From 5cfe15ba6e20d6a1d5ad7a50ca5d79a08ba969a3 Mon Sep 17 00:00:00 2001 From: StephanDeHoop Date: Thu, 16 Jan 2025 08:33:07 +0000 Subject: [PATCH 1/5] add difference everest ert data models in docs --- docs/everest/development.rst | 67 ++++++++++++++++++++++ docs/everest/images/Everest_vs_Ert_01.png | Bin 0 -> 230386 bytes docs/everest/images/Everest_vs_Ert_02.png | Bin 0 -> 108057 bytes 3 files changed, 67 insertions(+) create mode 100644 docs/everest/images/Everest_vs_Ert_01.png create mode 100644 docs/everest/images/Everest_vs_Ert_02.png diff --git a/docs/everest/development.rst b/docs/everest/development.rst index 9247b4424f7..cc93a9119a3 100644 --- a/docs/everest/development.rst +++ b/docs/everest/development.rst @@ -18,6 +18,8 @@ client component. :width: 700px :alt: Everest architecture + Everest architecture + Every time an optimization instance is ran by a user, the client component of the application spawns an instance of the server component, which is started either on a cluster node using LSF (when the `queue_system` is defined to be *lsf*) or on the @@ -52,3 +54,68 @@ long as the optimization process is running. * - POST - '/stop' - Signal everest optimization run termination. It will be called by the client when the optimization needs to be terminated in the middle of the run + + +Everest vs. Ert data models +=========================== +Everest uses Ert for running an experiment, but instead of submitting an `ensemble` (i.e., as in Ert) to the queue we submit +a `batch` in Everest. `Batches` are in principle very similar to `ensembles`, but they have some key differences. +The purpose of this section is to explain these key differences from a developer point-of-view. + +In Ert, an `ensemble` contains a number of `realizations` which are a set of `model parameters` which Ert attempts to history match to some data. +These `model parameters` are sampled from a certain distribution after creating the `ensemble`. In Everest we are not history matching our `model parameters`, +but instead try to find an optimal strategy on how to operate the particular model(s) in order to maximize some objective (i.e., we optimize for a set of `controls`). +Simply put, the optimization algorithm is iteratively updating our controls until we reach some convergence criteria (i.e., have obtained the optimal strategy). + +In order to perform the optimization we need to get the sensitivity of the `objective function` to these `controls` or `optimization variables`. +It is important to understand how are current controls are performing. Did we improve our strategy and therefore our objective function value? +Note: `updated controls` after one optimization iteration become `current controls` for the next iteration. This means in Everest there will be a +distinction between running a forward model for `current controls` or `perturbed controls`. The forward model doesn’t care for which type of controls +it is currently running, but the optimizer will handle the results slightly different. + +If we perform robust optimization (i.e., don’t have a single deterministic underlying model) a `batch` contains a certain number of `realizations` +(similar to `realizations` in Ert except static and denoted with ``) and each `realization` contains a certain number of `simulations` +(i.e., forward model runs). The `simulations` are either evaluating the objective function value for `current controls` and/or for +each `perturbed controls`. This means that the hierarchy of the output in Everest and Ert are different (Fig 2). + +.. figure:: images/Everest_vs_Ert_01.png + :align: center + :width: 700px + :alt: Everest vs. Ert data models + + Difference between `ensemble` in Ert and `batch` in Everest. + A `realization` in Everest refers to a static model configuration which doesn’t change during the optimization, + but is different from model to model. While `realization` in Ert means set of model parameters which are going to be history + matched to certain data. Particularly, in Ert the `model parameters` or `realizations` are actually the objective of the optimization, + while in Everest the optimization objective is finding the correct controls such that a certain objective is maximized or minimized. + Since Everest still uses Ert to submit the `batch` (i.e., `ensemble`) to the queue, the Everest runs (i.e., for a we run a set of controls) + are mapped to Ert `realizations` accordingly. After collecting the results for each Ert `realization` they are mapped back to Everest structure. + +The mapping from data models in Everest to Ert is the same as flatten a 2D array (i.e., from index based on `` and `simulation` in Everest to +index based on `realization` Ert). Ultimately, Ert is submitting the forward model runs to the queue and is agnostic about the meaning of each run. +Only when the data is collected back in Everest is the meaning of each run attributed. + +In Ert each `ensemble` is exactly one step in the history matching algorithm and `realizations` have continuity from one iteration to the next. +For example, `model parameter set 0` are smoothly(?) changing from prior to posterior over the course of the history matching. +This is not the case for `simluations` in Everest and highlights another key difference. A `batch` can contain several different configurations (Fig 3) +and `simulation 0` for ` = 0` can be either `current` or a `perturbed control` hence there is no continuity from one `batch` to the next. `` +is continuous from one `batch` to the next since they are not changing at all over the course of the optimization. + +.. figure:: images/Everest_vs_Ert_02.png + :align: center + :width: 700px + :alt: Other `batch` configurations Everest + + Two other possible configurations of Everest `batches` in the context of gradient-based optimization algorithms (i.e., `optpp_q_newton`). + Option (A) can occur when the option `speculative` is set to False, this means that before proceeding with the optimization the update set of controls is + first evaluated if it actually improves the objective function value (hence the `batch` contains only `current controls` for each forward model run for each + ``). Option (B) can occur when the `current controls` are already evaluated in a previous `batch` and no update to current controls has occurred yet. + In the context of gradient-free optimization methods a `batch` can also contain multiple current control forward model runs (ask Pieter how batches change + depending on the optimization algorithm, I wrote this now based on his old figure, but pretty sure the terminology is wrong here). + +Perhaps an attempt could be made to improve the terminology on the Everest side, at least from a developer point of view. `Simulation` +is meaningful to the user and perhaps should be kept as is, but `simulation` to a Everest developer is not helpful since in the code `realization` +is mapped to `simulation` and vice versa. Also, `` is misleading in several ways, it’s not the same as a `realization` in Ert since it’s static +and furthermore we could be optimizing anything which has nothing to do with “GEO”. Perhaps we could change `` to `` +to emphasize the fact these model parameters are not changing over the course of the optimization. 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zAz;19GYWDW`6EH4j{th}expvJeNxwQLBXflsP84C<79Xz&J0I^i*Z=%~WkiR_ literal 0 HcmV?d00001 From 0682e1bbf13063aae96a780a305efb5506306be3 Mon Sep 17 00:00:00 2001 From: StephanDeHoop Date: Tue, 21 Jan 2025 09:39:18 +0000 Subject: [PATCH 2/5] Simplify docs and add prelimenary equation --- docs/everest/development.rst | 113 +++++++++++----------- docs/everest/images/Everest_vs_Ert_01.png | Bin 230386 -> 116239 bytes docs/everest/images/Everest_vs_Ert_02.png | Bin 108057 -> 141566 bytes docs/everest/images/Everest_vs_Ert_03.png | Bin 0 -> 111308 bytes 4 files changed, 59 insertions(+), 54 deletions(-) create mode 100644 docs/everest/images/Everest_vs_Ert_03.png diff --git a/docs/everest/development.rst b/docs/everest/development.rst index cc93a9119a3..1ca817f6343 100644 --- a/docs/everest/development.rst +++ b/docs/everest/development.rst @@ -56,66 +56,71 @@ long as the optimization process is running. - Signal everest optimization run termination. It will be called by the client when the optimization needs to be terminated in the middle of the run -Everest vs. Ert data models +EVEREST vs. ERT data models =========================== -Everest uses Ert for running an experiment, but instead of submitting an `ensemble` (i.e., as in Ert) to the queue we submit -a `batch` in Everest. `Batches` are in principle very similar to `ensembles`, but they have some key differences. -The purpose of this section is to explain these key differences from a developer point-of-view. - -In Ert, an `ensemble` contains a number of `realizations` which are a set of `model parameters` which Ert attempts to history match to some data. -These `model parameters` are sampled from a certain distribution after creating the `ensemble`. In Everest we are not history matching our `model parameters`, -but instead try to find an optimal strategy on how to operate the particular model(s) in order to maximize some objective (i.e., we optimize for a set of `controls`). -Simply put, the optimization algorithm is iteratively updating our controls until we reach some convergence criteria (i.e., have obtained the optimal strategy). - -In order to perform the optimization we need to get the sensitivity of the `objective function` to these `controls` or `optimization variables`. -It is important to understand how are current controls are performing. Did we improve our strategy and therefore our objective function value? -Note: `updated controls` after one optimization iteration become `current controls` for the next iteration. This means in Everest there will be a -distinction between running a forward model for `current controls` or `perturbed controls`. The forward model doesn’t care for which type of controls -it is currently running, but the optimizer will handle the results slightly different. - -If we perform robust optimization (i.e., don’t have a single deterministic underlying model) a `batch` contains a certain number of `realizations` -(similar to `realizations` in Ert except static and denoted with ``) and each `realization` contains a certain number of `simulations` -(i.e., forward model runs). The `simulations` are either evaluating the objective function value for `current controls` and/or for -each `perturbed controls`. This means that the hierarchy of the output in Everest and Ert are different (Fig 2). +EVEREST uses ERT for running an experiment, but instead of submitting an `ensemble` (ERT) to the queue we submit +a `batch` in EVEREST. `Batches` are in principle very similar to `ensembles`, ERT queue system doesn't treat them differently, +but they have some hierarchical differences in terms of the meaning behind the data. +ERT history matches `realizations` (i.e., `model parameters`) to data, hence an `ensemble` contains a number of `realizations`. +EVEREST optimizes a set of `controls` and assumes static (i.e., unchanging) `realizations`. +In terms of collecting the results of forward model runs, there is a distinction between `unperturbed controls` +(i.e., current `objective function` value) and `perturbed controls` (i.e., required to calculate the `gradient`). +Furthermore, when performing robust optimization (i.e., multiple static `realizations`) a `batch` contains a +certain number of `realizations` (denoted by ``) and each `realization` contains a number of `simulations` +(i.e., forward model runs). These `simulations` are forward model runs for either `unperturbed controls` and/or +`perturbed controls`. This is the key differences between the hierarchical data model of EVEREST and ERT (Fig 3). .. figure:: images/Everest_vs_Ert_01.png :align: center :width: 700px - :alt: Everest vs. Ert data models - - Difference between `ensemble` in Ert and `batch` in Everest. - A `realization` in Everest refers to a static model configuration which doesn’t change during the optimization, - but is different from model to model. While `realization` in Ert means set of model parameters which are going to be history - matched to certain data. Particularly, in Ert the `model parameters` or `realizations` are actually the objective of the optimization, - while in Everest the optimization objective is finding the correct controls such that a certain objective is maximized or minimized. - Since Everest still uses Ert to submit the `batch` (i.e., `ensemble`) to the queue, the Everest runs (i.e., for a we run a set of controls) - are mapped to Ert `realizations` accordingly. After collecting the results for each Ert `realization` they are mapped back to Everest structure. - -The mapping from data models in Everest to Ert is the same as flatten a 2D array (i.e., from index based on `` and `simulation` in Everest to -index based on `realization` Ert). Ultimately, Ert is submitting the forward model runs to the queue and is agnostic about the meaning of each run. -Only when the data is collected back in Everest is the meaning of each run attributed. - -In Ert each `ensemble` is exactly one step in the history matching algorithm and `realizations` have continuity from one iteration to the next. -For example, `model parameter set 0` are smoothly(?) changing from prior to posterior over the course of the history matching. -This is not the case for `simluations` in Everest and highlights another key difference. A `batch` can contain several different configurations (Fig 3) -and `simulation 0` for ` = 0` can be either `current` or a `perturbed control` hence there is no continuity from one `batch` to the next. `` -is continuous from one `batch` to the next since they are not changing at all over the course of the optimization. + :alt: EVEREST vs. ERT data models + + Difference between `ensemble` in ERT and `batch` in EVEREST. .. figure:: images/Everest_vs_Ert_02.png :align: center :width: 700px - :alt: Other `batch` configurations Everest - - Two other possible configurations of Everest `batches` in the context of gradient-based optimization algorithms (i.e., `optpp_q_newton`). - Option (A) can occur when the option `speculative` is set to False, this means that before proceeding with the optimization the update set of controls is - first evaluated if it actually improves the objective function value (hence the `batch` contains only `current controls` for each forward model run for each - ``). Option (B) can occur when the `current controls` are already evaluated in a previous `batch` and no update to current controls has occurred yet. - In the context of gradient-free optimization methods a `batch` can also contain multiple current control forward model runs (ask Pieter how batches change - depending on the optimization algorithm, I wrote this now based on his old figure, but pretty sure the terminology is wrong here). - -Perhaps an attempt could be made to improve the terminology on the Everest side, at least from a developer point of view. `Simulation` -is meaningful to the user and perhaps should be kept as is, but `simulation` to a Everest developer is not helpful since in the code `realization` -is mapped to `simulation` and vice versa. Also, `` is misleading in several ways, it’s not the same as a `realization` in Ert since it’s static -and furthermore we could be optimizing anything which has nothing to do with “GEO”. Perhaps we could change `` to `` -to emphasize the fact these model parameters are not changing over the course of the optimization. And for developers it would be much clearer to -change `simulation` to `ert_realization` or `forward_model_evaluation`. + :alt: Additional explanation of Fig 3 + + Different meaning of `realization` and `simulation`. + +As is evident from the image above, in terms of execution in the queue `realization` (ERT) and `simulation` (EVEREST) are synonymous. +This means that ERT queue system is agnostic about the meaning of each run only when the data is collected back in EVEREST (`GEN_DATA`) is meaning +of each run attributed. +The mapping from data models in EVEREST to ERT is the same as flattening a 2D array (i.e., from a `` and `pertubation` based index in EVEREST to +`realization` in ERT). + +Explicitly this means: + +.. math:: + + r(g, p) = g, + +if `batch` only has `unperturbed controls`, + +.. math:: + + r(g, p) = p + g * P, + +if `batch` only has `perturbed controls`, + +.. math:: + + r(g, p) = g * (p<0) + (p + g * P + G) (p>=0), + +if `batch` has `unperturbed` and `perturbed controls`, where `r` is the ERT `realization_id` (0, ..., `R` - 1), `g` is the `` (0, ..., `G` - 1), `p` is `pertubation_id` (-1, 0, ..., `P` - 1), `R` +is the total number of ERT `realizations`, `G` is the total number of static `model_realizations`, `P` is the total number of pertubations. +NOTE: `p = -1` for `unperturbed controls`, and `p = 0, ..., P - 1` for `perturbed controls`. +**THIS IS MY SUGGESTION AND CURRENTLY NOT HOW IT WORKS AND ONLY VALID FOR GRADIENT BASED OPTIMIZATION ALGORITHMS I GUESS? +If we don't want `p` to be negative we need to use a flag (e.g., `is_pertubation`)** + +Another thing to note is that continuity for `realizations` between `ensemble` exists; however, this is not the case for `simulations` in `batches`. +A `batch` can contain several different configurations (Fig 5) and `simulation 0` for ` = 0` can be either `unperturbed` +or `perturbed controls`. `` is continuous from one `batch` to the next since they are not changing at all over the course of the optimization. + +.. figure:: images/Everest_vs_Ert_03.png + :align: center + :width: 700px + :alt: Other `batch` configurations EVEREST + + Two other possible configurations of EVEREST `batches` in the context of gradient-based optimization algorithms (i.e., `optpp_q_newton`). diff --git a/docs/everest/images/Everest_vs_Ert_01.png b/docs/everest/images/Everest_vs_Ert_01.png index 5df5024567e4a75b4fc9df13f3ad428e9c58f29f..8289e0348c6555c8d396a8e54af3fee0d68654eb 100644 GIT binary patch literal 116239 zcmeGEWl&sQ)ISR1Zh_z)EVu-B3(#nSOK^90O>hk~?g{Sh9^47;5ZocS!<^3ZzB4s< zzRjmwxBgTC-CgJO-e>LQ>$g_;Cq-#=6e1KTC@6H0jHEIY6dVo|6ifsX0`Q5wKh81m z587E-S{$lkoa6v_17|L#AO;0h{T=nm5FU7s>>#7%3urD%!f_lvdNs6g@ z=pARmCw%Ff3;r7V^4Gx~=__1VYisLB?<(X+7a0^5Dk{DLdEIkwoxR*x*jh-JN;Z_9 zwYaWrt}vx>I9#lv*1Fw65ridcj)4=>XGC)Z+;PQ zJB~FTh#9)S+#@yrJDx9>oUteRz>7B@=v=PfxyFo6(6GELSb2YUz7k=asdoNZ%KgLXZCI9AwHL5JQs(hl?aP?SXdNFS%QuksSvV!hK@i6=rrhTcDB}j(TA@O zXX)~tR6t!C?FZd!*H_pY`*q>e59Tom^eR%sM+=DJy$92Uq~<6luP@IRi{{zze0!s* 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zV%MWjO`E;%3zY_d&{&j=GT^B@f>LVTNY9WUev~zoHq1}j_0w%o#<vLj8~9n)aiUUT7YEE$0X5uzIEpn+$&)n@sS&$7 zOpK;+)Iq+zeYyHoj72JyLUgovxep@D4;BRuD&-H1LOxaWOh`+A{&Vt+%;QAd4b9xl ljEICl=Kjwg6!$pq!jN~WDr?^JUS_HFmbRW&(N(+9{{kwFaKHcn literal 0 HcmV?d00001 From de52b1f33dbab2df4def1cd4b6e7a0367d0d2371 Mon Sep 17 00:00:00 2001 From: StephanDeHoop Date: Thu, 23 Jan 2025 09:02:28 +0000 Subject: [PATCH 3/5] Update batch hierarchy for gradient-free methods --- docs/everest/development.rst | 3 ++- docs/everest/images/Everest_vs_Ert_03.png | Bin 111308 -> 166688 bytes 2 files changed, 2 insertions(+), 1 deletion(-) diff --git a/docs/everest/development.rst b/docs/everest/development.rst index 1ca817f6343..e807f6dd00c 100644 --- a/docs/everest/development.rst +++ b/docs/everest/development.rst @@ -123,4 +123,5 @@ or `perturbed controls`. `` is continuous from one `batch` to the next s :width: 700px :alt: Other `batch` configurations EVEREST - Two other possible configurations of EVEREST `batches` in the context of gradient-based optimization algorithms (i.e., `optpp_q_newton`). + Three other possible configurations of EVEREST `batches` in the context of gradient-based (i.e., `optpp_q_newton`) + and gradient-free (i.e., **WHICH ONE DO WE SUPPORT?**) optimization algorithms. diff --git a/docs/everest/images/Everest_vs_Ert_03.png b/docs/everest/images/Everest_vs_Ert_03.png index af9d01bb571f82a73d72f812ea870af1d372717b..3113a4d51c3392b6c43596edc726a4b99e897818 100644 GIT binary patch literal 166688 zcmd?Q1ydYb)HaH{6Wj(1?(T!T6N0;YaCZ+5K@yyx!QI{6ZP1{>-Q`ZsIq$vi{Rdyw z*Hu$9-7~#s?sh;_RFtGqkqD6>ARtg>WhB%fAYd3EAfQ$e;J{b#7gmdph0h5i2qiTJ-`5Ydl4Je$7ahQ61u_8aXP+;+=A>3GsY-6jX^xT3*WBA&yU z0_oIu7rmF4>#5X$4#EqiHuux~I?E{?f3Nq8uJ_k?$%_wFAf;L4IJLN&{yg!B!{J0q z#|5~;S0u1T08IZ1mX{UCA4A9$O~~V*kmGxW`1(gVAV1*s*f51gnKadIrB-z$iQ0NN zf!y_Ip6=;SanACJlZ#8i@luuU+3)m+x99UxuU?e>rCCAC&;LE%oGX_khJYg+ z6hUv{U<&+G~)C>QFYYvsnbkt;`=|XtHv0-h-t<*QW9|MszU+CVyrN6$R_fKhm{)b+ODjh6aV(Dz~5dA`Q!dSCG)J%(Dm9;G9c1QvpJ 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zV%MWjO`E;%3zY_d&{&j=GT^B@f>LVTNY9WUev~zoHq1}j_0w%o#<vLj8~9n)aiUUT7YEE$0X5uzIEpn+$&)n@sS&$7 zOpK;+)Iq+zeYyHoj72JyLUgovxep@D4;BRuD&-H1LOxaWOh`+A{&Vt+%;QAd4b9xl ljEICl=Kjwg6!$pq!jN~WDr?^JUS_HFmbRW&(N(+9{{kwFaKHcn From 85e9b55a221f863ebdce9ceb89eff9380e422554 Mon Sep 17 00:00:00 2001 From: StephanDeHoop Date: Thu, 23 Jan 2025 13:54:29 +0000 Subject: [PATCH 4/5] Remove explicit equation and refer to ropt for mapping (this is still being discussed on how it's implemented) --- docs/everest/development.rst | 32 ++++++-------------------------- 1 file changed, 6 insertions(+), 26 deletions(-) diff --git a/docs/everest/development.rst b/docs/everest/development.rst index e807f6dd00c..b7728e9f4fa 100644 --- a/docs/everest/development.rst +++ b/docs/everest/development.rst @@ -87,32 +87,12 @@ certain number of `realizations` (denoted by ``) and each `realization` As is evident from the image above, in terms of execution in the queue `realization` (ERT) and `simulation` (EVEREST) are synonymous. This means that ERT queue system is agnostic about the meaning of each run only when the data is collected back in EVEREST (`GEN_DATA`) is meaning of each run attributed. -The mapping from data models in EVEREST to ERT is the same as flattening a 2D array (i.e., from a `` and `pertubation` based index in EVEREST to -`realization` in ERT). - -Explicitly this means: - -.. math:: - - r(g, p) = g, - -if `batch` only has `unperturbed controls`, - -.. math:: - - r(g, p) = p + g * P, - -if `batch` only has `perturbed controls`, - -.. math:: - - r(g, p) = g * (p<0) + (p + g * P + G) (p>=0), - -if `batch` has `unperturbed` and `perturbed controls`, where `r` is the ERT `realization_id` (0, ..., `R` - 1), `g` is the `` (0, ..., `G` - 1), `p` is `pertubation_id` (-1, 0, ..., `P` - 1), `R` -is the total number of ERT `realizations`, `G` is the total number of static `model_realizations`, `P` is the total number of pertubations. -NOTE: `p = -1` for `unperturbed controls`, and `p = 0, ..., P - 1` for `perturbed controls`. -**THIS IS MY SUGGESTION AND CURRENTLY NOT HOW IT WORKS AND ONLY VALID FOR GRADIENT BASED OPTIMIZATION ALGORITHMS I GUESS? -If we don't want `p` to be negative we need to use a flag (e.g., `is_pertubation`)** +The mapping from data models in EVEREST and ERT is done in the `ropt` library, it maps from `realization` (ERT) to `` and `pertubation` (EVEREST) and vice versa. +`Batches` in EVEREST can contain several different configurations depending on the algorithm used. Gradient-based algorithms can have a single function +evaluation (`unperturbed controls`) per ``, a set of `perturbed controls` per `` to evaluate the gradient, or both. +Derivative-free methods can have several function evaluations per `` and no `perturbed controls`. +**NOTE**: the optimizer may decide that some ``s are not needed, these are then skipped and the mapping from `ropt` +should reflect this (i.e., less ``s in the batch results than expected). Another thing to note is that continuity for `realizations` between `ensemble` exists; however, this is not the case for `simulations` in `batches`. A `batch` can contain several different configurations (Fig 5) and `simulation 0` for ` = 0` can be either `unperturbed` From 0c0f820bac2b32c25a9fbe6f1d46ecb46c52c3cf Mon Sep 17 00:00:00 2001 From: StephanDeHoop Date: Thu, 23 Jan 2025 15:12:42 +0000 Subject: [PATCH 5/5] Fix error in building docs --- docs/everest/development.rst | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/docs/everest/development.rst b/docs/everest/development.rst index b7728e9f4fa..c81d3affd26 100644 --- a/docs/everest/development.rst +++ b/docs/everest/development.rst @@ -91,8 +91,8 @@ The mapping from data models in EVEREST and ERT is done in the `ropt` library, i `Batches` in EVEREST can contain several different configurations depending on the algorithm used. Gradient-based algorithms can have a single function evaluation (`unperturbed controls`) per ``, a set of `perturbed controls` per `` to evaluate the gradient, or both. Derivative-free methods can have several function evaluations per `` and no `perturbed controls`. -**NOTE**: the optimizer may decide that some ``s are not needed, these are then skipped and the mapping from `ropt` -should reflect this (i.e., less ``s in the batch results than expected). +**NOTE:** the optimizer may decide that some `` are not needed, these are then skipped and the mapping from `ropt` +should reflect this (i.e., less `` in the batch results than expected). Another thing to note is that continuity for `realizations` between `ensemble` exists; however, this is not the case for `simulations` in `batches`. A `batch` can contain several different configurations (Fig 5) and `simulation 0` for ` = 0` can be either `unperturbed`

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