Morten Skovdal

Catherine Campbell

Claudius Madanhire

C Nyamukapa

Challenges faced by elderly guardians in sustaining the adherence to antiretroviral therapy in HIV-infected children in Zimbabwe

Keywords: Pediatric AIDS, Anti-retroviral therapy, Adherence, Elderly guardians, Zimbabwe Pediatric AIDS, Anti-retroviral therapy, Adherence, Elderly guardians, Zimbabwe

Challenges faced by elderly guardians in sustaining the adherence to antiretroviral therapy in HIV-infected children in Zimbabwe

Introduction

The last couple of years has seen unprecedented progress in the availability of antiretroviral therapy (ART) in sub-Saharan Africa (WHO et al., 2009), not least in Zimbabwe where both primary care and hospitalbased HIV services are reaching some of the most isolated communities. As a result 215,000 (57% of those living with AIDS) people are estimated to access ART in Zimbabwe, with more people enrolling every day (UNAIDS, 2010). A survey carried out in 2008 estimates that 13% of all patients receiving HIV care from the health services in Zimbabwe are between 0-19 years of age, of which 33% are aged 0-4, 25%, 5-9 years; 25%, 10-14 years; and 17%, 15-19 years [START_REF] Ferrand | Survey of children accessing HIV services in a high prevalence setting: time for adolescents to count?[END_REF]. As the fight against HIV and AIDS continues, it is imperative that anti-retroviral drugs (ARVs) are accessible and that virologic suppression remains high. Virologic efficacy, or 'good adherence', is better achieved if patients stick to their ART regimen for 95% of the time [START_REF] Attaran | Adherence to HAART: Africans take medicines more faithfully than North Americans[END_REF]. Good adherence minimises resistance to affordable first line drugs [START_REF] Barth | Long-Term Outcome of Children Receiving Antiretroviral Treatment in Rural South Africa: Substantial Virologic Failure on First-Line Treatment[END_REF][START_REF] Tanuri | Prevalence of mutations related to HIV-1 antiretroviral resistance in Brazilian patients failing HAART[END_REF] but has been identified as a challenge for HIV-infected children and adolescents on ART [START_REF] Arrive | Assessment of adherence to Highly Active Antiretroviral Therapy in a Cohort of African HIV-infected Children in Abidjan, Cote d'Ivoire[END_REF][START_REF] Bikaako-Kajura | Disclosure of HIV status and adherence to daily drug regimens among HIVinfected children in Uganda[END_REF][START_REF] Fassinou | Highly active antiretroviral therapies among HIV-1-infected children in Abidjan, Cote d'Ivoire[END_REF][START_REF] Ferrand | Survey of children accessing HIV services in a high prevalence setting: time for adolescents to count?[END_REF][START_REF] Polisset | Correlates of Adherence to Antiretroviral Therapy in HIV-infected Children in Lomé, Togo, West Africa[END_REF]Weigel et al., 2009). The Zimbabwean survey carried out by Ferrand and colleagues (2010) asked clinic staff from all ART distribution centres in the country about common problems faced by children and adolescents in accessing and adhering to HIV care regimens and they found that for young children (0-5), malnutrition, lack of appropriate drugs, erratic drug taking and hospital visits, orphanhood and caregiver issues (e.g. elderly, ill or unsupportive caregivers) were major issues. For adolescents (10)(11)(12)(13)(14)(15)(16)(17)(18)(19), erratic drug taking was also a major obstacle to adherence, as was lack of disclosure and poor use of counselling and psychosocial services (ibid.). In our interest to develop an understanding of these obstacles to ART adherence, as well as to investigate the interconnections between obstacles to ART adherence and the age of children's caregivers, our focus will be on children who have outlived their parents and live with elderly guardians who also act as their treatment partner.

Although elderly people have been found to play a crucial role in providing care and support for adults infected with HIV [START_REF] Knodel | Older persons AIDS knowledge and willingness to provide care in an impoverished nation: Evidence from Cambodia[END_REF], this is the first study to explore the caring role of elderly guardians in sustaining the ART adherence of children. Whilst many studies have already explored the care and living arrangements of orphaned children and their elderly guardians, these studies have a tendency to highlight how elderly guardians selflessly take on the burden of caring for orphaned and AIDS-affected children -often at the expense of their own health and well-being. Elderly guardians for example are reported to struggle with financial difficulties [START_REF] Nyambedha | Retirement lost" -the new role of the elderly as caretakers for orphans in Western Kenya[END_REF] and suffer from psychosocial distress (Boon, James et al., 2010;Boon, Ruiter et al., 2010;[START_REF] Oburu | Parenting stress and self-reported discipline strategies of Kenyan caregiving grandmothers[END_REF][START_REF] Oburu | Stress related factors among primary and part-time caregiving grandmothers of Kenyan grandchildren[END_REF][START_REF] Ssengonzi | The Plight of Older Persons as Caregivers to People Infected/Affected by HIV/AIDS: Evidence from Uganda[END_REF].

Whilst this may certainly also the case of the elderly guardians reported on in this study, we believe there is a need to be realistic about the capabilities of elderly guardians, particularly in relation to their role in sustaining the ART adherence of HIV-infected children under their care -often under very difficult conditions. Furthermore, we also believe that such studies only give one side of the story. In line with dominant ideological understandings of children as passive beings in need of adult support and guidance (cf. [START_REF] Hutchby | Children and social competence : arenas of action[END_REF][START_REF] James | Theorising Childhood[END_REF], they generally begin with the view of children as the problem ('the burden') and adults as the solution ('the burdened'). We seek to move beyond this conception, building on studies that have highlighted the reciprocity of care and support within such households [START_REF] Abebe | Livelihoods, care and the familial relations of AIDS-affected children in eastern Africa AIDS Care[END_REF][START_REF] Skovdal | Children Caring for their 'Caregivers': Exploring the caring arrangements in households affected by AIDS in Western Kenya[END_REF]. Although elderly guardians do -very importantly -provide orphaned and HIV-infected children with a home and place of belonging, their old age, illness and fragility may limit their contribution towards income and food generation, meaning they depend on the contribution and active participation of children in sustaining their livelihoods [START_REF] Skovdal | Children Caring for their 'Caregivers': Exploring the caring arrangements in households affected by AIDS in Western Kenya[END_REF] and in some cases health [START_REF] Skovdal | Young Carers as Social Actors: Coping Strategies of Children Caring for Ailing or Ageing Guardians in[END_REF]. In this paper we extend this line of thinking by examining the challenges faced by elderly guardians -and the participation of children themselves -in sustaining the ART adherence of HIV-infected children under their foster care. This is an area of concern only previously mentioned in passing (e.g. [START_REF] Ferrand | Survey of children accessing HIV services in a high prevalence setting: time for adolescents to count?[END_REF][START_REF] Meyers | Challenges to Pediatric HIV Care and Treatment in South Africa[END_REF][START_REF] Raiman | A 4-pronged approach to addressing antiretroviral adherence in children: a paediatric pharmacy perspective[END_REF]. In South Africa for example, both [START_REF] Meyers | Challenges to Pediatric HIV Care and Treatment in South Africa[END_REF] and [START_REF] Raiman | A 4-pronged approach to addressing antiretroviral adherence in children: a paediatric pharmacy perspective[END_REF] report that elderly guardians have practical difficulties in administering ARV medication due to their often compromised visual and arithmetic skills. It is against this background, and in our interest to support elderly guardians in their efforts to sustain ART adherence in children, that we seek to explore the question: What challenges do elderly guardians face in contexts of poverty, AIDS and limited social services as they act as the treatment partners of children on ART? In exploring this question we hope gain knowledge that can help us develop programmes that better support the efforts of elderly guardians to provide the best possible treatment support for children under their care.

Methodology

This research forms part of an on-going study which was granted ethical approval from the Medical Research Council of Zimbabwe (A/681) and Imperial College London (ICREC_9_3_13). Informed and written consent was gathered from all research participants with the agreement that their identities would not be revealed. Pseudonyms have therefore been used throughout.

Study population and sampling

The study was conducted in three rural communities in Manicaland province of eastern Zimbabwe. The province is characterised by high levels of poverty and HIV. The three rural communities involved in this study are all served by rural hospitals or health clinics that distribute ART. This paper draws on the perspectives of 25 nurses and 8 elderly guardians of children enrolled onto an ART programme. Although 30 guardians of children with HIV participated in the larger study, these 8 guardians (the only guardians over the age of 50 participating in our study) all referred to their old age as a major physical and economic constraint to children's ART adherence and therefore qualify for inclusion in this paper. All guardians of HIV infected children were recruited through a mix of purposeful, snowball (using village community health workers) and opportunistic (self-selected informants) sampling, and in so doing actively sought to identify and recruit some elderly guardians. The age of the elderly guardians (all female) ranged from 52 to 79, with a mean age of 61.

The 25 nurses participating in this study were recruited from three health clinics on the basis of their willingness to participate. The nurses had a variety of experiences and came with different backgrounds, right from primary care and outreach programmes to HIV testing, ART distribution and palliative care. The nurses tend to live within the compounds of the hospital and have received training on the administration of pediatric ART. The mean age of the 25 nurses (11 female, 14 male) participating in the study was 40.

Data collection and analysis

The data for this study were collected in October and November 2009. Four experienced fieldworkers conducted 26 in-depth interviews (18 with nurses and 8 with elderly guardians) and one focus group discussion with 7 nurses in the local Shona language. The semi-structured topic guides used for the elderly guardians covered informants' personal background, their experiences of AIDS, stigma and being a treatment partners as well as problems and facilitators of children's ART access and adherence.

Individual and group nurse interviews used the same topic guide. The topic guide asked the nurses about their experiences in providing HIV treatment for children, including barriers and facilitators, as well as their interaction with treatment partners. With permission, the interviews were digitally recorded, transcribed and translated into English by the fieldworkers. To ensure the accuracy of transcription and translation, 20% of all transcripts were randomly selected to undergo a quality check, including back-translation. No inaccuracies were identified. Individual and group interview transcripts of both nurses and elderly guardians were imported into the qualitative software package Atlas.Ti for thematic content analysis [START_REF] Flick | An introduction to qualitative research[END_REF]. Data collected by the two study methods and populations were pooled and analysed collectively to provide a holistic and comprehensive understanding of patterns (core themes) within the data corpus [START_REF] Braun | Using thematic analysis in psychology[END_REF]. Using a social constructionist perspective, we sought to map out features of the collectively constructed representational field which shaped both nurses' and carers' understandings and actions, rather than seeking to document attitudes conceived of as properties of individuals [START_REF] Gaskell | Attitudes, social representations and beyond[END_REF]. Following the steps of Attride-Stirling's (2001) thematic network analysis, text segments were first coded with an interpretative title, which were clustered into higher order, or primary themes. The outcome of this thematic network analysis is presented in Figure 1 which highlights the three primary themes that address the research question of this paper and make up the structure of our presentation of findings.

Findings

To contextualise our findings within the lived realities of the elderly guardians acting as treatment partners, Table 1 gives detail to the household composition of each of the eight guardians participating in this study.

What Table 1 encapsulates is the impact of AIDS on families and the fostering role of older people.

Several guardians had lost their husbands and children and were now the foster parents of their grandchildren, young nieces or nephews. As such, the context, however as we will now illustrate, HIV-infected children who live with their elderly guardians may face some barriers to optimal ARV adherence that are unique to their care arrangements. "There are those who want to take advantage of these children. Maybe the child will be the only grandchild but he sends him to the grinding mill, to fetch water, to water the garden some being used that you feel pity. There was a case of a sick child, the grandmother was saying a-aah she works she even goes to fetch firewood. That child died". Peter (35), nurse

Poverty and struggles

In response to the poverty and struggles faced by ARV users in this context, a number of nongovernmental organisations were reported to provide them with food aid, minimising pressures on sick children to engage in income and food generating work as well as providing them with a nutritious diet to support their treatment.

"Some of these children are being cared for by elderly guardians so they lack food but now that problem has been met because a lot of organizations are distributing food in the area." Jackie (36), nurse

Immobility

We have already alluded to the fact that some elderly guardians are unable to generate food and income.

But in what other ways can their immobility compromise children's adherence to ART? ARV users in this context need to go for check-ups and pick up their drugs at a designated health facility on a monthly basis.

Each visit costs US$1, covering the administrative costs of the check-up, which, coupled with potential transport costs, can be a challenge for elderly people to pay. But the immobility of some elderly guardians can also prevent them from accompanying their children to their monthly review dates. This is particularly the case with young children who need to be carried.

"Like me at my age, to think of going there on foot or even to carry the child on my back, it's impossible and for me to get someone to help me take the child to hospital can be impossible.

I can't even think of carrying a child on my back at my age, if I fail to get anyone who is kind enough to help me then there is little I can do." Cellestine (54), elderly guardian

Other times the sheer distance can present as a barrier for guardians who had difficulties walking . They were also more likely to get sick and be periodically immobile, preventing the child from attending its review date and pick up ARV supplies.

"The health of some of the children who stay with their elderly guardians is not good and sometimes elderly people are not strong enough to come with the children to the clinic the distance might be near but the elderly guardian can't walk the child will end up defaulting."

James (36), nurse "Elderly women, they would be coming from far away. For her to send the child by herself is not possible, for her to send someone with the child is difficult because of the issue of stigma, so they will remain at home and miss the appointment. If there was a mobile facility that would have helped". Roselyn (57), nurse However, carers repeatedly stated the strength of their commitment to ensure that children attended their monthly hospital review on the appropriate date.

"The only thing that can stop me would be if I fall ill myself, and I fail to find anyone to escort my child to the hospital, As for my chores at home, I will always leave them behind and take the child to hospital first". Nokutenda (79), elderly guardian For some elderly guardians, remembering the review dates was a bigger challenge than immobility.

Poor memory and comprehension

Several informants spoke of difficulties faced by elderly guardians in remembering drug review dates, often only returning to the hospital when drugs had run out, resulting in a delay in the child's treatment.

"Sometimes these children will be staying with their grandmothers and these grandmothers are old, so they may sometimes forget the review dates, when they are and how many weeks they should wait before going back, so they end up forgetting everything, hence they delay to go and get the children's pills." Marjorie ( 53), elderly guardian Forgetfulness can also mean that children are not reminded to take their drugs at a regular interval, with elderly guardians forgetting when the last drug was taken. This, coupled with a relative complicated administration of drugs may compromise the child's treatment.

"Some of them stay with grandparents who are too old, and who confuse the time or the drugs the child has to take." Evelyn ( 29), nurse

However it may happen that other household members, and children in particular, play an active role in helping their elderly guardians remember when drugs need to be taken.

"The other children are the ones who are reminding me, they remind him to take the tablets and they also tell me that he has taken the tablets and is now leaving for school. properly administer the drugs may be challenging. She doesn't even know the name of the drug she is giving the child. You would have told her the name but she won't remember it, she just knows that the child is on medication, that's very tricky. Some of them they do not have middle aged people close to them to help them with taking care of the child, so its really a challenge." Collin ( 27), nurse

As Miriam, a nurse, explains, many of the elderly guardians are aware of their limitations and actively remind the child to be alert to the advice they receive from the nurses.

"It's a challenge when you have to explain to an elderly person, especially the issue of food, I would be emphasizing what the child needs to eat, I will explain it to the child and I will do it thoroughly so that the child will remind the grandmother. The grandmother will actually call the child to be alert to what we will be saying, you realize that the child is now taking responsibility for all this yet it should be the other way round. When it comes to food, you will hear that the child has eaten only once, yet s/he should be eating more often than this for the drugs, so it's a challenge, there is a problem with elderly caregivers". Judith (34), nurse

Discussion

The findings presented in this paper first and foremost highlight the commitment and dedication elderly people show in ensuring that children under their care comply with the rigid treatment regimens associated with ART, an asset previously identified in Cambodia (Williams et al., 2008) and Thailand [START_REF] Knodel | Older-age Parents and the AIDS Epidemic in Thailand: Changing Impacts in the Era of Antiretroviral Therapy: UNESCAP[END_REF]. In this paper we have illustrated that deteriorating mobility, It is clear that elderly people are ideally positioned as treatment partners to HIV-infected children by virtue of their living arrangements and associated emotional attachments [START_REF] Knodel | Older-age Parents and the AIDS Epidemic in Thailand: Changing Impacts in the Era of Antiretroviral Therapy: UNESCAP[END_REF]. However, as some of the age-related challenges identified in this paper highlight, such arrangements could benefit from externally facilitated support. HelpAge International ( 2008) recommend that elderly carers, especially women, must have access to legal advice, financial support and literacy programmes, so that they can access entitlements for themselves and those in their care. Failure to address the rights of elderly people will undermine their ability to care for others. One way to address poverty-related challenges is to make financial support, such as social pensions, available. Social pensions targeting poor and vulnerable elderly people have been identified as a feasible intervention alleviating the impact of HIV and AIDS [START_REF] Kakwani | Poverty among the elderly in Sub-Saharan Africa and the role of social pensions[END_REF][START_REF] Schatz | Caring and Contributing: The Role of Older Women in Rural South African Multi-generational Households in the HIV/AIDS Era[END_REF]. In the form of monthly cash transfers to elderly people, these could enable already committed elderly guardians to use the funds on direct and indirect expenses related to the treatment of children under their care (such as drugs or hospital fees, transport and food). Child-support grants, as seen in South Africa [START_REF] Case | The reach and impact of Child Support Grants: evidence from KwaZulu-Natal[END_REF] and Kenya [START_REF] Bryant | kenya's cash transfer program: protecting the health and human rights of orphans and vulnerable children[END_REF] also have the potential to alleviate some of the poverty-related challenges identified in this paper as barriers to the adherence of ART in children.

In addition to cash transfers, social networks can be strengthened through community mobilisation and community health workers (or hospital outreach staff) to monitor and address the challenges specific to households with elderly guardians [START_REF] Skovdal | Community-based capital cash transfer to support orphans in Western Kenya: A consumer perspective[END_REF][START_REF] Skovdal | Building orphan competent communities: experiences from a community-based capital cash transfer initiative in Kenya Health Policy and Planning[END_REF]. Such services could encompass providing elderly people with locally appropriate mobility aids and alarmed watches to ensure timely follow-ups and medication as well as making information about treatment regimens accessible to elderly people. One of our informants also highlighted the need for mobile health services, suggesting the need to decentralise ART dispensing to local clinics as well as making outreach staff mobile so that they can reach immobile elderly guardians out in the communities.

This study also highlighted that in contexts of limited support, the agentic capabilities of older children in the household, whether HIV positive themselves or the older siblings of a younger positive child, need to be acknowledged as contributing agents to ART adherence. Not only will such an acknowledgement of children's agency help us move beyond unhelpful understandings of children as a 'burden' to 'burdened' elderly people, it also highlights the importance of involving older children in the treatment process.

To conclude, we believe that much more needs to be done to advocate for policies and programmes that take heed of some of the challenges and resources identified in this paper and promote more family/household centred HIV services that take account of the elderlies' needs for support, and Referee: 1 Comments to the Author This study presents qualitative research done to identify the challenges faced by elderly caregivers supporting adherence to the anti-retroviral therapy (ART) for HIV infected children in Zimbabwe. In order to understand the scope of the problem and develop strategies to maintain adequate treatment of HIV infected children, this question is of importance. The results of this study are of particular relevance for the African countries where elderly guardians provide care for the HIV infected children living in poverty.

The following deficiencies must be addressed:

(1) "Introduction" has to expand the description of the previous publications in this field and their deficiencies, and the innovation in the approach and importance of the present study. Please, clarify the main goal and provide a solid "Introduction' (do not divide the introduction into parts). Combine "Introduction" and "Elderly caregivers in Africa". Avoid repetition by combining the sentences with the same information.

The introduction has been re-written and the two sections have been combined. Repeating sentences have been omitted and we believe the paper now has a more solid introduction with more literature and contextual information framing the contribution of this paper.

(2) "Methodology" must focus on the description of the study population, sampling, and questionnaire must also be presented. Please, clarify the definition of elderly people. Because all of them were grandparents of HIV infected children and age ranges varied widely, I suggest using "grandparents' to describe the study group.

We have provided an overview of the topic guide in the text.

We have expanded on the description of the study population and how they were sampled.

In the first draft of this paper we drew on the perspectives of all child carers (treatment partners of children on ART) participating in the on-going study. However, as the reviewers wanted more detail about the household compositions of elderly guardians and suggest we refer to them as grandparents, we found it impossible to incorporate all 30 child carers in the study. We have therefore decided to only include the perspectives of the 8 elderly guardians participating in the study. The sample size of the study has therefore changed from 25 nurses and 30 child carers to 25 nurses to 8 elderly guardians. We do not feel that grandparent is the appropriate term as two of the elderly guardians are the aunts of the children.

Although the authors presented the type of questions posed to identify the challenges for the grandparents, there is no information regarding the questions the nurses were asked.

We have now added: "The topic guide asked the nurses about their experiences in providing HIV treatment for children, including barriers and facilitators, as well as their interaction with treatment partners." We can't add more details because of he word limit.

Sentences starting with "HIV prevalence….. and Recent years….." should be used in "Introduction".

These sentences have now been moved to the introduction (3) "Findings" should include description of the studied population (gender, age mean +/-SD, level of poverty, employment of the family members, number of children with AIDS in the family, availability of parents, age of infected children, biological relationship with children, type of therapy used).

A table has been produced outlining the details we have of each household. The type of therapy used has been described in the beginning of the findings section (e. The results that were obtained from the two studied populations (grandparents and nurses) must be separated. They can not be combined because the populations are different, and therefore see the problem from different perspectives.

We are of the opinion that populations can be combined in order to identify common themes and patterns across data collection methods and populations. As both nurses and grandmothers spoke about the limitations of old age, we have combined their perspectives to report on more holistic themes that transcend the perspectives of individuals or small groups of people. We appreciate there are different opinions on how to analyse qualitative research and we have therefore constructed a clarifying sentence that outlines our social constructionist way of doing it:

"Data collected by the two study methods and populations were pooled and analysed collectively to provide a holistic and comprehensive understanding of patterns (core themes) within the data corpus [START_REF] Braun | Using thematic analysis in psychology[END_REF]. Using a social constructionist perspective, we sought to map out features of the collectively constructed representational field which shaped both nurses' and carers' understandings and actions, rather than seeking to document attitudes conceived of as properties of individuals [START_REF] Gaskell | Attitudes, social representations and beyond[END_REF]."

Please describe the activities in the focus group. Such as, it is not clear how many did not participate in the discussion.

All 7 participants of the group discussion participated actively in the interview. We have clarified that only 7 nurses participated through a group discussion. There were no particular activities in the nurse group discussions, so we have further highlighted that a topic guide was used and what questions it covered:

"Four experienced fieldworkers conducted 26 in-depth interviews (18 with nurses and 8 with the grandmothers) and one focus group discussion with 7 nurses in the local Shona language. The semistructured topic guides used for the grandmothers covered informants' personal background, their experiences of AIDS, stigma and being a treatment partners as well as problems and facilitators of children's ART access and adherence. Individual and group nurse interviews used the same topic guide. The topic guide asked the nurses about their experiences in providing HIV treatment for children, including barriers and facilitators, as well as their interaction with treatment partners."

(4) "Implications for policy and practice" should be started with the conclusion, otherwise it is not clear what results can be used to change the policy and improve the practice. No study is needed to identify the financial, literacy, and health problems faced by the grandparents of HIV infected children in Zimbabwe. I think the discussion has to emphasize the main point -How to decrease the impact of the grandparent's financial, literacy, and health problems on adherence to the medical treatment of the HIV infected children. Because eliminating poverty in a country is a long-term process, however, HIV infected children need adequate treatment today, and cannot wait for tomorrow.

It would be important to identify the participants' opinion regarding their needs to maintain adherence to ART treatment of their grandchildren or foster children. No such question was discussed in the present study.

We think that strategies to maintain adherence were presented in the findings section, albeit we acknowledge that these were not flagged up adequately in the discussion. The quotes selected for the manuscript were carefully selected to illustrate both their needs and potential strategies. Whilst this was not always possible to find such a quote, many of the quotes in this paper do point to what e.g. elderly treatment partners would suggest is a strategy to overcome their obstacle (although sometimes this would only be implicit) (e.g. the need for food aid, need for health facilities to be closer to them, mobile facilities, increased involvement of the children etc.). We acknowledge that we do not explicitly say what the recommendations were, but we have amended the discussion accordingly: e.g. "Whilst the obstacles faced by elderly treatment partners took precedence in this paper, our informants did also highlight potential strategies that can help elderly people in their role as treatment partners of HIV infected children. These included the distribution of food aid, the involvement of other children within the household, stigma-reduction programmes, increasing social support networks, mobile health facilities and improving the physical access to health facilities (e.g. constructing satellite clinics to reduce the distance)."

Overall main deficiencies of the present study: (i) participants were not selected randomly and this decries external validity of the present study;

We use a social constructionist conceptual framework -the extent to which the findings would be generalisable to other populations would have to be assessed by a skilled observer on a case by case basis (Flyvberg (2001) Making social science matter : why social inquiry fails and how it can count again), and we use the criterion of PUBLIC ACCOUNTABILITY (Bauer and Gaskell, Chapter 19) rather than validity and reliability. We don't have the word count to explain this at length, however we believe that most social scientists should be familiar with a social constructionist perspective, and that having identified it in our methodology section, no more detailed discussion is necessary (ii) no separation of results between the two groups of participants (grandparents and nurses); (iii) no discussion regarding the participants (grandparents and nurses) vision regarding what would be really helpful for the families in poverty, low literacy, and poor health to provide adequate care of the HIV infected children.

The Table presented at the start of the findings section seeks to provide more contextual background.

Referee: 2 Comments to the Author Article Summary:

This article details a qualitative study of challenges to maintaining antiretroviral adherence in African children in the care of elderly guardians. The identification of these challenges was based on interviews and focus groups conducted with caregivers and nurses in 3 communities in Zimbabwe. Elderly caregivers play an important role in caring for children with HIV in many African nations, particularly for orphaned children. However, the age and living conditions of these caregivers give rise to several specific difficulties with respect to maintaining adequate adherence to ARV medications. The primary challenges to adherence identified were poverty, immobility, poor memory and inadequate understanding of medication instructions on the part of caregivers.

This study collected interview and focus group responses from thirty caregivers and twenty-five nurses from three rural communities in Zimbabwe. Respondents ranged in age from 39 to 79 with a mean age of 54. This information was collected by four field researchers over the course of two months. Discussions were conducted in the local language and subsequently translated into English. Themes from the interviews were derived through software analysis. The authors suggest a need for increased social services, such as mobile clinics, social pensions, or child welfare grants in order to address the needs of this population. The authors also encourage further development of the positive resource of older children as assistants in the process of maintaining medication adherence. This manuscript has the potential to contribute to the literature, however some important albeit minor concerns need to be addressed as follows:

Minor Concerns:

1.

More specific demographic information would be helpful. Caregiver age range and mean age are provided, however, one is left to assume all caregivers are female. It would be helpful to include caregiver gender, caregiver health status (HIV+ vs other medical condition, if known), how many HIV+ children vs total children each caregiver is responsible for, and the age ranges of these children. Given the age range, mode of caregiver age might be more meaningful, or at the very least to indicate how many were over age 50.

At the start of the findings section we have included a table that depicts most of the household demographics the reviewer is looking for. Unfortunately, we do not have information about the caregiver health status.

It would also be helpful to know when referring to older children, what age is meant by "older". It would also be important to know how many from each community participated to know whether any one community was more open than the other(s). Similar nurse demographics would be informative.

We have included nurse age and gender demographics in the methodology. We have defined "older" as over the age of 10, which also reflects the findings of Ferrand as referenced in the introduction.

2.

Given it is reported 30 guardians and 25 nurses participated, it would be helpful to know how many of each were interviewed individually versus participated in a focus group. How many focus groups were conducted in each community and how many participants were in each focus group? Given the mention of stigma, how difficult was focus group recruitment and how open was the discussion? "Four experienced fieldworkers conducted 26 in-depth interviews (18 with nurses and 8 with the grandmothers) and one focus group discussion with 7 nurses in the local Shona language." As the one focus group discussion involved in this study was with nurses who have been trained on AIDS and work with AIDS patients on a daily basis, stigma was not an issue for the group discussion.

3.

Were the three communities included in the study analyzed for any differences between them? If so, were any differences noted?

No, the communities were not analysed for any difference. We included three communities to optimise variety in responses and to get increase the number of participants. The frame of reference of the paper is Manicaland province, all Shona and rural or semi-rural as specified in the methodology section. We do not feel that a fine grained analysis of differences between the communities would have yielded helpful findings in light of research question/objectives.

4.

Similarly, were there any differences observed by recruitment method?

No, this was not the focus of the paper. The aim was to generate patterns of core themes across methods and participants.

5.

There's always a question of accuracy given transcription and translation between languages. It is noted that interviews were conducted in the native language and translated into English. Was backtranslation from English to the native language conducted to assure intended meaning of the communication was upheld in the translation? Yes, due to word limits this was not previously included. We have now included a sentence in the methodology which reads: "To ensure the accuracy of translation, two interviews were randomly selected and back-translated. No inaccuracies were identified."

6.

Figure 1 is not discussed in the text. While the figure poses challenges elderly caregivers face, HIV+ children working and children helping with medications appear to be solutions to the problems and should be depicted differently as real-life accommodations to the challenges faced. A second Figure could include these factors as accommodations that occur in reality with an overlay of proposed solutions to address the problems.

That's a really good point. We are now referring to Figure 1 in the manuscript text and have dotted the two boxes in figure 1 that mention facilitators to the challenges faced by elderly guardians. We felt this was the best way to distinguish between challenges and facilitators and keep 1 figure.

7.

The authors discuss the potential positive effects of older children assisting in promoting adherence. It would be helpful to know how many reported assistance from children who are HIV+ versus siblings who are negative.

I am afraid we do not have any numbers to highlight this. However, to help the reader make his or her own judgement on this topic, we have tried to provide some more detail to the households in which grandparents are caring for children on ART. See newly added table for example.

8.

NGO is not defined (page 5).

Apologies. As we only refer to NGOs once, we have spelled out the abbreviation.

9.

Minor grammatical and typographical errors need to be addressed throughout.

We have had the manuscript proof read.

Figure 1 :

 1 Figure 1: Thematic Network: Challenges (lessened by children's agency, see dotted boxes) faced by elderly guardians in sustaining the ART adherence of HIV-infected children

  Households made up of elderly guardians and orphaned or AIDS-affected children are particularly vulnerable to poverty and related struggles. Such struggles are faced by Joanna who is too old to work and has difficulties paying for the schooling and clothing of her foster children. "The problems I am facing are monetary including schooling and clothing. All these are solely my responsibility at the same time I am too old to work and this has been a major problem." Joanna (52), elderly guardian Such struggles often result in children having to carry the responsibility of contributing to household sustenance, and on occasion HIV-infected and ill children are forced to engage in work. As one nurse commented:

  memory and comprehension of complicated treatment plans, coupled with poverty in a context of limited social welfare and government support, make it difficult for some elderly guardians to sustain the treatment of HIV-infected children in their care -presenting a key adherence challenge for pediatric ART. Our findings suggest that support for elderly carers should address both their poverty-and age-related needs. Although these needs are interrelated, they require different interventions. Age-related challenges refer to their physical limitations (deteriorating memory, comprehension and mobility) and poverty-related challenges refer to their lack of income and food, encouraging the infected child to engage in income generating activities beyond what is good for them. Whilst the obstacles faced by elderly guardians took precedence in this paper, our informants did also highlight potential strategies that can help elderly people in their role as treatment partners children. These included the distribution of food aid, encouraging children's agency and participation in the treatment, stigma-reduction programmes, increasing social support networks, mobile health facilities and improving the physical access to health facilities (e.g. constructing satellite clinics to reduce the distance).

(

  over the age of 10 as defined by[START_REF] Ferrand | Survey of children accessing HIV services in a high prevalence setting: time for adolescents to count?[END_REF] play a key role in coping with some of the challenges identified in this paper. Older children help with income and food generating activities and remind guardians about review dates and the administration of drugs. Therefore, to ensure the adherence to ART in children under the care of elderly guardian, there is an urgent need for HIV services to embrace and involve both the children and their elderly guardians in HIV programme planning. Older children living

  Figure 1: Thematic Network: Challenges (lessened by children's agency, see dotted boxes) faced by elderly guardians in sustaining the ART adherence of HIV-infected children

Table 1 : Household characteristics of elderly guardians providing HIV care for children Overall

 1 Table1were all on first line treatment (combination of Stavudine, Lamivudine and Nevirapine) which was administered on a weight-based system in tablets. Although syrups, which are more easily administered to younger children, are available, these are hard to access in Zimbabwe. , great progress has been made in achieving children's access and adherence to ART in this

	table also highlights that elderly people were not

  Most of the children we have who are on ART are cared for by people who are very old. We give people adherence calendars that they should fill in, but what if she can't write, if she is 80 something years? The child may be around 4 or 6 years, so for her to write down and to

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" Nokutenda (79), elderly guardian "No I do not forget to give the child the drugs. Even if I was to forget this kid himself would remind me. You hear him saying: 'Granny its six o'clock, time for my medication'." Violet (67), elderly guardian A number of nurses spoke of problems in communicating with elderly guardians. They said the guardians often failed to understand the complexity of the child's treatment regimen. Nurses go to great lengths to ensure that treatment partners -people (often household members) who play a primary role in facilitating the treatment regimen of ARV users -have the tools and knowledge to facilitate child adherence, but these are often inadequate for elderly people who are unable to read and write. This is particularly a problem if the child in their care is still young and cannot assist keeping notes in their treatment schedule. "

  acknowledge and enhance the agency of older children as active and responsible contributors to ART adherence. However, as this is a new area of research, further investigation into the prevalence of elderly treatment partners and the obstacles they are facing (e.g. segregated by the children's age) as well as the experiences of elderly guardians in other settings is urgently needed.

	Weigel, R., Makwiza, I., Nyirenda, J., Chiunguzeni, D., Phiri, S., & Theobald, S. (2009).
	Supporting children to adhere to anti-retroviral therapy in urban Malawi: multi method
	insights. BMC Pediatrics, 9(1), 45.
	WHO, UNAIDS, & UNICEF. (2009). Towards Universal Access: Scaling up priority HIV/AIDs
	interventions in the health sector -Progress Report (September 2009 ed.). Geneva:
	WHO Press http://www.who.int/hiv/pub/tuapr_2009_en.pdf (retrieved 12/05/2010).
	Williams, N., Knodel, J., Kim, S., Puch, S., & Saengtienchai, C. (2008). Overlooked potential:
	older-age parents in the era of ART. AIDS Care, 20(10), 1169-1176.
	F o F o F o
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	e e r e e r e e r
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	O n O n O n
	l y l y l y

Table 1 : Household characteristics of elderly guardians providing HIV care for children Elderly guardian Household characteristics

 1

		Nokutenda stays with five grandchildren between the ages
		4-12, of which one 8-year-old girl is HIV positive and on
	Nokutenda, age 79	ART. Nokutenda's husband, two sons and two daughters
		have all died and she survives through subsistence
		farming.
		Sandra lives with her husband, mother-in-law, her sick
		daughter and her three grandchildren. One of her
	Sandra, age 59	grandchildren, a 16-year-old girl, is HIV positive and on
		ART. She and her husband provide for the household
		through subsistence farming.
		Carolyn stays with her brother and 6-year-old niece, both
		of who are HIV positive and on ART. The 6-year-old girl is
	Carolyn, age 56	separated from her older brother who is under the care of
		other family members. Carolyn's son occasionally visits
		and supports Carolyn financially.
		Tanyaradzwa stays with her two great grandchildren,
		including a 4-year-old boy is HIV positive and on ART.
	Tanyaradzwa, age 70	Her grandson (children's father) has gone to the city to look for work. Their mother has died. Her 10-year-old
		great granddaughter helps Tanyaradzwa to care for the
		boy.
		Violet stays with three grandsons, the oldest being 12,
		HIV positive and on ART. Her son (father of her
	Violet, age 67	grandchildren) and her daughter-in-law have both died.
		Violet sustains the household through subsistence
		farming.
		Joanna stays her ill son, her sister's 20-year-old daughter
	Joanna, age 52	and her brother's three boys, of which a 9-year-old is HIV positive and on ART. Joanna works as a sales woman,
		buying and selling goods at the local market.
		Cellestine and her husband live with their three
	Cellestine, age 54	grandchildren, of which the oldest, a 11-year-old girl is HIV positive and on ART. Cellestine and husband provide
		for the household through subsistence farming.
		Marjorie stays with her 6-year-old grandson who is HIV
		positive and on ART. She stayed with her daughter and
	Marjorie, age 53	son-in-law until recently when they both died. Marjorie
		sustains herself and her grandson through subsistence
		farming.

  g. "The HIV positive children on ART in the eight households described in Table1are all on first line treatment (combination of Stavudine, Lamivudine and Nevirapine) which are administered on a weight-based system in tablets. Although syrups, which are more easily administered to younger children, are available, these are more difficult to access and are limited in Zimbabwe".)

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http://mc.manuscriptcentral.com/ac-phm-vcy Health Sciences

http://mc.manuscriptcentral.com/ac-phm-vcy


Jesús A López Fernández

Miguel López Portugués

José Manuel

Taboada Varela

Henry J Rice

Fernando Obelleiro

Miguel López
email: mlopez@tsc.uniovi.es

López Fernández

J A López-Fernández

J M Taboada

HP-FASS: A Hybrid Parallel Fast Acoustic Scattering Solver. CMMSE-09

Keywords: Acoustic scattering, Boundary Elements Method, Fast Multipole Method, MPI, OpenMP Acoustic scattering, Boundary Elements Method, Fast Multipole Method, MPI, OpenMP AMS Subject Classification: 45-04, 45J05

come   L'archive ouverte pluridisciplinaire

Introduction

Low-frequency methods, such as the Boundary Elements Method (BEM) [START_REF] Wu | Boundary Element Acoustics[END_REF], offer the potential of precise numerical formulations of acoustic scattering problems. The use of the BEM on this type of problems yields a linear system of N equations with N unknowns. The computational cost associated to the direct solution is O N 3 in terms of time and O N 2 in terms of memory, meanwhile, iterative solvers have an O N 2 time cost per iteration. Hence, if the number of iterations to achieve certain residual error is smaller than N , which is usually the case, a reduction in the execution time is achieved. Among the different iterative solvers applied to acoustic scattering, it is worth mentioning the Generalized Minimum Residual (GMRES) method [START_REF] Saad | GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems[END_REF] which is a very robust solver for this type of problems [START_REF] Marburg | Performance of iterative solvers for acoustic problems. Part I. Solvers and effect of diagonal preconditioning[END_REF]. The complexity, O N 2 per iteration, of iterative solvers is due to the computation of a Matrix-Vector Product (MVP). The Fast Multipole Method (FMM) [START_REF] Rokhlin | Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions[END_REF] and its multilevel version, known as Multilevel Fast Multipole Algorithm (MLFMA) [START_REF] Greengard | Accelerating fast multipole methods for the Helmholtz equation at low frequencies[END_REF][START_REF] Song | Multilevel Fast-Multipole Algorithm for Solving Combined Field Integral Equations of Electromagnetic Scattering[END_REF], avoid matrix explicit calculation yielding a dramatic reduction in the MVP time without significantly affecting to BEM's iterative solution accuracy. The FMM and the MLFMA reduce the cost per iteration to O N 1.5 and to O (N log (N )), respectively.

For the last decade, the interest in using parallel computing to decrease solution times and to enlarge the scope of application of BEM solvers has been growing. On the one hand, the parallel implementation of a direct solver (based on LU factorisation) presented in [START_REF] Geng | Massively Parallel Computation for Acoustical Scattering Problems using Boundary Element Methods[END_REF] achieved quite good scalability, although, its application to big size problems was limited by the high computational cost of the solver. On the other hand, the different approaches reported of the parallel MLFMA achieve poor scalability results due to the interdependence of the data required by the different hierarchy levels [START_REF] Waltz | Massively Parallel Fast Multipole Method Solutions of Large Electromagnetic Scattering Problems[END_REF].

In this work, a hybrid parallel scattering solver tool is developed. The GMRES is chosen, as iterative solver, due to its robustness for solving acoustic scattering problems [START_REF] Marburg | Performance of iterative solvers for acoustic problems. Part I. Solvers and effect of diagonal preconditioning[END_REF]. No preconditioners have been used in the iterative solution. In addition, the FMM is employed as accelerator for the MVP that each GMRES iteration requires.

This paper is organised as follows. In section §2, the statement of the acoustic scattering problem is posed. Section §3 is a brief summary of the FMM. Section §4 describes the hybrid parallel implementation of the FMM developed in this work. Section §5 is a benchmarking study of the proposed implementation. Finally, some conclusions are discussed in section §6.

Statement of the Physical Problem

The physical problem studied here consists in predicting the acoustic pressure on the space that surrounds a 3-D obstacle on which an incident acoustic wave is impinging. This problem may be possed in terms of the integral form of the Helmholtz equation [START_REF] Wu | Boundary Element Acoustics[END_REF] which is also known as Conventional Boundary Integral Equation (CBIE). Thus, the solution of the scattering problem may be carried out into two steps. First, the CBIE applied to the surface S of the obstacle is solved. This yields the distribution of the pressure over S. Second, the solution provided by the first step is used to obtain the scattered pressure by means of the application of the CBIE to the acoustic domain. This work is focused on the first step for solving the scattering problem, that is usually the one that demands more computational resources. In order to overcome the non-uniqueness difficulty [START_REF] Burton | The Application of Integral Equation Methods to the Numerical Solution of Some Exterior Boundary-Value Problems[END_REF] that appears in the CBIE at resonant frequencies, the CBIE is linearly combined with its normal derivative yielding the Burton and Miller equation [START_REF] Burton | The Application of Integral Equation Methods to the Numerical Solution of Some Exterior Boundary-Value Problems[END_REF]. By means of the BEM, the Burton and Miller equation may be discretised over S and formulated in terms of a linear system of equations in the form:

K p = g, ( 1
)
where K is the coupling or system matrix, p is the coefficients vector, and g is the excitation vector which is related to the incident pressure.

In the sequel, the surface of the obstacle is considered to be discretised into N elements which means that the linear system in (1) has N unknowns. In order to produce accurate results, S must be discretised into 6 to 10 elements per linear wavelength (λ). Since the wavelength and the frequency f are in inverse proportion, N increases, for a given obstacle, with the frequency squared f 2 .

Accelerated Iterative Solution

The solution of equation ( 1) may be achieved by means of an iterative method such as the GMRES [START_REF] Saad | GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems[END_REF] which is an extension of Minimum Residual (MINRES), designed to solve nonsymmetric linear systems. The GMRES is, among the Krylov solvers, the most efficient for solving acoustic scattering problems [START_REF] Marburg | Performance of iterative solvers for acoustic problems. Part I. Solvers and effect of diagonal preconditioning[END_REF]. The most time consuming operation in Krylov solvers is the MVP to get the iterate and the residual error (stop condition). The GMRES computes the residual norm without the iterate having to be explicitly formed, which reduces to one the number of MVPs per iteration. Once the residual norm is deemed small enough, the iterate is formed.

The FMM computes a MVP without explicit calculation of the matrix. The algorithm requires grouping the N elements of the problem into N g groups and then it efficiently calculates far interactions, between non-neighboring groups, by means of the Addition Theorems and plane wave decomposition [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF]. Near interactions, between elements that pertain to the same group or to neighboring groups, do not satisfy the conditions associated with the transformations described by the Addition Theorems [START_REF] Rokhlin | Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions[END_REF]. As a consequence, near (or local) interactions must be computed directly evaluating its corresponding part of the system matrix. The whole procedure may be summarised as follows:

(1) Setup: the geometry is divided into groups of elements. This step is also known as spatial grouping and it pertains to the setup of the algorithm. Nonetheless, it is an important step because the way the geometry division is performed may yield important computational savings. The use of octtree structures in combination with hierarchical numbering [START_REF] Nail | Data Structures, Optimal Choice of Parameters, and Complexity Results for Generalized Multilevel Fast Multipole Methods in d Dimensions[END_REF] produces very efficient spatial ordering calculations needed in the FMM. (2) Near interactions: the radiation of the elements of each group over elements pertaining to itself and to nearby groups is evaluated directly, see Fig. 1. (3) Far interactions: the radiation of the elements of each group over elements pertaining to far groups is efficiently computed by means of the following operations: a) Aggregation: the radiation produced by the elements of each group is represented by a multipole expansion placed at the group centre, see the schematic drawing in Fig. 1. b) Translation: the above-mentioned multipole representation is translated from each source group centre to each observation group centre, see Fig. 1. Source group translated contributions are added together at the centre of each observation group. The translation operation is diagonalised which leads to the major computational saving of the method [START_REF] Rokhlin | Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions[END_REF]. c) Disaggregation: the radiation produced by the observation group centre multipole at every observation element within the group, see Fig. 1, is computed. (4) Near-Far interactions addition: near and far interactions are added together yielding the MVP without explicit formation of the system matrix.

The number of multipoles (L) that are necessary to represent the field as an expansion at each group centre and the average number of near elements to each one (N c ) are proportional to the wavelength size of the group [START_REF] Rokhlin | Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions[END_REF]. The computational cost savings produced when using FMM instead of direct MVP comes from the fact that instead of having to compute the interaction between every pair of elements, O N 2 operations, only the interactions between far groups, O L 2 N 2 g operations, and between close elements, O N g N 2 c operations, are computed. It is possible to show that the optimum number of groups increases proportionally to √ N [START_REF] Rokhlin | Diagonal Forms of Translation Operators for the Helmholtz Equation in Three Dimensions[END_REF], yielding a cost O N 1.5 . In this work, the efficient computation of elements-groups relations by means of oct-tree structures is exploited in order to select a group size that diminishes as possible the execution time while requiring an O(N ) memory cost.

Parallel Implementation

The software presented in this work (HP-FASS, Hybrid Parallel Fast Acoustic Scattering Solver ) has two main computational goals: low memory footprint and high scalability. To achieve these goals we developed a hybrid parallel implementation using OpenMP [START_REF][END_REF] threads and the Message Passing Interface (MPI) [START_REF]Message Passing Interface Forum, Message-Passing Interface Standard[END_REF][START_REF]MPI-2: Extensions to the Message-Passing Interface[END_REF].

To achieve a low memory footprint two different actions were taken. To begin with, data structures of size O(N 1.5 ), typical of common FMM implementations, are completely avoided which diminishes the memory cost to O(N ). Nonetheless, this memory saving strategy may increase the constant factor in the time cost. In addition, the parallel implementation has been carried out paying special attention to reduce as much as possible data replication among processes. First, the computation associated to the aggregation and disaggregation steps is distributed using groups, minimising the replication of the geometry data. Furthermore, the computation associated to the translation step is distributed using integration points, avoiding any replication of the translation operator.

High scalability is the second goal of the parallel implementation proposed in here. Since the group size is critical to minimise the execution time and improve the scalability, the following model is used to calculate the time cost of each MVP: where N is the number of unknowns and p is the number of processors (cores).

T M V P (N, p) = T near (N, p) + T aggregation (N, p) + T translation (N, p) + T disaggregation (N, p) + T addition (N, p), (2)
In order to obtain each term of the equation ( 2), a model that includes both computation and communication has been used. The computation time is calculated taking into account the number of operations associated to each step of the algorithm and the correspondent machine-dependent constant factors, the latter estimated using empirical data. The communications time is calculated taking into consideration the number and the size of the messages, together with the estimated network latency (α) and the estimated transfer time per byte (β).

The time related to the near interactions has been estimated by the following equation:

T near (N, p) = K near Ng i=1 n i • n near i p + α + β N p , (3)
where N g is the number of groups, n i is the number of elements in the i th group, n near i is the number of elements in the neighbourhood of the i th group and K near is the near interactions constant factor.

The time related to the far interactions has been approximated by means of the following set of equations:

T aggregation (N, p) = K agg k l N p + (p -1) α + β k l N g p , (4a)
T translation (N, p) = K trans k l Ng i=1 g f ar i p + (p -1) α + β k l N g p , (4b)
T disaggregation (N, p) = K dis k l N p + α + β N p , (4c)
where k l is the number of integration points (related to the square of the number of multipoles), g f ar i is the number of distant groups to the i th group, and • is the ceiling function. K agg , K trans , and K dis are the constant factors for the aggregation, translation, and disaggregation steps, respectively.

Finally, the time taken by the addition of near and far interactions follows:

T addition (N, p) = K add N, (5)
where K add is the addition constant factor. In the setup stage (at runtime), the algorithm exhaustively tests different group sizes and precisely calculates the time cost of the MVP for the resulting spatial groupings. In this manner, the optimum group size, in the sense of minimizing the time cost, is chosen for an arbitrary problem solved in any number of processors. This technique results in an adaptive group size as opposed to the most common fixed group size. It is worth mentioning that taking advantage of the oct-tree theory [START_REF] Nail | Data Structures, Optimal Choice of Parameters, and Complexity Results for Generalized Multilevel Fast Multipole Methods in d Dimensions[END_REF], the time cost of searching for the optimum group size is O(N ).

A methodical procedure [START_REF] Foster | Designing and Building Parallel Programs: Concepts and Tools for Parallel Software Engineering[END_REF] has been followed to design and develop this software, taking into account partitioning, load balancing, and communications. In addition, HP-FASS has been developed using the masteronly programming model in which only one thread per process (master thread) communicates. Thus, MPI communications take place only outside of OpenMP parallel regions, avoiding possible A domain decomposition technique taken at two levels is used in this work. In aggregation, disaggregation, and near interactions, both coarse (MPI) and fine (OpenMP) partitions use the group as minimum computing entity. However, in the translation step a new approach is provided: fine partition is taken at group level whereas coarse partition is taken using integration points. This technique is scalable since both groups and integration points scale with √ N . It is worth mentioning that there is not redundant computation, resulting in scalable codes, and there is minimum data replication, which yields a very low memory footprint.

Load balancing is accomplished at a very low cost thanks to the oct-tree based space subdivision in addition to a hierarchical numbering. As above-mentioned, an optimal geometry subdivision trying different group sizes is performed during the setup of the algorithm. Then, the computational cost associated to each group is computed at runtime taking advantage of the oct-tree structure. Finally, a simple algorithm is used to distribute consecutive groups and integration points among processes, achieving a well-balanced workload at a very low cost. There are also two different levels to deal with load balancing: the MPI coarse level, explained above, and the OpenMP fine level. Coarse level uses a static load-balancing scheme whereas fine level group assignment is done on demand thanks to OpenMP dynamic and guided schedulers. This dynamic scheme easily corrects runtime imbalances among threads inside a process.

The main interchanges of data, those performed at the end of aggregation and translation, are carried out by means of point-to-point non-blocking communications that take place between neighbour processes. Since the communication partners do not change over time, the processes are organised as a static ring structure, allowing an efficient interchange of the data computed by each process. In addition, the MPI collectives are used as little as possible with the aim of limiting the impact of global communications in the scalability. Global communications that are performed throughout the algorithm have a static tree structure, where one process is the root and the other ones are leafs. MPI Bcast is used to distribute groups data during the setup step and to provide the residual error (stop condition) at the end of each iteration. MPI Scatterv is employed to spread the geometry information during the setup just as to distribute, at the beginning of each iteration, the updated pressure vectors. Finally, MPI Gatherv is used to collect computed data (far interactions, near interactions, and solution pressure vectors once the desired residual error is reached).

Results

The results shown in this work have been obtained using the CMI cluster, which is part of the Clúster de Modelización Científica (Universidad de Oviedo -Spain). CMI consists of 32 nodes (with Red Hat EL 4 Update 7) linked through a 20 Gbps InfiniBand network. Each node contains 2 AMD Opteron quad-core processors at 2.3 GHz and 32 GB of RAM, resulting in a total of 256 cores and 1 TB of RAM. Furthermore, we have used Intel MKL 10.0.011 for linear algebra operations, HP-MPI 2.3 middleware for communications over InfiniBand, and AMD/GCC 4.2 to compile our source code. It is also worth mentioning that single precision complex arithmetic is used in our codes.

The nodes in the CMI cluster have a cache-coherent Non-Uniform Memory Access architecture (ccNUMA), where each CPU socket (4 cores) has an integrated memory controller and can access the local memory faster than the remote mem- ory. Therefore, each node may be viewed as two different nodes linked trough a HyperTransport [START_REF][END_REF] link. We have taken this into consideration and we have carried out the parallel executions using 2 MPI processes per node and 4 OpenMP threads per process, taking into account the CPU binding to maximise memory bandwidth and to minimise latency.

Since the availability of the cluster for benchmarking is limited, we decided to analyse rather small problems. The geometry of the obstacle analysed corresponds to an aircraft model of an Airbus A300 series used in the wind tunnel test. This aircraft has one wing and a type of wing-shaped pylon to stand the model. The geometry has been used to obtain the distribution of the acoustic pressure at two different frequencies: 500 Hz and 1 kHz. As a consequence, two different meshes have been used to model the geometry. One of them consists of 253836 triangles (500 Hz frequency) and the other one consists of 1009392 (1 kHz frequency) triangles. Each triangle corresponds to one unknown in the problem.

Figure 2 shows the acoustic pressure on the wing of the aircraft. The engine noise is modelled by means of two point sources placed over the wing. As we have shown earlier in the paper, the FMM algorithm consists of two parts: an initial setup and standard iterations. The initial setup is only performed once at the beginning of the problem solution whereas the number of required iterations is very dependent upon the geometry, requiring some tens of iterations in the case of an aircraft like the one shown. As a consequence, the setup usually takes much less time than the iterative solution. Hence, we chose to focus the study of the scalability on the MVP, that is the most computationally demanding operation of each iteration.

Table 1 shows some benchmarking data and the group size (the cube shaped group side measured in wavelengths) when using the adaptive group size algorithm. It is worth mentioning that 80 and 92 GMRES iterations were necessary to achieve a residual error smaller than 10 -2 , for the problems at 500 Hz and 1 kHz, respectively. In this work, the residual error of the predicted total pressure on the aircraft, used to stop the GMRES algorithm, is defined as follows:

= Kp (i) -g 2 g 2 , (6)
where • 2 is the Euclidean norm (or 2-norm), p (i) is the iterate at step (i), K is the system matrix, and g is the excitation vector. From Table 1, it is possible to see that, for a given group size, a low increase of the memory use is produced with increasing the processors count. In addition, since the memory used by the FMM implemented in here is related in inverse proportion to the group size, the memory use may diminish when increasing the processors count if the group size also increases. It is interesting to note that the direct solution of these problems -253836 and 1009392 unknowns-would require an approximate memory of 480 GB and 7.4 TB, respectively. Meanwhile, the HP-FASS requires no more than 1.6 GB and 7.5 GB, respectively.

E p = T 1 p T p (7)
Figure 4 shows the parallel efficiency of the MVP for both problems analysed in this study. For the whole cluster usage and moderate size problems, the tool developed in this work provides an efficiency around 60% (using the adaptive group size implementation). It is also worth mentioning the steep drop in the efficiency from 1 core to 8 cores. For the execution on 1 core, the only thread has all the level 3 cache (2 MB) and the full memory bandwidth (10.7 GB/s) of the CPU. Meanwhile, for the execution on 8 cores, 4 threads have to share the level 3 cache and the memory bandwidth of each CPU. This reduction in cache and memory bandwidth per thread has an impact over the translation step that is memory bounded.

Conclusions

A hybrid implementation of the FMM applied to acoustic scattering problems is presented in this work. The developed software tool incorporates some techniques suggested by the authors that diminish as possible the constant factor of the FMM time cost while requiring an O(N ) memory cost. In addition, the memory footprint grows very slowly with increasing processors counts thanks to a low data replication. The scalability of the above-mentioned tool is demonstrated by means of execution times (MVP and global solution times) and efficiency metrics. The combination of the FMM as speedup for the MVP required by an iterative method with the highly scalable implementation shown in this paper results on an interesting engineering tool for noise control applications.

Figure 1 .

 1 Figure 1. Illustration of near interactions vs. far interactions computed by FMM.

  inter-node network bandwidth. As a result, this programming model is well suited to hybrid systems (clusters of multi-core nodes).

Figure 2 .

 2 Figure 2. Real part of the pressure on the wing of an aircraft model. Frequency: 500 Hz.

Figure 3 .

 3 (a) shows the global solution time for the small problem whereas Figure3.(b) shows the global solution time for the big problem. Results for fixed group size vs. adaptive group size are shown in the plots. It is interesting to note the differences among both implementations of HP-FASS. Using the whole cluster, the new implementation with the adaptive group size selection reduces time-to-solution from more than 11 hours to less than 5 minutes for the small problem, and from almost 93 hours to less than 38 minutes for the big one.

Figure 3 .

 3 Figure 3. Global solution time of adaptive vs. fixed group size. (a) N = 253836. (b) N = 1009392.

Figure 4 .

 4 Figure 4. MVP efficiency of adaptive vs. fixed group size. (a) N = 253836. (b) N = 1009392.

Table 1 .

 1 MVP time, global solution time, group size, and total memory consumption for HP-FASS

			253836 unknowns			1009392 unknowns
		MVP	Global	Group	Cluster	MVP	Global	Group	Cluster
	Cores		Solution	Size	Mem. use		Solution	Size	Mem. use
		[s]	[s]	[λ]	[MB]	[s]	[s]	[λ]	[MB]
	1	501.7	41046.6	1.01	1571	3563.2	334226.3	1.40	7286
	8	70.2	5767.0	1.01	1603	496.0	46631.7	1.40	7397
	16	36.4	3003.3	1.01	1618	253.4	23862.3	1.40	7441
	32	19.8	1642.5	1.01	1647	132.8	12544.0	1.40	7529
	64	10.9	903.4	1.19	1396	74.1	7034.1	1.40	7705
	96	7.7	642.4	1.20	1409	52.5	4988.0	1.49	7367
	128	6.0	503.5	1.20	1445	40.9	3883.3	1.60	6714
	160	5.0	419.8	1.20	1480	34.0	3226.9	1.70	6226
	192	4.2	353.9	1.20	1516	29.4	2795.7	1.70	6322
	224	3.8	318.9	1.30	1352	25.9	2468.7	1.70	6418
	256	3.3	278.2	1.30	1379	23.5	2243.6	1.70	6515

Acknowledgements

This work has been partially supported by the "Ministerio de Ciencia e Innovación" from Spain and by FEDER, under the research project TEC2008-01638/TEC (IN-VEMTA), and by infraestructure project EQP06-015 co-financed by the European Union. The Airbus A300 series geometry has been provided by the research project GRD1-2001-40147 financed by the European Union. Many thanks are due to Dr. Dimas Suárez-Rodríguez, Dr. Ángel Gutiérrez-Rodríguez, and Dr. José Ranilla-Pastor for their technical support and for the chance given to the authors of using the Clúster de Modelización Científica (Universidad de Oviedo -Spain) in a dedicated way.