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d94a3c2a4629163ea243d984a86e2db820d72206 | subsection | 165 | 239 | Stage 3 | For Stage 3, the power converters of LEIR are upgraded to allow light ion acceleration up to 440 MeV/u. This energy upgrade of the LEIR machine does not have any impact on the BioLEIR machine operation.The power converter upgrade for Stage 3 has no impact on the controls system. | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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d5d2fe1284692a323c617948b78766902a4747f4 | subsection | 166 | 239 | Optional upgrade to interleaved operation | In order to be able to interleave BioLEIR and LHC/NA cycles, LINAC5, the LEIR machine and their connecting beamlines undergo a number of modifications:the switching magnet ITE.BHN20 needs to be replaced and its power supply modified to support pulse-to-pulse modulation.
LEIR injection elements (magnetic and electrosta... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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dd444ce440f55129144e4893d48c12b94cd135af | subsection | 167 | 239 | LEIR's magnetic cycle | The proposed magnetic cycle is similar to the one currently used for the LHC pilot or the North Area, completed with an extended flat top for a 2.4 second long slow extraction, see figure REF . The field is first ramped to the injection energy. After a single injection and a short (\displaystyle \sim 300 ms) cooling ti... | {
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69573df31ca45cdb6c3c59e3efc22ea5cc19a150 | subsection | 168 | 239 | Preliminary machine development periods | The slow extraction section will not be installed in LEIR before LS2 at the earliest.
However, as Oxygen is used as support gas for Xenon and Lead operation, it is already possible to carry out some machine studies with Oxygen in 2017.In order to benchmark the expected radiation levels during the running of LINAC5 for ... | {
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bbea22af741eee791dff7190fcd8fe8f7b5e3b30 | subsection | 169 | 239 | Control System | The BioLEIR control system will use standard tools and frameworks already implemented in the existing accelerator complex. In particular, no development of new hardware or software components is foreseen, but rather adaptation or extension of existing controls solutions is favoured.
As part of the control system, an in... | {
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ec5093c9db3f4e92016dbf82750385187b78fd56 | subsection | 170 | 239 | Optional upgrade to interleaved operations | It is possible to change between heavy-ions and light-ions cycles in a matter of seconds, once the LEIR transfer line is upgraded to support multiplexing and therefore interleaved operations (pulse-to-pulse modulation, PPM - see section REF ).
The standard BE-CO stack supports multiplexing, and such multiplexing is exp... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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fee697c95597a58ac7859aecac635ec8520cc507 | subsection | 171 | 239 | Implementation | Standard BE-CO hardware and software tools are used to control the new machine equipment as well as magnets needed for the new slow extraction from LEIR and for the beamlines towards the experimental areas (see chapters and for details on equipment to be controlled). The same holds true for each of the three beam stop... | {
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e8d211ff2d2264938a63a7a4c878ebe2f2d8c275 | subsection | 172 | 239 | Dumping the beam | The decision to dump a beam may be taken at almost any time. In such a case, no beam shall reach any of the BioLEIR irradiation areas. Lead ions can be safely dumped inside the LEIR machine, or an extraction kicker between LEIR and the PS machine can be used to dump the beam on a concrete block between the two machines... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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f144008fe8415e51c9be709f529e1766b0603c3d | subsection | 173 | 239 | Dedicated control room | It is expected that a dedicated, local BioLEIR control room be established in the vicinity of the experimental area. We presume that two new workstations and four screens are needed for local controls applications. | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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50c87c6c3a8c14ce241c753abb4acb306c3d34fe | subsection | 174 | 239 | Robotic table for samples | As described in section REF , it is planned to have a robotic table that positions and exchanges cell samples in irradiation room H1 and V. The automation system is provided by EN-STI-ECE group and is expected to be an industrial system, comprising equipment and its specific controls that need to be integrated in the c... | {
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e88436b3012a2f130d49b3435d0f86f304097663 | subsection | 175 | 239 | Timing system and cycling of the machine | One of the most challenging control system modifications concerns the timing system ensuring synchronisation of equipment elements and beam scheduling (sequencing).In Stage 1 the timing system is expanded to control the three new beamlines. Stage 2 requires a bigger adaptation. First, the control system has to be exten... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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be1ec62d33f6b03240c754ae71ad73f7559eb211 | subsection | 176 | 239 | Open analogue signal information system – OASIS | It is estimated that the following analogue system set-up is needed to control BioLEIR:up to 32 analog inputs (sources) to monitor.
possibility to monitor up to 16 of them in parallel (i.e. 4 oscilloscopes each with 4 digitizers).
the fastest signals to monitor (fastest sampling) are needed for beam position monitors... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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e14f43d04fe8e9bd17b511ae9802d72b51c71ea7 | subsection | 177 | 239 | Injector controls architecture – InCA | It is expected that the high level controls infrastructure (e.g. settings management) is integrated into the LEIR operation configuration. The settings are configured in a standard way using knobs and working sets. | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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54d821109353e06ea11403bc35ca03b59a8da495 | subsection | 178 | 239 | Status display | For the moment no new status displays (fixed display) are requested. Should such a fixed status display become useful, it can be developed relatively rapidly based on existing software frameworks. | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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f1281b6c1956945b0c435de5278cdb08c5e1d9ac | subsection | 179 | 239 | Other controls services | It is assumed that other standard controls services are used as needed. These may include: the controls configuration database, the machine data logging service, the alarm service and monitoring of equipment. | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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8ba35ced23576c8670bbe9d2b4c82a3f62a03355 | subsection | 180 | 239 | Outstanding questions and issues | Part of the transfer line from LINAC5 to LEIR is shared with LINAC4, which accelerates negative polarity particles (H\displaystyle ^-). This aspect may present stringent operational limitations due to necessary polarity switching of the transfer line magnets when switching from one beam to the other. The magnet polarit... | {
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6492b99b15e7f4c47c6cde05a9bddc1f5d803fd9 | subsection | 181 | 239 | Radiation Safety and Radiation Protection | This chapter is dedicated to radiation safety and radiation protection aspects of the BioLEIR facility.
After defining the parameters of the scenarios that have been used for the radiation protection assessment in section , the estimate for the prompt radiation and the shielding requirements derived from it are present... | {
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1c3415cd381168dd97b4e4da78a589247393352d | subsection | 182 | 239 | Scenarios considered for the radiation protection assessment | The particle species and beam intensities used for the radiation protection assessment are presented in table REF . Once the LEIR power converters are upgraded, the BioLEIR facility can provide beams fulfilling the beam specification requirements as described in table REF . These intensities correspond to the maximum i... | {
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d739bcf0050f70619ed195b0328bbdfbd0d36089 | subsection | 183 | 239 | Prompt radiation and derived shielding requirements | FLUKA Monte Carlo simulations , have been performed to assess the prompt and residual ambient dose equivalent rates for both target cases and all particle species defined in table REF .The prompt ambient dose equivalent rate distribution for Argon ions is shown in figure REF for phantom and cell culture targets. The pr... | {
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7c51d354c29450bc5fa62b6deb7b1d7a3c1692b4 | subsection | 184 | 239 | Air activation | The desired high beam intensities and energies (see table REF ) produce air activation in the irradiation rooms at levels requiring
a ventilation system to provide dynamic confinement and a flush before access. The radioactivity that is released by the ventilation system via a dedicated exhaust has to be monitored as d... | {
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a3ed083deafd8e7c45c865883d8dc0c4fc696257 | subsection | 185 | 239 | Residual radiation | Based on the results of the FLUKA Monte Carlo simulations, the residual ambient dose equivalent rates at 40 cm from the phantom and the beam dump are estimated to be below 15 \displaystyle \mu Sv/h after one hour of cool-down time after Argon ions operation. This means that for access during Argon ions operation, the i... | {
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ffcc71e1648bdc9ff84758077bb34a77078dffb8 | subsection | 186 | 239 | Radiation safety requirements for the personnel protection system | The switchyard areas and the irradiation rooms have to be considered as primary beam areas, notably for the Important Safety Elements (EIS-b) requirements, due to the high prompt ambient dose equivalent rates (see section ). This implies that these areas are not accessible during beam operations.Before access to an irr... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
"G. Roy",
"S. Schuh"
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"physics.ins-det",
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f462d43561b0aa3d39ab4e069bd60c3731ea0ac9 | subsection | 187 | 239 | Radiation protection monitoring system | Based on the preliminary BioLEIR beamlines and shielding layout, the following monitors have to be installed to provide adequate radiation protection monitoring:Ionization chambers of type IG5–H20 to monitor neutron stray radiation:
a total of 10 IG5–H20 monitors are needed: 3 around the LEIR shielding, 1 above the roo... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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f15da9b6bc8ab1d9a675fc12f4ccaedcbaefb344 | subsection | 188 | 239 | Safety Aspects | BioLEIR is similar in scope and use to other experimental facilities at CERN, for example ELENA at the Antiproton Decelerator. The beam and radiation hazard is the main concern. Other risks are identified primarily as industrial risks that are already covered by CERN rules and practices, and taken into account in the d... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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] | [
"physics.ins-det",
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] | 2,018 | en | Physics | [
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908026af6fc8ff6df95da9a834c052d145140102 | subsection | 189 | 239 | Operational safety of the ion sources and the linacs | In Stage 1 of the facility, LINAC3 with its source is used to provide ions for BioLEIR, and its safety aspects are already covered. The modification of this frontend to produce lighter ions introduces new chemical substances. The associated risks have to be assessed and mitigated. It is expected that these risks are ea... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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2b34295aa84b56f0eb5cf7cf6e442eea8250de07 | subsection | 190 | 239 | Operational safety of the transfer lines (from LEIR to experimental areas) | As expressed above, BioLEIR is similar to other CERN beam facilities, and to be designed, built, commissioned, and operated according to CERN rules and practices. Most of the operational procedures are very similar to the ones already applied around beam facilities at CERN.The main issues to be considered for personal ... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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45eb2a8fc3170c0860f3d00baa26a13c35245468 | subsection | 191 | 239 | Operational safety of the transfer lines (from LEIR to experimental areas) | Detection of abnormally high temperatures provokes powering off the equipment, and possibly generates a technical alarm.A starting fire during presence of personnel would immediately be detected by the smell and tackled by means of a fire extinguisher (to be made available). No fire detection system is currently forese... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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"physics.ins-det",
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82bb9364c702b01b1b4de91ad42c5e3c6f316cc1 | subsection | 192 | 239 | Operational safety of the experimental area | The experimental area consists of three independently accessible irradiation points: H1, H2 and V, as well as a Biolab, and the shielded switchyard areas 1 and 2 (see detailed descriptions in chapter ).H2 houses propane-based gases for dosimetry development. This area needs a flammable gas detection system with alarm. ... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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] | [
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0c53b604f68d92106617f9cae6517d977429da3d | subsection | 193 | 239 | Operational safety specifics of the Biolab | A new risk potentially imported by BioLEIR users is the chemical hazard. This is obviously the main risk in the Biolab. The CERN regulation concerning chemical hazards is SR-C. In particular, dedicated chemical cabinets are provided for users in the Biolab (according to Safety Guideline C-1-0-1). Three different cabine... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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f354432279a8dc3952ac1a41f7d76f4e51d1c966 | subsection | 194 | 239 | Personnel access - the Personnel Protection System PPS | This section introduces the principles of the Personnel Protection System (PPS) that secures the facility by means of access control and safety interlock functions.
The PPS regulates the accesses of personnel to areas where a potential hazard may be present, by means of controlled physical barriers such as access point... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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80845cb786f79a98e40d3d93cc5e85c073d950dc | subsection | 195 | 239 | Sectorization | The accessible areas of the BioLEIR facility are divided into 3 zones, according to their radiological classification from chapter :ZONE 1: this area includes Switchyards 1 and 2, where the ion beam is extracted from the LEIR accelerator and dispatched towards the three irradiation rooms. The RP analysis(chapter ) show... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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4d7fafd8f30d6db460a13e86a49fb7a96c8eab26 | subsection | 196 | 239 | Access modes and operation | An important strategy for the risks mitigation consists on the alternation of specific exploitation modes:when the absence of radiological hazards is ensured, the ACCESS mode is allowed;
when the absence of personnel inside the radiological areas is ensured, by means of a PATROL mode, then BEAM mode is allowed.The dif... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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041c180133bde9afa5832bc180fd64193b6a8965 | subsection | 197 | 239 | Main principle of protection | This section summarizes the main principles of the BioLEIR PPS:The BioLEIR facility is completely independent from the LEIR accelerator. The PPS ensures autonomously the safety of the facility and interlocks the LEIR accelerator EISs when needed.
The protection of BioLEIR is based on the following 4 safety elements:
... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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c2c1d14ea1a2a09ba96923216bfa8b4feab62452 | subsection | 198 | 239 | Resource estimate of the personal protection system | Table REF shows the estimated material cost for equipment and external services provided by contractors, while table REF shows the estimated CERN personnel needed.
It shall be noted that the provision and installation of the two escape doors of Zone 1 and the three access doors to the irradiation rooms are not included... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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e67ca880749dc063f823e926c887f0bf1541b393 | subsection | 199 | 239 | Installation Planning | This chapter summarizes the foreseen installation schedule for the BioLEIR facility, taking into account the following aspects:CERN machine schedules with respective scheduled stops that can be used to install BioLEIR elements;
design and production lead times for the different elements;
minimum times necessary for i... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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1ce1baee0bffa9f6a7a78734bb213e8e081b2414 | subsection | 200 | 239 | CERN schedules until 2035 | Figure REF shows the planned schedule for the overall CERN accelerator complex. Beyond LS2 in 2019/2020, the schedule is only indicative.
[Figure: The overall CERN machine schedule.] | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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6e5d4402b47be9802f4a08a2705c5131d2e79305 | subsection | 201 | 239 | BioLEIR installation scenario | This section lists the different possible installation periods, the installation works that could be foreseen in any specific period and the resulting possible operation scenarios.The following assumptions and inputs have been used to establish the proposed schedule:Identification of alternative space for storage of ma... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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5e8004d8920d4c4df9fb7bb364158b263117b3d9 | subsection | 202 | 239 | BioLEIR installation scenario | Include LINAC5 into the access control and safety system of the PS complex.Stage 2 (2023- ): Running with ions from LINAC5 up to 250 MeV/u2023 Run periodOperate BioLEIR using LINAC5 with all light ion species available.YETS 2023/2024 (2 Months)Deploy the LEIR power converter upgrade.Stage 3 (2024- ): Running with ions ... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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5aa0d2df1c9a6ff0b82974680907a35dea1bd0d9 | subsection | 203 | 239 | Risk Assessment | This chapter lists and describes all identifiable risks for the BioLEIR project, as it is known and defined in this document. Details of proposals and ideas to mitigate a particular risk are elaborated where this information is already available. For some areas, we can only establish a list of issues at this point.This... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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55b44ad41b57c533babf24930c76841f08f31cc8 | subsection | 204 | 239 | Safety | The risks pertaining to the safety of personnel and of the installation are essentially: exposure to ionizing radiation, fire or explosion from flammable gases, chemical hazards, electrical hazards, environmental risks, heavy handling.They are generally well understood at CERN and appropriate mitigation measures can be... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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ef2689919112915f56156a856edd14f12a05eef9 | subsection | 205 | 239 | Schedule | Several aspects in this study assume best-case scenarios. A few additional detailed studies have been suggested in chapter . They might reveal technical issues that may require more work and potentially a schedule delay.
In particular, the risk that the ageing infrastructure that BioLEIR plans to use, would require an ... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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ee705879a9738b76bc442a1ee56c69c7bdf30aed | subsection | 206 | 239 | Cost | The cost of the facility has been evaluated in a very short time with minimal resources. In most areas, high-level engineering estimates have been given and no detailed studies have been performed. This aspect is reflected in the large uncertainty of the cost of the order of \displaystyle \pm 30%.Since BioLEIR relies l... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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3a5068d0c1181a2ac2dc73f2b8c92e3e379e6fde | subsection | 207 | 239 | Undesired beam characteristics | It is essential to have accurate knowledge of the beam characteristics all along the irradiation process such that irradiation with unwanted beam (i.e. incorrect energy, incorrect intensity, incorrect ion species) can be avoided.Since LEIR will be used for both, heavy ions for the North Area and LHC physics programmes ... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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bbf4e181af6bc289787938495806b854319ff3ce | subsection | 208 | 239 | Application of new technologies | In general there are very few risks linked to the application of new technologies for BioLEIR. One of the main design criteria was to reuse proven and tested technologies in order to ensure minimum cost, maximum operability and availability for beams to the experimental areas.However, in the case of LINAC5 we have chos... | {
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1b8e647c377fe8cdff02939731431422e8298f1a | subsection | 209 | 239 | Ageing infrastructure | This feasibility study assumes that building 150 and the LINAC2 area can be re-used with only small investments into the electrical, cooling and piping infrastructures, as well as the structure of the buildings.However, due to the age of the infrastructure, the risk exists that more investment has to be considered. In ... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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48783b820099d3b9ec52ddd9fc3fb39c6255d6de | subsection | 210 | 239 | Availability of hall 150 | Before any new facility can be installed in building 150, the material stored in the hall needs to be evacuated. The material is slightly radioactive. A new and appropriate storage area for this material needs to be identified or constructed. This represents potentially a schedule risk for BioLEIR. | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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564b04cc9d22a1e06870ad13436ada76270314c3 | subsection | 211 | 239 | Concrete radiation shielding in switchyard | The radiation protection shielding for the BioLEIR switchyard structures should be built using precast concrete blocks. However, due to space constraints in the hall, the structures are of non-standard shapes with relatively large spans, and are to be built in areas with limited access and with very thick roof slabs fo... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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7851ab9fffefe28934f95e4f08a0682da3deb3f9 | subsection | 212 | 239 | Points Flagged for Further Study and Investigation | This chapter lists and describes some topics that have been raised during discussions in the preparation of this document. These specific topics require significant ad-hoc studies, potential Research and Development activities and specific measures that shall be investigated and considered when the study becomes a proj... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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f6b65ca3a757cc3864dc37a081c22867a4c633d9 | subsection | 213 | 239 | Beam Requirements | The requirements for the BioLEIR beams shall be reviewed and refined with the biomedical community, taking due account of the various upstream instruments and techniques being used to produce the beams. Indeed, some of these techniques might have an adverse effect on some beam parameters and we need guidance and valida... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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be082d9b54be4ef30ab79cf43ab7a9b8231c7ff4 | subsection | 214 | 239 | Machine developments in 2017 and 2018 | In order to benchmark the expected radiation levels during the running of LINAC5 for BioLEIR, a radiation measurement campaign should be scheduled in 2017 or 2018 during the running of LINAC3 with Oxygen and Lead ions.In order to better understand the beam dynamics with light ions in LEIR, we recommend to initiate in 2... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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1f0b6d1c2d3a97e53282e7d386e37823f3e79aff | subsection | 215 | 239 | Frontend and linac | The choice of PMQs within the Quasi-Alvarez high-energy accelerating modules reduces the power consumption and complexity of the linac since there is no feed-through for electrical power or cooling channels. However, the PMQs will have to withstand significant radiation levels from particle losses and the materials use... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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55b73b6202993a8105c5aaa12517829f6bfa3409 | subsection | 216 | 239 | Transfer line from LINAC5 to LEIR injection | A very preliminary analysis of the injection transfer beamline has been carried out. However, no actual detailed design could be produced in the time frame of this study. The baseline design (see chapter ) of the injection transfer line from LINAC5 to LEIR includes a short section (\displaystyle < 20 m and about 6 magn... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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09a20c5706faf62223b0cf512a56abb304471ac0 | subsection | 217 | 239 | Beamlines to the BioLEIR experimental area | Apart from the element design and the associated integration, the main unresolved technical issue appears to be the required uniformity at the target, which is strongly linked to the collimation and beam purity which can be expected. This needs design and calculations, for both the achievable beam purity and for the be... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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ba1d5c99cc5c149ff89aefd4a37060e29f5ea868 | subsection | 218 | 239 | Biomedical motivation for a biomedical research facility hosted at CERN | Bleddyn JonesCERN is internationally recognised for probing the nature of matter and its interactions, by studying the behaviour of sub-atomic particles, and for playing a major role in establishing the most precise physics model available to us today - the Standard Model of particle physics. Fundamental research in pa... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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baca1e409da23f99282b4f38f69eb7806716783a | subsection | 219 | 239 | Brief history of medical applications that followed advances in radiation physics | The discovery of X-rays and natural radioactivity were soon followed by their application in medicine. Diagnostic imaging became widespread within a few decades, and therapeutic advances in cancer irradiation, as an alternative to surgery, became increasingly important and remain so. Technical developments, notably the... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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29f1edd97e57356114e8b2cfc49fbec4b0cdd506 | subsection | 220 | 239 | Brief history of medical applications that followed advances in radiation physics | The ionisation events are shown as red dots.][Figure: Examples of cell survival data expressed as the Surviving Fraction (SF) from the data of Furusawa et. al. 2000 , for three cell types (V79, HSG and T-1) exposed to increasing doses of Carbon ions at different LETs in [keV/\displaystyle \mu m]. The curves and lines a... | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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98bf87c1f28103c2c0b0d984a882b4c9cf3afd7b | subsection | 221 | 239 | Brief history of medical applications that followed advances in radiation physics | Improving our knowledge of RBE and its prediction from LET, dose, and perhaps other beam characteristics such as average inter-track distances, and its relation to radiobiological characteristics connected to the baseline radiosensitivity characteristics for low LET radiation (reflected in the intrinsic ability to repa... | {
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9cef23d9fd67310b2ade877728c494b4cc860baa | subsection | 222 | 239 | Radiobiological influences in hadrontherapy | Physicists have made major contributions to radiobiological modelling. Initial power law models proved too inflexible in biological systems and were replaced by simple target theory models linked to Poisson statistics. Presently, the linear-quadratic (LQ) model is used to guide many clinical decisions. It is based on t... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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c3277b9feab84076c667de5a68cea27a490d95f3 | subsection | 223 | 239 | Radiobiological influences in hadrontherapy | It will be important to determine which are the best definition for which precise situation. Such data sets where not only the physics is studied closely, but also the biological effects are studied with high throughput to improve the data confidence limits, could only be achievable in a well-organised and resourced la... | {
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7cf08f67a79f0f75a0a0e651e23bac3a052e1d9e | subsection | 224 | 239 | Radiobiological influences in nuclear medicine | A significant number of cancer treatments involve the administration of unsealed radionuclides which are introduced into the body either orally or intravenously. The first such therapies (developed in the 1940's) usually involved iodine-131 or phosphorus-32 (both of which are \displaystyle \beta emitters), typically fo... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
"S. Ghithan",
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d032cfd265aed83acf6e8f3b61c2d0ef6aee3187 | subsection | 225 | 239 | Radiobiological influences in nuclear medicine | This is a consideration which opens additional avenues for other useful and clinically-relevant research and is achievable on the proposed modifications to the CERN LEIR synchrotron. | {
"cite_spans": []
} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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f6c72427f16a77ca680a9c13680875536d03d83c | subsection | 226 | 239 | Diagnostics and detectors | It is self-evident that a test bed facility such as BioLEIR at CERN would need to incorporate state-of-the-art and innovative diagnostic instruments, coupled with extensive Monte Carlo FLUKA and GEANT based simulations. To study beam ballistics extensively and correlate beam track separations with LET and the way RBE m... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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e5555aadacfa6db9e831626427a856fd61bc4d58 | subsection | 227 | 239 | IT support challenges | To plan, record and analyse the extensive simulation and experimental data that would be associated with the BioLEIR project, the data handling and storage facilities of CERN would furnish open access results to participating Universities, that will also hold data on the detailed biological characteristics of the cell ... | {
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d56a52dd6aaca418468e23892edf49c8dc4554d6 | subsection | 228 | 239 | Potential interest of availability of light ions to the CERN research programme and the existing CERN users' community | Marco SilariIn addition to the necessity of light ions for biomedical research, light ions could also be of interest to the CERN research programme and the CERN user community at large. A first, non-comprehensive feedback and interest has been gauged. | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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f8c0e47b9bdb4671b7147b262c86670b2af4e7cd | subsection | 229 | 239 | Detector development and testing | Advances in radiation therapy technologies call for an increasing reliability and accuracy in dosimetric techniques. In hadrontherapy, it also implies the verification of the position of the Bragg peak in a humanoid phantom. GEMPix is a novel detector developed at CERN by coupling two CERN technologies, the Gas Electro... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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75cd256c85301dacfade7492536af707e22b4a26 | subsection | 230 | 239 | Nuclear physics studies and benchmarking of Monte Carlo codes | The availability of light ion beams is a fundamental condition for the investigation of the fragmentation process at intermediate/low energies, which plays a critical role in hadrontherapy. The possibility of carrying out experiments aimed at the measurement of double differential cross sections, such as fragment spect... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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4a6f7ee94941b73c3a09d13a942e02ee837a7952 | subsection | 231 | 239 | Radiation damage to electronics and microelectronics studies | The user community interested in radiation effects on electronics would clearly profit from a several hundred MeV/u light ion beam for both practical and radiation hardness assurance reasons. Standard heavy ion ground test facilities in Europe (e.g. UCL in Louvain-la-Neuve, Belgium, and RADEF in Jyväskylä, Finland) pro... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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052194eac3a1cc67467495ea257b78189acb0203 | subsection | 232 | 239 | Shielding data and induced radioactivity in accelerator and beamline components | There is a growing number of proton and light ion therapy facilities in operation or at the planning stage worldwide. The light ion presently used in clinical practice is \displaystyle ^{12}C. However, different ions (anything in the range from protons to Oxygen) may be considered in the future. A synchrotron generatin... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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e7561bb9b5799d6bfc9e947a3afbeb098e581b43 | subsection | 233 | 239 | Real-time tumour tracking and dose delivery in hadrontherapy | Dose delivery in hadrontherapy follows more or less the same protocol as conventional (electron and photon) radiation therapy that is as follows: identification of the target volume with proper imaging techniques, referencing the tumour location with markers in the patient's body, patient set-up with Computer Tomograph... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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7486f343feb3e24b5f48c52f05ed91cf961b9f82 | subsection | 234 | 239 | Control irradiator choice for the BioLEIR project | Bleddyn JonesThe study of the relative biological effectiveness (RBE) of different radiation qualities is important, since the RBE concept is used in the dose prescription process of hadrontherapy. RBE is defined as:RBE = \frac{Dose~of~reference~radiation~(low~LET)}{Dose~of~hadronic~test-radiation~(higher~LET)},such th... | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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6960442a8991f05a7390b599c616f8c554b1941a | subsection | 235 | 239 | Choice of the ideal control irradiator | As demonstrated in the section above, the choice of the control irradiation has an important effect on the measured RBE ratio. For example, Hall and Gaccia quote typical LET values of 2 keV/\displaystyle \mu m for 250 keV irradiation (but the filtration is unspecified), 0.2 keV/\displaystyle \mu m for 60-Co, 4.7 keV/\... | {
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e543c81c10992ffc166a257844598a4439b8a8eb | subsection | 236 | 239 | Choice of the ideal control irradiator | The measurements were performed on only a single cell line. | {
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} | 10.23731/CYRM-2017-001 | 1802.07724 | Feasibility Study for BioLEIR | [
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b2760310588578e93edbe15d3a8769abd9c1c332 | subsection | 237 | 239 | Discussion | The literature contains many contributions which concentrate on the influence of photon energies in the low-keV range on RBE. However, these papers contribute mainly to the field of radiation protection, because this class of X-rays are used extensively in routine diagnostic medicine. Such work has confirmed that RBE v... | {
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d28308e73d81d9601e5f21f04a9d1d4dba269ed2 | subsection | 238 | 239 | Discussion | However, a broad electron beam operating at say 8 – 10 MeV would give a satisfactory, uniform dose for radiobiological experiments, yet could not be used for beam ballistics combined with radiobiology comparisons.There is interest in some clinical particle therapy centres across Europe, including international laborato... | {
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f9d26f9267e84018f92b671ee7a15d8d7273e248 | abstract | 0 | 39 | Abstract | This work is concerned with the development of a space-time adaptive
numerical method, based on a rigorous a posteriori error bound, for the
semilinear heat equation with a general local Lipschitz reaction term whose
solution may blow-up in finite time. More specifically, conditional a
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7a212edddfb4d47c42e2c45b5fdc6df352d1481e | subsection | 1 | 39 | Introduction | Let \displaystyle \Omega \subset \mathbb {R}^d with d=2 or d=3 be a bounded polyhedral domain and consider the problem\begin{aligned}u_t - a\Delta {u}-f(u) &= 0 \qquad && \text{in } \Omega , \mbox{ } t > 0 \mbox{,} \\ u &=0 \mbox{ } && \text{on } \partial \Omega , \mbox{ } t > 0 \mbox{,} \\ u(\cdot ,0) &=u_0 \mbox{ } &... | {
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18b00d521414ab399207733b8a361437f4a8958b | subsection | 2 | 39 | Introduction | For nonlinear problems, the situation is more complicated since the existence of a solution to an appropriate error equation (and, thus, of an error bound) usually requires that either the data or the discretization parameters are sufficiently small. As a result, a posteriori error estimators for nonlinear problems ten... | {
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6f675d4b099d216ff60873805ec8ac2397af91c0 | subsection | 3 | 39 | Introduction | Firstly, we significantly broaden the range of nonlinearities under consideration; specifically, the (possibly) nonlinear reaction term f:\bar{\Omega } \times [0,T_\infty )\times \mathbb {R}\rightarrow \mathbb {R} is assumed to be continuous and to satisfy the local Lipschitz estimate|f(x,t,v)-f(x,t,w)| \le \mathcal {L... | {
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ac2811b80d07ba280fc885e37fec1362b4425719 | subsection | 4 | 39 | Outline | We begin in Section 2 by outlining the IMEX discretization of (REF ) then in Section 3 we introduce several auxiliary results which will be used in the error analysis. In Section 4, we derive the conditional a posteriori error bound and in Section 5 we propose a general adaptive algorithm, applicable to both blow-up an... | {
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061ed03955603095a3d1d982ea775c5f753da357 | subsection | 5 | 39 | Notation | As they will be used frequently in this work, we denote the L^2 inner product on \Omega by (\cdot ,\cdot ) and the L^{\infty } norm on \Omega by ||\cdot ||. | {
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5d22972da7ed2f2c11f3fbfbe1cd771917030f8e | subsection | 6 | 39 | Discretization | Consider a shape-regular mesh \mathcal {T}=\lbrace K\rbrace of \Omega with K denoting a generic element of diameter h_K that is constructed via affine mappings F_{K}:\widehat{K}\rightarrow K with non-singular Jacobian where \widehat{K} is the d-dimensional reference simplex or the d-dimensional reference cube. The mesh... | {
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ffe3099f1ac29c4a22fbe8149c2d06d3d2faa087 | subsection | 7 | 39 | Discretization | Furthermore, if we let \mathcal {T}_0 denote an initial spatial mesh of \Omega associated with the first time node t_0=0 then to each additional time node t_m, m=1,\dots ,M, we associate a spatial mesh \mathcal {T}_m which is assumed to have been obtained from \mathcal {T}_{m-1} by local refinement and/or coarsening. W... | {
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483067fbbed7ca7fe10f73d540434c1281bb3c60 | subsection | 8 | 39 | Preliminaries | Before we proceed with the error analysis, we require some auxiliary results and some additional notation. Our first result is a maximum principle for a related parabolic equation.Theorem 3.1
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... | |
df83c4334d3705ba58adc4372c8085faf185d4d5 | subsection | 9 | 39 | Preliminaries | As we are primarily interested in blow-up in the elliptic regime, we elect to use the simpler error bound of Theorem REF but if one is interested in the singularly preturbed regime, the spatial estimators should be replaced with the full estimator from . | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00211-015-0763-0",
"end": 255,
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"raw": "Alan Demlow and Natalia Kopteva, Maximum-norm a posteriori error estimates for singularly perturbed elliptic reaction-diffusion problems, Num... | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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963d9b8bc577db85617621597e582606cd49a893 | subsection | 10 | 39 | Error Analysis | In the forthcoming error analysis, on each time step m, we will work with the Bochner space C(\bar{I}_m;L^{\infty }(\Omega )) – the space of all continuous functions v:\bar{I}_m \rightarrow L^{\infty }(\Omega ) equipped with the norm\begin{aligned}||v||_m := \sup _{t \in \bar{I}_m} ||v(t)||.
\end{aligned}In what follow... | {
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{
"arxiv_id": "",
"doi": "10.1090/s0025-5718-06-01858-8",
"end": 392,
"openalex_id": "https://openalex.org/W2030685662",
"raw": "Omar Lakkis and Charalambos Makridakis, Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic prob... | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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bb8432e09c91cb1734374eb3818e008c33bea81a | subsection | 11 | 39 | Error Analysis | \max _{K \in \mathcal {T}_m} \eta _S^m \big |_K,
\end{aligned}holds with \displaystyle {h}_m := \min _{K \in \mathcal {T}_m} h_K denoting the minimum mesh-size, C_{\infty } the constant of Theorem REF and where \eta _S^m is the primary space estimator given by\begin{aligned}\eta _S^m \big |_K:= h_K^2 a^{-1} ||A^m + a\D... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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eea4a1a2d9ee2db73f06cfa94c7b9a0344dbbd17 | subsection | 12 | 39 | Fixed Point Argument | We now seek to show that (REF ) has a unique solution \rho \in \mathcal {B}_m where \mathcal {B}_m is the closed ball of radius \delta _m \psi _m centered on zero in the ||\cdot ||_m norm and where \delta _m \in [1,\infty ) is a parameter to be determined with \psi _m chosen such that||\rho (t_{m-1})|| + \int _{I_m} ||... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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27bd945d795c14eebbee432dd22c8d4163b09b1e | subsection | 13 | 39 | Fixed Point Argument | \end{aligned}Using (REF ) with the monotonicity of \mathcal {L} and recalling (REF ) we obtain\begin{aligned}
||\Phi _m(v)(t)|| & \le \psi _m + \int _{t_{m-1}}^t ||v(s)||\mathcal {L}(s,||v(s)||+||U(s)||+||\epsilon (s)||,||U(s)||+||\epsilon (s)||) \,\text{d}s.
\end{aligned}In order to bound this further, we require anot... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.... | |
89c5a9350cc1a6e0fb1d8b254c3db332f9596dfa | subsection | 14 | 39 | Fixed Point Argument | \end{aligned}Thus we obtain that property (1) is satisfied if1+ \delta _m \int _{I_m} L(s, \delta _m) \,\text{d}s \le \delta _m.To show that property (2) holds, let v_1, v_2 \in \mathcal {B}_m then the definition of \Phi _m implies that\begin{aligned}(\Phi _m(v_1)-\Phi _m(v_2))(t) & = \int _{t_{m-1}}^t\mathrm {e}^{(t-s... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... | |
3b4503672d201bc0b12e79bf047a828bcfa22233 | subsection | 15 | 39 | Fixed Point Argument | As we have choice over the value \delta _m can take in (\star ) then for practical reasons we choose \delta _m \in [1,\infty ) to be, if it exists, the smallest root of the function \varphi _m:[1,\infty ) \rightarrow \mathbb {R} defined by\varphi _m(\delta ) := 1+ \delta \!\left[\int _{I_m} L(s, \delta ) \,\text{d}s -1... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... | |
cd04dd2dbf921674aaf8b5e3fd3d364d2ae77042 | subsection | 16 | 39 | Computable Error Bound | Now that \delta _m has been defined, we must characterize \psi _m from (REF ) in an a posteriori fashion in order to obtain a fully computable error bound. To do this, we must first estimate the term ||\rho (t_{m-1})|| along with the remaining terms containing \epsilon in (REF ).In the previous subsection, we deduced t... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.025... | |
55cc9babb27c795d4dea3d1524b5224253205667 | subsection | 17 | 39 | Computable Error Bound | To this end, we rewrite \rho as \rho = e - \epsilon and utilize Lemma REF to obtain||\rho (0)|| \le ||e(0)||+ ||\epsilon (0)|| \le \eta _I,where \eta _I is the initial condition estimator given by\eta _I := ||e(0)|| + C_{\infty }\log (1 \slash {h}_0)\!\max _{K \in \mathcal {T}_0} \eta _S^0 \big |_K.With regards to the ... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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383f67908b63a3abc2c9419fd08719ba8f9fee61 | subsection | 18 | 39 | Computable Error Bound | As before, {C}_\infty is a constant that is independent of the maximum mesh-size, a, u and U but may be dependent upon the size of the domain \Omega as well as the number of refinement levels between \mathcal {T}_{m-1} and \mathcal {T}_m.
}
\begin{}
We note the observation made in Corollary 2.9 of \cite {DLM09} that \e... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... | |
e8123d0e642dd046d88b4b8cbfa1a93bafe7775a | subsection | 19 | 39 | Computable Error Bound | \end{equation}
\end{}
\begin{}
Since \delta _1,\ldots ,\delta _M exist then by the exposition of the previous subsection we have
\begin{equation}
||\rho ||_m \le r_m\psi _m,
\end{equation}
for any 1 \le m \le M. Noting the decomposition e = \rho + \epsilon , we obtain that
\begin{equation}
\max _{1 \le m \le M} ||e||_m... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... | |
c5afa6829254e6631437db738f2201987c6fdfed | subsection | 20 | 39 | Computable Error Bound | \end{equation}
Therefore, \delta _m = 1 regardless of the size of the time step length k_m and so the \emph {a posteriori} error bound of Theorem \ref {maintheorem} holds unconditionally. The stated result then follows from Theorem \ref {maintheorem} by applying \psi _m recursively and upon noting that r_m = 1.
\end{}
... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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916f4fa51b4e52b71a99d62129d2ef02db1f53b2 | subsection | 21 | 39 | Adaptive Algorithm for Corollary | The nature of the data in Corollary means that the algorithm in this subsection will be for a fixed-time problem. As such, the inputs to the algorithm include the data, the domain \Omega , the final time T, a coarse initial mesh \mathcal {T}_0 of \Omega and an unrefined initial time step length k_1 \le T.Suppose that w... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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... | |
e0d9b7165c387a27b545404521dbe8847602c632 | subsection | 22 | 39 | Adaptive Algorithm for Corollary | After this is done, we set k_{m+1} = k_{m} and proceed to the next interval unless the total time t_{m+1} = t_m + k_{m+1} would surpass the final time T in which case we set k_{m+1} = T - t_{m}. When the final time is reached, we halt our computations.All that remains is to deal with the coarse grid and time step lengt... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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881eb4e5403a480e5f8bfb760f7d632459bb4ec0 | subsection | 23 | 39 | Adaptive Algorithm for Theorem | In this subsection, we modify the adaptive algorithm of the previous subsection for adaptivity under the full estimator of Theorem REF ; we do this in such a way that the algorithm of the previous subsection is recovered when f is independent of u. In fact, the only changes that need to be made to the algorithm of the ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1137/16m106073x",
"end": 1392,
"openalex_id": "https://openalex.org/W1524366444",
"raw": "Andrea Cangiani, Emmanuil H. Georgoulis, Irene Kyza, and Stephen Metcalfe, Adaptivity and blow-up detection for nonlinear evolution problems, SIAM ... | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0c3d0651555ba809739395904f181cd7ae4f50db | subsection | 24 | 39 | Adaptive Algorithm for Theorem | On the other hand, the space derivative estimator {\mathop {\eta }\limits ^{\vbox to -.45ex{\hspace{-3.1875pt}\hbox{$\mathbin {\hbox{\scalebox {$${$\m@th .5\bullet $}}}}$\vss }}}_S^m is part of the term \psi _m which is affected by {r}_m and so we ask that
\begin{equation}
\begin{aligned}
\widetilde{r}_{m-1}^{\,-1}{\ma... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
0.007729255128651857,
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5cdfc6d9b23080d75ef3034f0b2bcef7f796b55a | subsection | 25 | 39 | Adaptive Algorithm for Theorem | So upon defining
\begin{equation}
\begin{aligned}\alpha _m := \max \!\left\lbrace 1, \, \widetilde{r}^{\,-1}_{m-1}k^{-1}_m \int _{I_m} \mathcal {L}(s,||U(s)||,||U(s)||+\xi _m) \,\text{d}s\right\rbrace \!,
\end{aligned}
\end{equation}
we combine (\ref {spaceadap1}), (\ref {spaceadap2}) and (\ref {spaceadap3}) to define ... | {
"cite_spans": []
} | 1802.07757 | Pointwise a posteriori error bounds for blow-up in the semilinear heat
equation | [
"Irene Kyza",
"Stephen Metcalfe"
] | [
"math.NA"
] | 2,018 | en | Mathematics | [
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0.015070276334881783,
-0.013338148593902588,... |
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