Space-Dependent Sobolev Gradients as a Regularization for Inverse Radiative Transfer Problems

Diffuse optical tomography problems rely on the solution of an optimization problem for which the dimension of the parameter space is usually large. Thus, gradient-type optimizers are likely to be used, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, along with the adjoint-state method to compute the cost function gradient. Usually, theL2-inner product is chosen within the extraction procedure (i.e., in the definition of the relationship between the cost function gradient and the directional derivative of the cost function) while alternative inner products that act as regularization can be used. This paper presents some results based on space-dependent Sobolev inner products and shows that this method acts as an efficient low-pass filter on the cost function gradient. Numerical results indicate that the use of Sobolev gradients can be particularly attractive in the context of inverse problems, particularly because of the simplicity of this regularization, since a single additional diffusion equation is to be solved, and also because the quality of the solution is smoothly varying with respect to the regularization parameter.

Download Full-text

An Overlooked Issue of Variational Data Assimilation

Monthly Weather Review ◽

10.1175/mwr-d-14-00404.1 ◽

2015 ◽

Vol 143 (10) ◽

pp. 3925-3930 ◽

Cited By ~ 7

Author(s):

Benjamin Ménétrier ◽

Thomas Auligné

Keyword(s):

Data Assimilation ◽

Cost Function ◽

Control Variable ◽

Variational Data Assimilation ◽

Inner Product ◽

Background Error ◽

Error Covariance ◽

Definition Of ◽

Linear Unbiased Estimate ◽

The Cost

Abstract The control variable transform (CVT) is a keystone of variational data assimilation. In publications using such a technique, the background term of the transformed cost function is defined as a canonical inner product of the transformed control variable with itself. However, it is shown in this paper that this practical definition of the cost function is not correct if the CVT uses a square root of the background error covariance matrix that is not square. Fortunately, it is then shown that there is a manifold of the control space for which this flaw has no impact, and that most minimizers used in practice precisely work in this manifold. It is also shown that both correct and practical transformed cost functions have the same minimum. This explains more rigorously why the CVT is working in practice. The case of a singular is finally detailed, showing that the practical cost function still reaches the best linear unbiased estimate (BLUE).

Download Full-text

Evolutionarily stable mutation rate in a periodically changing environment.

Genetics ◽

10.1093/genetics/121.1.163 ◽

1989 ◽

Vol 121 (1) ◽

pp. 163-174 ◽

Cited By ~ 13

Author(s):

K Ishii ◽

H Matsuda ◽

Y Iwasa ◽

A Sasaki

Keyword(s):

Mutation Rate ◽

Selection Pressure ◽

Sine Wave ◽

Fluctuating Selection ◽

Pass Filter ◽

Low Pass Filter ◽

Low Pass ◽

Haploid Population ◽

The Cost ◽

Evolutionarily Stable

Abstract Evolution of mutation rate controlled by a neutral modifier is studied for a locus with two alleles under temporally fluctuating selection pressure. A general formula is derived to calculate the evolutionarily stable mutation rate mu(ess) in an infinitely large haploid population, and following results are obtained. (I) For any fluctuation, periodic or random: (1) if the recombination rate r per generation between the modifier and the main locus is 0, mu(ess) is the same as the optimal mutation rate mu(op) which maximizes the long-term geometric average of population fitness; and (2) for any r, if the strength s of selection per generation is very large, mu(ess) is equal to the reciprocal of the average number T of generations (duration time) during which one allele is persistently favored than the other. (II) For a periodic fluctuation in the limit of small s and r, mu(ess)T is a function of sT and rT with properties: (1) for a given sT, mu(ess)T decreases with increasing rT; (2) for sT </= 1, mu(ess)T is almost independent of sT, and depends on rT as mu(ess)T & 1.6 for rT < 1 and mu(ess)T & 6/rT for rT > 1; and (3) for sT >/= 1, and for a given rT, mu(ess)T decreases with increasing sT to a certain minimum less than 1, and then increases to 1 asymptotically in the limit of large sT. (III) For a fluctuation consisting of multiple Fourier components (i.e., sine wave components), the component with the longest period is the most effective in determining mu(ess) (low pass filter effect). (IV) When the cost c of preventing mutation is positive, the modifier is nonneutral, and mu(ess) becomes larger than in the case of neutral modifier under the same selection pressure acting at the main locus. The value of c which makes mu(ess) equal to mu(op) of the neutral modifier case is calculated. It is argued that this value gives a critical cost such that, so long as the actual cost exceeds this value, the evolution rate at the main locus must be smaller than its mutation rate mu(ess).

Download Full-text

Sobolev Gradients for the Möbius Energy

Archive for Rational Mechanics and Analysis ◽

10.1007/s00205-021-01680-1 ◽

2021 ◽

Author(s):

Philipp Reiter ◽

Henrik Schumacher

Keyword(s):

Optimization Methods ◽

Inner Product ◽

Stationary Points ◽

Inner Products ◽

Sobolev Gradients ◽

Gradient Based ◽

Standard Techniques ◽

Sobolev Inner Products

AbstractAiming to optimize the shape of closed embedded curves within prescribed isotopy classes, we use a gradient-based approach to approximate stationary points of the Möbius energy. The gradients are computed with respect to Sobolev inner products similar to the $$W^{3/2,2}$$ W 3 / 2 , 2 -inner product. This leads to optimization methods that are significantly more efficient and robust than standard techniques based on $$L^2$$ L 2 -gradients.

Download Full-text

Design Optimization of Low-Pass Filter with Exponential Transmission Lines Using Differential Evolutionary Algorithm

Journal of Intelligent Systems with Applications ◽

10.54856/jiswa.201812030 ◽

2018 ◽

pp. 93-97

Author(s):

Aysu Belen ◽

Filiz Gunes ◽

Mehmet Ali Belen ◽

Mahdi Ranjbar Moule

Keyword(s):

Evolutionary Algorithm ◽

Design Optimization ◽

Transmission Lines ◽

Heuristic Algorithms ◽

Pass Filter ◽

Low Pass Filter ◽

Simpler Algorithm ◽

Differential Evolutionary Algorithm ◽

Low Pass ◽

The Cost

In this work, Differential Evolutionary Algorithm (DEA), a novel and commonly used optimization algorithm in engineering problems, is applied for the design optimization of a low pass filter with exponential transmission lines. Basically the principle of DEA is similar to genetic algorithms techniques, however compare to meta-heuristic algorithms it has a much simpler algorithm structure and higher stability compare to its counterpart algorithms. For design optimization of low pass filter with exponential transmission lines, each of the transmission lines width and variation with its length are taken as an optimization variable for DEA. Firstly the unit microstrip transmission line model is chosen. After that, the optimal value of widths and lengths are obtained via DEA. The cost function of the DEA is based on the calculation of scattering parameters of candidate's solutions crossed the requested frequency bandwidth.

Download Full-text

Accurate spatial directional derivative: Antialiasing Kirchhoff migration

Geophysics ◽

10.1190/1.1759467 ◽

2004 ◽

Vol 69 (3) ◽

pp. 811-816 ◽

Cited By ~ 1

Author(s):

Chengshu Wang

Keyword(s):

Directional Derivative ◽

Tangent Plane ◽

Image Point ◽

Pass Filter ◽

Low Pass Filter ◽

Kirchhoff Migration ◽

Low Pass ◽

Plane Direction ◽

Correct Direction ◽

True Direction

Applying a low‐pass filter to each input trace before performing migration is a common antialiasing method used in Kirchhoff 3D migration. The spatial directional derivative that gives the tangent plane direction of the reflector is key in determining the antialiasing frequency limits. In this paper, we present the formula of the traveltime gradient for the diffraction hyperboloid. We also deduce the formula of the vertical traveltime gradient for the reflection ellipsoid, which can be used to construct the antialiasing low‐pass filter after migration. The migration stretch is easily computed from the ratio of gradient expressions obtained for the reflection ellipsoid and the diffraction hyperboloid. The gradient formula includes magnitude and direction. The correct direction of the gradient function is from an image point to the dip‐moveout image point (ρ* direction) on the surface. The gradient is not in the direction from the image point to the midpoint (ρ direction) as has been previously concluded. The correct spatial directional derivative along ρ is given, and it is slightly less than the gradient norm. We also prove that the true direction of the tangent plane on the reflection ellipsoid can be determined by the gradient. We introduce the directional derivative value along ρ that can be approximately applied to calculate the high‐frequency limit. Compared with the maximum directional derivative value (gradient norm), the directional derivative is relatively accurate and is much easier to calculate for use in antialiasing. The different results from the new method and the former method are compared. Numerical examples compare the performance of the different formulae.

Download Full-text

Structural Damage Detection Using Inner Product Vector and Low Pass Filter Technique

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.204-208.2942 ◽

2012 ◽

Vol 204-208 ◽

pp. 2942-2946 ◽

Cited By ~ 3

Author(s):

Le Wang ◽

Zhi Chun Yang

Keyword(s):

Damage Detection ◽

White Noise ◽

Structural Damage ◽

Inner Product ◽

Pass Filter ◽

Low Pass Filter ◽

Filter Technique ◽

White Noise Excitation ◽

Low Pass ◽

Band Pass

The inner product vector (IPV) was proposed in our previous research as a damage detection algorithm which uses cross correlation functions between vibration responses under band pass white noise excitation. It was verified that the IPV of a structure is a weighted summation of the mode shapes of the structure and the weighing factors for each modes are different. Structural damage detection examples in our previous research shown that the damage location can be correctly detected if the bandwidth of the band pass white noise excitation only covers the fundamental natural frequency of the structure. However, the bandwidth of the band pass white noise excitation may cover first several natural frequencies of the structure in practice. This paper just investigates the damage detection method using IPV under the situation when the bandwidth of the band pass white noise excitation may cover first several natural frequencies of the structure, i.e. the hybrid method using IPV and low pass filter technique. Firstly, the theory of IPVs and the damage detection method using IPV are reviewed shortly. Then, the hybrid method using IPV and low pass filter technique is introduced. Finally, experimental damage detection example of a shear frame structure is presented to illustrate the method proposed in this paper.

Download Full-text