commit 2117859a70703cc2d8e02a6cf0835a537648a92e
parent bf003f2cd11b230948ccd480e0146ea6edf6f924
Author: Ivan Gankevich <igankevich@ya.ru>
Date: Tue, 13 Jun 2017 10:21:56 +0300
Translate NIT verification.
Diffstat:
| arma-thesis-ru.org | | | 76 | +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++--- |
| arma-thesis.org | | | 223 | +++++++++++++++++++++++++++++++++++-------------------------------------------- |
2 files changed, 173 insertions(+), 126 deletions(-)
diff --git a/arma-thesis-ru.org b/arma-thesis-ru.org
@@ -1333,6 +1333,9 @@ eqref:eq-solution-2d-full до
* Численные методы и результаты экспериментов
** Форма АКФ для разных волновых профилей
+:PROPERTIES:
+:CUSTOM_ID: sec-wave-acfs
+:END:
**** Аналитический метод.
Прямой способ нахождения АКФ, соответствующей заданному профилю морской волны,
состоит в применении теоремы Винера---Хинчина. Согласно этой теореме
@@ -1443,7 +1446,7 @@ eqref:eq-solution-2d-full до
- Перенести максимум получившейся функции в начало координат, используя свойства
тригонометрических функций для сдвига фазы.
-** Дополните льные формулы, методы и алгоритмы для модели АРСС
+** Дополнительные формулы, методы и алгоритмы для модели АРСС
:PROPERTIES:
:CUSTOM_ID: sec:arma-algorithms
:END:
@@ -1791,7 +1794,6 @@ for (i in seq(0, 4)) {
поверхности. Одинаковая степень соответствия для подъема поверхности получается
из-за того, что это характеристика поверхности (и соответствующего процесса АР
или СС), и она не зависит от типа волн.
-
*** Верификация полей потенциалов скоростей
:PROPERTIES:
:CUSTOM_ID: sec:compare-formulae
@@ -1917,6 +1919,72 @@ arma.plot_velocity(
#+RESULTS: fig-velocity-field-2d
[[file:build/large-and-small-amplitude-velocity-field-comparison-ru.pdf]]
+*** Верификация нелинейного безынерционного преобразования
+Для того чтобы измерить влияение НБП на форму результирующей взволнованной
+пверхности, было сгенерировано три реализации:
+- реализация с Гауссовым распределением (без НБП),
+- реализация с распределением на основе ряда Грама---Шарлье
+ (с\nbsp{}\(\gamma_1=2.25,\gamma_2=0.4\)), и
+- реализация с асимметричным нормальным распределением (с\nbsp{}\(\alpha=1\)).
+Начальные состояния ГПСЧ были заданы одинаковыми для всех запусков программы,
+чтобы модель АРСС выдавала одни и те же значения для каждой реализации. Было
+проведено два эксперимента: для стоячих и прогрессивных волн с АКФ, заданными
+формулами из разд.\nbsp{}[[#sec-wave-acfs]].
+
+Результаты экспериментов двояки: в то время как эксперимент показал, что
+применение НБП с распределением РГШ увеличивает крутизну волн, то же самое
+нельзя сказать об асимметричном нормальном распределении (рис.\nbsp{}[[fig-nit]]).
+Использование этого распределения приводит к взволнованной поверхности, в
+которой аппликаты всегда больше или равны нулю. Таким образом, асимметричное
+нормальное распределение не подходит для НБП. НБП увеличивает высоту и крутизну
+как прогрессивных, так и стоячих волн. Увеличение параметра либо асимметрии,
+либо эксцесса РГШ приводит в увеличению как высоты, так и крутизны волн. Ошибка
+аппроксимации АКФ (ур.\nbsp{}eqref:eq-nit-error) принимает значения от 0.20 для
+РГШ до 0.70 для асимметричного нормального распределения
+(табл.\nbsp{}[[tab-nit-error]]).
+
+#+name: fig-nit
+#+header: :width 5 :height 5 :pointsize 8
+#+begin_src R :file build/nit.pdf
+source(file.path("R", "nonlinear.R"))
+par(mfrow=c(2, 1), mar=c(4,4,4,0.5), family='serif')
+args <- list(
+ graphs=c('Гауссово', 'РГШ', 'АНР'),
+ linetypes=c('solid', 'dashed', 'dotted')
+)
+args$title <- 'Прогрессивные волны'
+arma.plot_nonlinear(file.path("build", "nit-propagating"), args)
+args$title <- 'Стоячие волны'
+arma.plot_nonlinear(file.path("build", "nit-standing"), args)
+#+end_src
+
+#+label: fig-nit
+#+caption: Срезы взволнованной поверхности с различными распределениями волновых аппликат (Гауссово, РГШ и асимметричное нормальное).
+#+RESULTS: fig-nit
+[[file:build/nit.pdf]]
+
+#+name: tab-nit-error
+#+caption: Ошибка аппроксимации АКФ (разность дисперсий) для различных распределений волновых аппликат.
+#+attr_latex: :booktabs t
+| Тип волн | Распределение | Ошибка | Колич. коэф. | Высота волн |
+|--------------+---------------+--------+--------------+-------------|
+| | | <r> | | <r> |
+| прогрессиные | Гауссово | | | 2,41 |
+| прогрессиные | РГШ | 0,20 | 2 | 2,75 |
+| прогрессиные | АНР | 0,70 | 3 | 1,37 |
+| стоячие | Гауссово | | | 1,73 |
+| стоячие | РГШ | 0,26 | 2 | 1,96 |
+| стоячие | АНР | 0,70 | 3 | 0,94 |
+
+Таким образом, единственный тестовый сценарий, который показал приемлемые
+результаты\nbsp{}--- это реализации с распределением на основе РГШ для
+прогрессивных и стоячих волн. Распределение АНР искажает взволнованную
+поверхность для обоих типов волн. Реализации с распределением на основе РГШ
+характеризуются большой ошибкой аппроксимации АКФ, что приводит к увеличению
+высоты волн. Причина большой ошибки заключается в неточность ошибка
+аппроксимации ряда Грама---Шарлье, которая не сходится для всевозможных
+функций\nbsp{}cite:wallace1958asymptotic. Несмотря на большую ошибку, изменение
+высоты волн невелико (табл.\nbsp{}[[tab-nit-error]]).
*** Нефизическая природа модели
Благодаря своей нефизической природе модель АРСС не включает в себя понятие
@@ -3468,7 +3536,9 @@ Emacs, предоставляющего вычислительное окруж
- <<<BLAS>>> :: Basic Linear Algebra Sub-programmes.
- <<<LAPACK>>> :: Linear Algebra Package.
- <<<DNS>>> :: Dynamic name resolution.
-- <<<HPC>>> :: High-performance computing.
+- <<<HPC>>> :: High-performance computing.
+- <<<РГШ>>> :: Распределение на основе ряда Грама---Шарлье.
+- <<<АНР>>> :: Асимметричное нормальное распределение.
#+begin_export latex
\input{postamble}
diff --git a/arma-thesis.org b/arma-thesis.org
@@ -740,10 +740,6 @@ fixed_point_iteration:Iteration=146, var_wn=0.0710238
WN variance = 0.0710238
Partition size = (34,13,13)
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-prfl dev_to_host_copy = 0us
-prfl fft = 67598us
-prfl second_function = 1780us
-prfl window_function = 673751us
'zeta.csv' -> 'zeta-none.csv'
Input file = /home/igankevich/workspace/arma-thesis/config/nit-propagating-gramcharlier
Output grid size = (200,40,40)
@@ -754,26 +750,26 @@ MA model = order=(20,10,10),acf.shape=(20,10,10),algorithm
Velocity potential solver name = N4arma8velocity21High_amplitude_solverIdEE
Velocity potential solver = wnmax=from (0,0) to (0,0.25) npoints (2,2),depth=12,domain=from (10,-12) to (10,3) npoints (1,128)
NIT transform = dist=gram_charlier,skewness=2.25,kurtosis=0.4,interpolation_nodes=100,interpolation_order=12,gram_charlier_order=20
-err = 0.997239
-err = 0.195639
-err = 0.204665
-err = 3.3227
-err = 4.20064
-err = 41.2299
-err = 47.0536
-err = 366.08
-err = 406.6
-err = 2335.93
-err = 2527.49
-err = 11726.2
-err = 12492.8
-err = 49079.2
-err = 51692.6
-err = 177911
-err = 185800
-err = 573975
-err = 595540
-err = 1.68124e+06
+err = 0.999006
+err = 0.171001
+err = 0.25565
+err = 0.275353
+err = 6.34477e+26
+err = 7.61373e+26
+err = 4.86898e+32
+err = 4.86898e+32
+err = 7.03444e+34
+err = 7.03444e+34
+err = 7.03444e+34
+err = 7.03444e+34
+err = 7.03444e+34
+err = 7.03444e+34
+err = 7.03444e+34
+err = 7.03447e+34
+err = 7.03447e+34
+err = 7.03807e+34
+err = 7.03808e+34
+err = 7.26744e+34
trim = 2
ACF variance = 1
fixed_point_iteration:Iteration=0, var_wn=0.541662
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WN variance = 0.0710238
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-prfl dev_to_host_copy = 0us
-prfl fft = 80945us
-prfl second_function = 1810us
-prfl window_function = 674201us
'zeta.csv' -> 'zeta-gramcharlier.csv'
Input file = /home/igankevich/workspace/arma-thesis/config/nit-propagating-skewnormal
Output grid size = (200,40,40)
@@ -940,27 +932,27 @@ MA model = order=(20,10,10),acf.shape=(20,10,10),algorithm
Velocity potential solver name = N4arma8velocity21High_amplitude_solverIdEE
Velocity potential solver = wnmax=from (0,0) to (0,0.25) npoints (2,2),depth=12,domain=from (10,-12) to (10,3) npoints (1,128)
NIT transform = dist=skew_normal,mean=0,stdev=1,alpha=1,interpolation_nodes=100,interpolation_order=12,gram_charlier_order=20
-err = 0.906446
-err = 0.711503
-err = 0.697377
-err = 1.8528
-err = 2.67306
-err = 40.4663
-err = 46.4794
-err = 364.63
-err = 404.774
-err = 2335.07
-err = 2527.48
-err = 11727.3
-err = 12493.9
-err = 49080.4
-err = 51693.7
-err = 177912
-err = 185801
-err = 573976
-err = 595541
-err = 1.68124e+06
-trim = 3
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+err = inf
+trim = 0
ACF variance = 1
fixed_point_iteration:Iteration=0, var_wn=0.541662
fixed_point_iteration:Iteration=1, var_wn=0.387581
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WN variance = 0.0710238
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-prfl dev_to_host_copy = 0us
-prfl fft = 74121us
-prfl second_function = 1833us
-prfl window_function = 693556us
'zeta.csv' -> 'zeta-skewnormal.csv'
NIT for standing waves
Input file = /home/igankevich/workspace/arma-thesis/config/nit-standing-none
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Partition size = (21,10,10)
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Zeta size = (192,32,32)
-prfl dev_to_host_copy = 0us
-prfl fft = 120726us
-prfl second_function = 2048us
-prfl window_function = 963006us
'zeta.csv' -> 'zeta-none.csv'
Input file = /home/igankevich/workspace/arma-thesis/config/nit-standing-gramcharlier
Output grid size = (200,40,40)
@@ -1146,36 +1130,32 @@ AR model = order=(7,7,7),acf.shape=(10,10,10)
Velocity potential solver name = N4arma8velocity21High_amplitude_solverIdEE
Velocity potential solver = wnmax=from (0,0) to (0,0.25) npoints (2,2),depth=12,domain=from (10,-12) to (10,3) npoints (1,128)
NIT transform = dist=gram_charlier,skewness=3.25,kurtosis=2.4,interpolation_nodes=100,interpolation_order=12,gram_charlier_order=20
-err = 0.99983
-err = 0.260174
-err = 0.292255
-err = 3.33387
-err = 4.27093
-err = 41.4813
-err = 47.2428
-err = 366.116
-err = 406.773
-err = 2336.92
-err = 2528.49
-err = 11727.9
-err = 12494.5
-err = 49081
-err = 51694.3
-err = 177913
-err = 185801
-err = 573976
-err = 595542
-err = 1.68124e+06
+err = 0.995366
+err = 0.472576
+err = 0.608796
+err = 0.61584
+err = 0.639818
+err = 12.4501
+err = 9.45159e+10
+err = 1.08018e+11
+err = 3.05905e+13
+err = 3.39774e+13
+err = 1.70414e+14
+err = 1.82818e+14
+err = 1.83166e+14
+err = 2.11267e+25
+err = 9.15985e+25
+err = 2.75794e+29
+err = 2.79745e+29
+err = 3.63216e+31
+err = 3.64474e+31
+err = 2.32998e+33
trim = 2
ACF variance = 1
WN variance = 0.000519022
Partition size = (21,10,10)
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-prfl dev_to_host_copy = 0us
-prfl fft = 88660us
-prfl second_function = 1828us
-prfl window_function = 720888us
'zeta.csv' -> 'zeta-gramcharlier.csv'
Input file = /home/igankevich/workspace/arma-thesis/config/nit-standing-skewnormal
Output grid size = (200,40,40)
@@ -1186,36 +1166,32 @@ AR model = order=(7,7,7),acf.shape=(10,10,10)
Velocity potential solver name = N4arma8velocity21High_amplitude_solverIdEE
Velocity potential solver = wnmax=from (0,0) to (0,0.25) npoints (2,2),depth=12,domain=from (10,-12) to (10,3) npoints (1,128)
NIT transform = dist=skew_normal,mean=0,stdev=1,alpha=1,interpolation_nodes=100,interpolation_order=12,gram_charlier_order=20
-err = 0.906446
-err = 0.711503
-err = 0.697377
-err = 1.8528
-err = 2.67306
-err = 40.4663
-err = 46.4794
-err = 364.63
-err = 404.774
-err = 2335.07
-err = 2527.48
-err = 11727.3
-err = 12493.9
-err = 49080.4
-err = 51693.7
-err = 177912
-err = 185801
-err = 573976
-err = 595541
-err = 1.68124e+06
-trim = 3
+err = 0.893011
+err = 0.269503
+err = 0.173329
+err = 0.0040605
+err = 2.82799e+09
+err = 4.3525e+25
+err = 4.3525e+25
+err = 7.2978e+33
+err = 2.7618e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+err = 3.06867e+44
+trim = 4
ACF variance = 1
WN variance = 0.000519022
Partition size = (21,10,10)
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Zeta size = (192,32,32)
-prfl dev_to_host_copy = 0us
-prfl fft = 86923us
-prfl second_function = 1889us
-prfl window_function = 842185us
'zeta.csv' -> 'zeta-skewnormal.csv'
#+end_example
@@ -2818,8 +2794,7 @@ arma.plot_velocity(
#+RESULTS: fig-velocity-field-2d
[[file:build/large-and-small-amplitude-velocity-field-comparison.pdf]]
-*** DONE Verification of nonlinear inertialess transformation
-CLOSED: [2017-06-12 Пн 21:05]
+*** Verification of nonlinear inertialess transformation
In order to measure the effect of NIT on the shape of the resulting wavy
surface, three realisations were generated:
- realisation with Gaussian distribution (without NIT),
@@ -2827,20 +2802,20 @@ surface, three realisations were generated:
\(\gamma_1=2.25,\gamma_2=0.4\)), and
- realisation with skew normal distribution (with \(\alpha=1\)).
The seed of PRNG was set to be the same for all progrmme executions to make ARMA
-model produce the same values for each realisation. There we two experiments: on
-for standing and one for propagating waves with ACFs similar to the ones in
+model produce the same values for each realisation. There we two experiments:
+for standing and propagating waves with ACFs given by formulae from
section\nbsp{}[[#sec-wave-acfs]].
The results of the experiments are twofold: while the experiment showed that
-applying NIT with GCS-based distribution makes wave profiles steeper, the same
-is not true for skew normal distribution (fig.\nbsp{}[[fig-nonlinear]]). Using this
+applying NIT with GCS-based distribution increases wave steepness, the same is
+not true for skew normal distribution (fig.\nbsp{}[[fig-nit]]). Using this
distribution results in wavy surface each \(z\)-coordinate of which is always
greater or equal to nought. So, skew normal distribution is unsuitable for NIT.
-NIT increases the wave height and wave steepness for both standing and
-propagating waves. Increasing either skewness or kurtosis parameter of GCS-based
-distribution increases both wave steepness and wave height. The error of ACF
+NIT increases the wave height and steepness of both standing and propagating
+waves. Increasing either skewness or kurtosis parameter of GCS-based
+distribution increases both wave height and steepness. The error of ACF
approximation (eq.\nbsp{}eqref:eq-nit-error) ranges from 0.20 for GCS-based
-distribution to 0.70 for skew normal distribution.
+distribution to 0.70 for skew normal distribution (table\nbsp{}[[tab-nit-error]]).
#+name: fig-nit
#+header: :width 5 :height 5 :pointsize 8
@@ -2848,7 +2823,7 @@ distribution to 0.70 for skew normal distribution.
source(file.path("R", "nonlinear.R"))
par(mfrow=c(2, 1), mar=c(4,4,4,0.5), family='serif')
args <- list(
- graphs=c('No transform', 'Gram---Charlier', 'Skew normal'),
+ graphs=c('Gaussian', 'Gram---Charlier', 'Skew normal'),
linetypes=c('solid', 'dashed', 'dotted')
)
args$title <- 'Propagating waves'
@@ -2863,7 +2838,8 @@ arma.plot_nonlinear(file.path("build", "nit-standing"), args)
[[file:build/nit.pdf]]
#+name: tab-nit-error
-#+caption: Errors of ACF approximations (differences in variances) for different wave elevation distributions.
+#+caption: Errors of ACF approximations (the difference of variances) for different wave elevation distributions.
+#+attr_latex: :booktabs t
| Wave type | Distribution | Error | No. of coef. | Wave height |
|-------------+--------------+-------+--------------+-------------|
| propagating | Gaussian | | | 2.41 |
@@ -2875,12 +2851,12 @@ arma.plot_nonlinear(file.path("build", "nit-standing"), args)
To summarise, the only test case that showed acceptable results is realisation
with GCS-based distribution for both standing and propagating waves. Skew normal
-distribution realisations have warped wavy surface for both types of waves.
-GCS-based realisations have large error of ACF approximation, which results in
-increase of wave height. The reason for the large error is that Gram---Charlier
-series are not accurate as they do not converge for all possible
+distribution warps wavy surface for both types of waves. GCS-based realisations
+have large error of ACF approximation, which results in increase of wave height.
+The reason for the large error is that approximations Gram---Charlier series are
+not accurate as they do not converge for all possible
functions\nbsp{}cite:wallace1958asymptotic. Despite the large error, the change
-in wave height is small.
+in wave height is small (table\nbsp{}[[tab-nit-error]]).
**** Wave height :noexport:
:PROPERTIES:
@@ -4307,6 +4283,7 @@ Basic Research (projects no.\nbsp{}\mbox{16-07-01111}, \mbox{16-07-00886},
- <<<DNS>>> :: Dynamic name resolution.
- <<<HPC>>> :: High-performance computing.
- <<<GCS>>> :: Gram---Charlier series.
+- <<<SN>>> :: Skew normal distribution.
- Master/slave node ::
- Principal/subordinate kernel ::
