data, and Extreme Learning Machine which is part of the artificial neural network method, are methods that can be used as a tool for forecasting problems. Holt Winter's Exponential Smoothing uses three times of smoothing on related data, which are level smoothing, trend smoothing, and season smoothing, while Extreme Learning Machine goes through three stages, which are normalization, training, and denormalization. In measuring the error rate in related forecasting, the symmetric Mean Absolute Percentage Error (sMAPE) value is used. The Holt Winter's Exponential Smoothing method Additive model produces a sMAPE value of 26.14%; while the Multiplicative model with the same method resulted in the sMAPE value of 25.69%. For the Extreme Learning Machine method, the sMAPE value is 49.85%. Based on the accuracy test using the sMAPE value, Holt Winter's Exponential Smoothing method Multiplicative model is the better method than Extreme Learning Machine

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45

Halik, Azhar, Rahmatjan Imin, Mamtimin Geni, Afang Jin, and Yangyang Mou. "Numerical Modeling for Discrete Multibody Interaction and Multifeild Coupling Dynamics Using the SPH Method." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/205976.

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Анотація:

Discrete multibody interaction and contact problems and the multiphase interactions such as the sand particles airflow interactions by Aeolian sand transport in the desert are modeled by using the different kernel smoothing lengths in SPH method. Each particle defines a particular kernel smoothing length such as larger smoothing length which is used to calculate continuous homogenous body. Some special smoothing lengths are used to approximate interaction between the discrete particles or objects in contact problems and in different field coupling problem. By introducing the Single Particle Model (SPM) and the Multiparticle Model (MPM), the velocity exchanging phenomena are discussed by using different elastic modules. Some characteristics of the SPM and MPM are evaluated. The results show that the new SPH method can effectively solve different discrete multibody correct contact and multiphase mutual interference problems. Finally, the new SPH numerical computation and simulation process are verified.

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46

Dong, Li, Bo Yu, and Yu Xiao. "A Spline Smoothing Newton Method for Semi-Infinite Minimax Problems." Journal of Applied Mathematics 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/852074.

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Анотація:

Based on discretization methods for solving semi-infinite programming problems, this paper presents a spline smoothing Newton method for semi-infinite minimax problems. The spline smoothing technique uses a smooth cubic spline instead of max function and only few components in the max function are computed; that is, it introduces an active set technique, so it is more efficient for solving large-scale minimax problems arising from the discretization of semi-infinite minimax problems. Numerical tests show that the new method is very efficient.

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47

Zhu, Xingwen, and Lixiang Zhang. "A Smoothing Process of Multicolor Relaxation for Solving Partial Differential Equation by Multigrid Method." Mathematical Problems in Engineering 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/490156.

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Анотація:

This paper is concerned with a novel methodology of smoothing analysis process of multicolor point relaxation by multigrid method for solving elliptically partial differential equations (PDEs). The objective was firstly focused on the two-color relaxation technique on the local Fourier analysis (LFA) and then generalized to the multicolor problem. As a key starting point of the problems under consideration, the mathematical constitutions among Fourier modes with various frequencies were constructed as a base to expand two-color to multicolor smoothing analyses. Two different invariant subspaces based on the 2h-harmonics for the two-color relaxation with two and four Fourier modes were constructed and successfully used in smoothing analysis process of Poisson’s equation for the two-color point Jacobi relaxation. Finally, the two-color smoothing analysis was generalized to the multicolor smoothing analysis problems by multigrid method based on the invariant subspaces constructed.

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48

Liu, Wenli, Xiaoni Chi, Qili Yang, and Ranran Cui. "Jacobian Consistency of a Smoothing Function for the Weighted Second-Order Cone Complementarity Problem." Mathematical Problems in Engineering 2021 (January 23, 2021): 1–11. http://dx.doi.org/10.1155/2021/6674520.

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In this paper, a weighted second-order cone (SOC) complementarity function and its smoothing function are presented. Then, we derive the computable formula for the Jacobian of the smoothing function and show its Jacobian consistency. Also, we estimate the distance between the subgradient of the weighted SOC complementarity function and the gradient of its smoothing function. These results will be critical to achieve the rapid convergence of smoothing methods for weighted SOC complementarity problems.

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49

Chi, Xiaoni, Zhongping Wan, and Zijun Hao. "The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/965931.

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Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of its smoothing function, which will help to adjust a parameter appropriately in smoothing methods.

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50

Bube, Kenneth P., and Robert T. Langan. "A continuation approach to regularization of ill-posed problems with application to crosswell-traveltime tomography." GEOPHYSICS 73, no. 5 (September 2008): VE337—VE351. http://dx.doi.org/10.1190/1.2969460.

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In most geometries in which seismic-traveltime tomography is applied (e.g., crosswell, surface-reflection, and VSP), determination of the slowness field using only traveltimes is not a well-conditioned problem. Nonuniqueness is common. Even when the slowness field is uniquely determined, small changes in measured traveltimes can cause large errors in the computed slowness field. A priori information often is available — well logs, initial rough estimates of slowness from structural geology, etc. — and can be incorporated into a traveltime-inversion algorithm by using penalty terms. To further regularize the problem, smoothing constraints also can be incorporated using penalty terms by penalizing derivatives of the slowness field. What weights to use on the penalty terms is a major decision, particularly the smoothing-penalty weights. We use a continuation approach in selecting the smoothing-penalty weights. Instead of using fixed smoothing-penalty weights, we decrease them step by step, using the slowness model computed with the previous, larger weights as the initial slowness model for the next step with the new, smaller weights. This continuation approach can solve synthetic problems more accurately than does one that uses fixed smoothing-penalty weights, and it appears to yield more features of interest in real-data applications of traveltime tomography. We have formulated guidelines for making the many choices needed to implement this continuation strategy effectively and have developed specific choices for crosswell-traveltime tomography.

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