Yousra Benserhir

Anne-Claire Salaün

Florence Geneste

Laurent Pichon

Anne Jolivet-Gougeon

Recent Developments for the Detection of Escherichia Coli Biosensors Based on Nano-Objects-A Review

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Abstract-Advances in nanotechnology have made it possible in recent years to develop strategies for rapid and sensitive detection of pathogenic bacteria using new nanomaterials and the development of electronic nano-sensors. The detection of bacteria still faces problems such as long analysis time and complexity of the process. An alternative method is the use of biosensors, which combines a biological recognition mechanism with a physical transduction technique. Thus, a number of methods and manufacturing technologies have been developed in order to achieve performance in sensitivity, detection limit, label-free detection or real-time analysis. This review aims to focus on the state of the art of biosensors for the recognition elements of Escherichia coli in label-free biosensors with a particular focus on the beneficial use of nanomaterials and nano-objects for detection. Among the recent related biosensors based on nanoobjects for E. coli detection, the technologies and measurement techniques are detailed by comparing their performances detection in terms of concentration range and detection limit. Detection becomes more sensitive and more flexible using nanoparticles as markers, and real-time electrical detection methods are dominant in comparison with optical ones. The lowest detection limit can be achieved for sensors based on metal (gold or silver nanoparticles) with optical detection techniques in contrast with electrical detection methods using measurement conductance. Index Terms-Biosensor; nanosensors; nanomaterials; bacteria detection; functionalization; electrical detection I.  INTRODUCTION nfectious diseases have been historically ranked on top as one of the major challenges for human survival due to the periodical epidemic emergence of old and new outbreaks [START_REF] Majowicz | Global incidence of human Shiga toxinproducing Escherichia coli infections and deaths: a systematic review and knowledge synthesis[END_REF]. Foodborne infections are frequently transmitted by food contamination or cross-contamination during production, processing or transportation. Despite continuous efforts to improve safe-handling and processing practices, outbreaks caused by foodborne pathogens continue to arise, posing an imminent risk that needs to be effectively addressed. Biosensors have been proposed as rapid and cost-effective analytical methods to monitor bacteria in samples of interest. A biosensor is an analytical device which integrates a biological recognition element with a physical transducer to generate a measurable signal proportional to concentration of M anuscript received; revised; accepted. Date of publication; date of current version . The associate editor coordinating the review of this article and approving it for publication was .(Corresponding author: Anne-Claire Salaün). Y. Benserhir, A-C Salaün and L. Pichon are with the Institut d'Electronique et de Télécommunications de Rennes (IETR), University of Rennes1, Rennes, France (e-mail: asalaun@univ-rennes1.fr. F. Geneste is with Institut des Sciences Chimiques de Rennes (ISCR) (e-mail: florence.geneste@univ-rennes1.fr). A. Jolivet-Gougeon is with the INSERM, INRA INRAE, Institut NUMECAN (Nutrition Metabolisms and Cancer) (e-mail: anne.gougeon@univ-rennes1.fr).

analytes. In the general scheme of a biosensor, the biological recognition element responds to the target compound and the transducer converts the biological response to a detectable signal, which can be measured electrochemically, optically, acoustically, mechanically, calorimetrically, or electronically, and then correlated with the analyte concentration. Different types of biosensors available as electrical, and optical devices can selectively detect biological species. A good biosensor must not only respond to low concentrations of analytes, but also have the ability to discriminate species based on the recognition molecules that are immobilized on its surface [START_REF] Majowicz | Global incidence of human Shiga toxinproducing Escherichia coli infections and deaths: a systematic review and knowledge synthesis[END_REF]. This review focuses on recent advances in the field of biosensors for the detection of pathogen bacteria and more specifically of Escherichia coli. Owing to the high interest of nanomaterials to enhance sensitivity and selectivity of biosensors, a special attention is made on the role of nano objects in biosensor. Thus, after a description of conventional methods for E. coli detection, alternative biosensor methods are presented with a brief reminder of the main transduction methods (optical and electrochemical). Then the role of nanomaterials or nano-object on biosensors performance is detailed and a comparison between the different biosensors is made in terms of concentration range and detection limit.

A. Standard methods for in vitro diagnostic

In the bacteriology laboratory, many methods exist for detecting bacteria, whether based on direct examination, cultures, detection/quantification of antigens/ toxins or specific antibodies, use of animal models or methods of molecular biology.

Visualization of bacteria under an optical microscope (Gram or Ziehl stains, immunofluorescence, immunostaining ...) [START_REF] Biswas | Gram staining and its molecular mechanism[END_REF] often lacks sensitivity and specificity, unless the latter is improved by use of specific and labeled antibodies, (e.g. with a fluorochrome). The electron microscope allows a more precise visualization of bacteria, internal or external, but it is not very specific [START_REF] Humbert | Electron microscopic improvement in the study of diarrheagenic Escherichia coli[END_REF] and not extensively accessible. The detection of bacterial virulence factors can also be demonstrated by cell cultures, studying the cytopathogenic effect (toxicity), the capacity of adhesion and/or cell invasion, but remains in the field of research. Culture methods are the most commonly used, but their interest is limited for bacteria requiring a long incubation time or for viable non culturable bacteria (under stress ...), and null for bacteria for which no culture medium already exists. Quantitative or semi-quantitative measurements of bacteria by bacterial counts on agar media or flow cytometry [START_REF] Sträuber | Viability states of bacteria-specific mechanisms of selected probes[END_REF] have also been proposed. Many methods based on the detection of antigens [START_REF] Marolda | Micromethods for the characterization of lipid A-core and O-antigen lipopolysaccharide[END_REF] or specific toxins [START_REF] Paletta | Shiga Toxin-Producing and Enteroaggregative Escherichia coli in Animal, Foods, and Humans: Pathogenicity Mechanisms, Detection Methods, and Epidemiology[END_REF] have been developed, in order to increase the specificity and selectivity of detection: agglutination of sensitized latex particles, rapid immunoenzymatic detection of antigens [START_REF] Jouve | Adhesion to and invasion of HeLa cells by pathogenic Escherichia coli carrying the afa-3 gene cluster are mediated by the AfaE and AfaD proteins, respectively[END_REF]. Another alternative consists in quantifying the serum antibodies (serology) generated by an in vivo stimulation by bacterial antigens [START_REF] Zhou | ALCA and many more: are they useful in the diagnosis of inflammatory bowel disease?[END_REF]. This method is very specific according to the antigens used in vitro , and many tests exist (e.g. Enzyme Linked Immunosorbent Assay: ELISA…) [START_REF] Boer | Methods for the detection and isolation of Shiga toxin-producing Escherichia coli[END_REF], but are limited by the sensitivity often linked to the kinetics of appearance and disappearance of the antibodies, and the results of which may vary depending to the bacterial antigens and cross reactions [START_REF] Chart | The serodiagnosis of infections caused by Verocytotoxin-producing Escherichia coli[END_REF]. Confirmation methods can be used to increase the specificity of the test, such as Western blot or immunoblot (different proteins are separated on a membrane which serves as a reaction support to specify against which specific proteins the antibodies are directed).

Numerous molecular biology methods have been developed, based on the detection of nucleic acids, often after molecular amplification, ranging from simple PCR (Polymerase Chain reaction) to recent NGS (Next generation Sequencing) methods [START_REF] Dunne | Epidemiology of transmissible diseases: array hybridization and next generation sequencing as universal nucleic acidmediated typing tools[END_REF]. Among these we can cite molecular hybridization and its variants (hybridization with signal amplification or bDNA), gene amplification or PCR/ multiplex PCR, DNA microarray [START_REF] Jahandeh | Uropathogenic Escherichia coli virulence genes: invaluable approaches for designing DNA microarray probes[END_REF], nucleotide sequencing and derivatives (NGS, syndromic panels ...). These methods made it possible to study whole microbiota under different conditions and compare them. Traditional methods of cloning a DNA sequence into a vector with replication in a living cell require days or weeks of work. The use of the PCR, in molecular diagnostics or research laboratories, is now accepted as the gold standard for detecting nucleic acids from a number of origins [START_REF] Mackay | Real-time PCR in virology[END_REF]. Different PCR methods exist, based on the RNA or DNA detection, whether qualitative or quantitative, in real time, simple or multiplex. These methods require small sample size and are extremely sensitive and rapid techniques [START_REF] Salman | Shunting microfluidic PCR device for rapid bacterial detection[END_REF], which do not require a bacterial culture step. Mutations in targeted sequence can lead to false negative results, and contamination (if the strict handling conditions are not respected) to false positive results. Real-time PCR analysis can be completed faster and simultaneously detect several species (multiplex), within several hours, but still requires specific equipment and reagents [START_REF] Ahmed | Biosensors for whole-cell bacterial detection[END_REF]. Gadsby et al [START_REF] Gadsby | Development of two real-time multiplex PCR assays for the detection and quantification of eight key bacterial pathogens in lower respiratory tract infections[END_REF] detected eight bacterial targets that could be reliably quantified from sputum samples by two real-time PCR assays.

These methods still face the issues such as extensive analysis time and process complexity. An alternative method for the detection of microorganisms is the use of biosensors, which combines a biological recognition mechanism with a physical transduction technique. In this way, number of methods and technologies of fabrication have been developed to aim the following performances (but difficult to meet simultaneously): high sensitivity, low detection limit, simultaneous detection and identification of different strains, real-time analysis, and especially portability and an ease of use for in-situ monitoring.

B. Optical biosensors

In comparison with biology detection methods previously mentioned, optical biosensors offer some possibilities for ease of use, rapid, portable, multiplexed, and cost-effective diagnosis. In this part, we review recent advances in optical biosensors for pathogen diagnosis, especially colorimetric and plasmonic sensors, and introduce technologies and strategies that constitute those optical biosensors with such sensing performances. Recent examples of optical biosensors are presented and their advantages and limitations are discussed.

Optical transducers represent a board class of common and popular signaling methods, all involving an optical change in the sensing layer, including absorption, transmittance, scattering, reflection, refraction, and emission, as a result of interactions between the analyte and the recognition element. Optical sensors are commonly based on colorimetry, UV absorption, fluorescence, chemiluminescence, surfaceenhanced Raman scattering (SERS), or surface plasmon resonance (SPR), which can be detected by the naked eye or an optical sensing instrument, with the advantages of versatility, no destruction, sensitivity, universality, and simultaneous detection of multiple targets [START_REF] Kaushik | Label-free detection of E scherichia coli bacteria by cascaded chirped long period gratings immunosensor[END_REF] [START_REF] Song | Recent progress in microfluidics-based biosensing[END_REF]. Several research teams have worked on the development of this type of sensor using substrates, based on materials that can be at the nanometer scale, for examples: paper, glass slide, polymer, multilayer on glass substrate, polystyrene microparticle, gold nanoparticles (AuNPs), gold nanowires (AuNWs), carbon nanotube, Au plate [START_REF] Yoo | Optical biosensors for the detection of pathogenic microorganisms[END_REF]. Colorimetric biosensors have attracted wide attention owing to their easy readout and fast visual detection through the naked eyes or low-cost and portable equipment, which can be used to detect the analytes based on the color variation [START_REF] Chen | Solutionphase synthesis of platinum nanoparticle-decorated metal-organic framework hybrid nanomaterials as biomimetic nanoenzymes for biosensing applications[END_REF] [START_REF] Zhang | Recent progress on the construction of nanozymesbased biosensors and their applications to food safety assay[END_REF]. Thus, Ji et.al. [START_REF] Ji | Real-time detection of bacterial contamination in dynamic aqueous environments using optical sensors[END_REF] developed a colorimetric sensor with positively-charged amine-terminated polyamidoamine dendrimers to capture bacteria and reported a detection limit of 1.0×10 4 cells/mL. Other teams have validated the detection of proteins [START_REF] You | Detection and identification of proteins using nanoparticle-fluorescent polymer 'chemical nose'sensors[END_REF], bacteria [START_REF] Phillips | Rapid and efficient identification of bacteria using gold-nanoparticlepoly (para-phenyleneethynylene) constructs[END_REF], viruses [START_REF] Bruno | Reovirus detection using immunomagnetic-fluorescent nanoparticle sandwich assays[END_REF], and cancer cells [START_REF] Bajaj | Detection and differentiation of normal, cancerous, and metastatic cells using nanoparticle-polymer sensor arrays[END_REF], with functionalized gold nanoparticles. In 2013, the hybrid systems that consist of Au NPs and surfaceassembled fluorescent probes were exploited for fluorescent turn-on sensing of liposaccharide (LPS). The results showed that the sensitivity and selectivity to LPS relied strongly on the binding affinity between fluorescent probes and Au NPs [START_REF] Gao | Exploring and exploiting the synergy of non-covalent interactions on the surface of gold nanoparticles for fluorescent turn-on sensing of bacterial lipopolysaccharide[END_REF]. Remaining in 2013, Zagorovsky et al [START_REF] Zagorovsky | A plasmonic DNAzyme strategy for point-of-care genetic detection of infectious pathogens[END_REF] used, for DNA detection, a colorimetric sensor modified with gold nanoparticles (GNPs) and this study showed a detection limit of 500 pM pathogenic DNA. The success of GNPs ranging in size from 1 nm to several hundred nanometers is due to their strong light-scattering properties. The intensity of light scattering is based on the size of the particle. Moreover, GNPs can be easily conjugated with protein(s) or modified DNA molecule(s) through sulfhydryl linkages. These properties make GNPs a useful tool for ultrasensitive molecular detection [START_REF] Chang | Rapid single cell detection of Staphylococcus aureus by aptamer-conjugated gold nanoparticles[END_REF]. One of the notable recent trends in biosensors is the development and application of plasmonic biosensors for pathogen detection. Two major plasmonic biosensors use surface plasmon resonance (SPR) and surface enhanced Raman spectroscopy (SERS), stable and sensitive detection tools that can detect low bacterial concentration [START_REF] Díaz-Amaya | Aptamerbased SERS biosensor for whole cell analytical detection of E. coli O157: H7[END_REF]. The detection of Escherichia coli (E. coli) O157:H7 by a surface plasmon resonance sensor (SPR) used three sample preparation methods of bacteria: untreated (viable), heatkilled and then soaked in 70% ethanol, and detergent-lysed.

The SPR detection surface consisted of a monoclonal antibody immobilized on a monolayer of self-assembled alkanethiols (SAM) terminated with -COOH and -OH on a gold surface. The limit of detection (LOD) of each method is determined by the minimum measurable shift in resonance wavelength corresponding to the specific binding of E. coli O157:H7 and subsequent binding of an antibody for amplification. Detergent-containing samples produce the lowest detection limit at 10 4 CFU/mL, while the detection limit was 10 5 CFU/mL for heat-killed samples and 10 6 CFU/mL for untreated samples, respectively [START_REF] Taylor | Comparison of E. coli O157: H7 preparation methods used for detection with surface plasmon resonance sensor[END_REF]. A fibreoptic sensor was developed and optimised for the simultaneous detection of E. coli, Listeriaand Salmonella in food. Streptavidin-coated optical waveguides were immobilized with biotinylated polyclonal antibodies and exposed to bacterial suspensions or enriched food samples for 2 hours, allowing a detection limit of 10 3 CFU/mL [START_REF] Ohk | Multiplex fiber optic biosensor for detection of Listeria monocytogenes, Escherichia coli O157: H7 and Salmonella enterica from ready-to-eat meat samples[END_REF].

C. Electrochemical biosensors

Electrochemical biosensors have been intensively studied and well developed for the detection of pathogenic bacteria. Some advantages of electrochemical biosensors over others include comparable sensitivity, low cost, fast response, and possibility to operate in turbid solution and to be miniaturized. According to the variable measured, these electrochemical biosensors can be classified into three classes: amperometric, potentiometric, impedimetric biosensors, for current, potential and impedance/capacitance measurement, respectively.

1) Amperometric biosensors

One of the most uses electrochemical detection techniques is the amperometric detection. Amperometric sensors monitor the changes in current or potential caused by the oxidation or reduction of the electrochemically active analyte in the electrochemical system. The redox reaction directly or indirectly involves the analyte, and a linear relationship between the analyte concentration and the measured current can be observed [START_REF] Riu | Electrochemical biosensors for the detection of pathogenic bacteria in food[END_REF], [START_REF] Xu | Electrochemical biosensors for rapid detection of Escherichia coli O157: H7[END_REF].

The following examples illustrate the different methods of manufacturing amperometric sensors for E. coli detection, as well as their detection limits. The development of an amperometric biosensing platform for the detection of pathogenic E. coli O157:H7 has been proposed by Dhull et al. [START_REF] Dhull | Label-free amperometric biosensor for Escherichia coli O157: H7 detection[END_REF]. The biosensor developed works on the principle of antibody-antigen interactions, where the antibody is covalently bound to the surface of the nickel oxide thin-film matrix prepared by sputtering. The sensors have a wide linear range of 10 7 to 10 1 cells/mL with a significantly low detection limit of 1 cell/mL. The biosensors were found to be highly specific and selective towards the target species, even in samples composed of several bacterial species.

In another study, a method for easy detection of E. coli has been established based on the enzymatic activity of intracellular -D-galactosidase (Gal) [START_REF] Noh | Facile electrochemical detection of Escherichia coli using redox cycling of the product generated by the intracellular β-d-galactosidase[END_REF]. The bacteria was first treated with a gene expression inducer (isopropyl--Dthiogalactopyranoside IPTG), leading to the increase of the expression level of Gal inside E. coli. Then, the hydrolysis of 1-naphthyl--D-galactopyronoside into 4-aminonaphthol by Gal occurred, facilitating by a permeabilization treatment of the bacteria. The electrochemical detection of 4aminonaphthol amplified by an electrochemical-chemical redox cycling led to a detection limit of 10 5 CFU/mL for 30 min of IPTG treatment. By this method, a detection limit of 1.0 CFU/mL was reached with a 4-hour IPTG treatment, without DNA amplification or immunoassay protocol [START_REF] Noh | Facile electrochemical detection of Escherichia coli using redox cycling of the product generated by the intracellular β-d-galactosidase[END_REF].

Another amperometric immunosensor has been developed for the rapid detection of heat-killed E. coli O157:H7 by Yan Li et al. [START_REF] Li | Amperometric immunosensor for the detection of Escherichia coli O157: H7 in food specimens[END_REF]. This immunosensor was prepared as follows. First, long chain, amine-terminated 11-amino-1undecanethiol alkanethiol hydrochloride (AUT) was selfassembled on a gold electrode surface to form an ordered, oriented, compact and stable monolayer with -NH2 functional groups that could immobilize massive gold nanoparticles. Then successive layers of AuNPs and a nanocomposite formed by the combination of chitosanmultiwalled carbon nanotubes and SiO2/thionine nanoparticles were added. Finally, E. coli O157:H7 antibody (anti-E. coli O157:H7) was covalently bound to the AuNP monolayer and its bioactivity was measured by amperometry. Under optimal conditions, the calibration curve for E. coli O157:H7 had a working range of 4.12×10 2 -4.12×10 5 CFU/mL, and the total analysis time was less than 45 minutes.

2) Potentiometric biosensors

Potentiometric biosensors use a high voltmeter to measure the electrical potential difference or electromotive force between the working and reference electrodes. The electrochemical process is non-faradic, with zero or negligible current in the electrochemical cell, and the potential difference is generated by the accumulation of charge density on the surface of working electrode. A Spanish team has demonstrated that the detection and labelfree identification of live bacteria in real complex samples can be performed in few minutes and directly, by a simple and selective method, using an aptamer-based potentiometric biosensor. This biosensor used signal-walled carbon nanotubes as excellent ion to electron transducer and covalently immobilized aptamers as biorecognition elements. A highly selective detection of live bacteria with an immediate linear response of up to 10 4 CFU/mL was validated with a detection limit of 6.0 CFU/mL [START_REF] Zelada-Guillén | Real-time potentiometric detection of bacteria in complex samples[END_REF]. In the same context a potentiometric sensor was developed by Tian et al. [START_REF] Tian | Label-Free Detection of E. coli O157: H7 DNA Using Light-Addressable Potentiometric Sensors with Highly Oriented ZnO Nanorod Arrays[END_REF]. This is a light-addressable potentiometric sensor (LAPS) deposited with highly oriented ZnO nano-arrays, used for the label-free detection of signal-stranded bacterial DNA. For sensitive detection of E. coli O157:H7 DNA, surface functionalization was achieved using covalent immobilization of the single-stranded DNA (ssDNA) of the specific probe on the sensor surface. The results indicate that distinct signal changes can be registered and recorded to detect the target E. coli ssDNA in the range concentration 10-10 5 CFU/mL. The lower detection limit of the target ssDNA corresponded to 1.0×10 2 CFU/mL of E. coli O157:H7 cells. Shaibani et al. [START_REF] Shaibani | Portable nanofiber-light addressable potentiometric sensor for rapid Escherichia coli detection in orange juice[END_REF] developed sensors for E. coli detection in orange juice, using a portable nanofiberlight addressable potentiometric sensor (LAPS). They chose a pH-sensitive electrospun poly(vinyl/alcohol)/poly (acrylic acid) (PVA/PAA) hydrogel nanofiber functionalized with Dmannose as sensitive layer. This sensor enabled real-time detection of the bacterial species in less than one hour, with a detection limit of 10 2 CFU/mL. A high selectivity of the biosensor for E. coli against the sugar fermenting Salmonella Typhimurium (S. Typhi) was obtained, attributed by the authors to the presence of D-mannose on the surface of the nanofiber.

3) Impedimetric biosensors a) Electrochemical impedance spectroscopy detection (EIS)

Impedance measurements have become increasingly popular as a label-free detection tool for many different types of biosensors, during the past few years. The biosensors based on impedance measurements are prepared on electrodes such that the electron transfer is selectively modulated by the analyte [START_REF] Chang | Electrochemical impedance spectroscopy[END_REF]. A highly sensitive electrochemical biosensor has been developed by Krupinska et al. [START_REF] Krupinska | Detection of low levels of Escherichia coli by electrochemical impedance spectroscopy and singular value decomposition[END_REF] for the detection of E. coli in water. The detection system is manufactured using two gold electrodes in a flow cell modified with polyclonal antibodies against E. coli. The results are promising and show that label-free detection of trace amounts of E. coli is certainly possible with a simple and straightforward surface modification of the electrode.

Always with the same detection method, the immobilization of antibodies at the gold electrode was carried out through a stable acyl amino ester intermediate generated by 1-ethyl-3-(3-dimethylaminopropyl) carbodiimide (EDC) and N-hydrosuccinimide (NHS), which could condense antibodies reproducibly and densely on the self-assembled monolayer (SAM). The resistance of the electrical transfer increases as a function of the immobilization of the antibodies and the adhesion of the E. coli cells at the electrode. This change is directly detected by electrochemical impedance spectroscopy in the presence of Fe(CN)63-/Fe(CN)64-as a redox probe [START_REF] Geng | Self-assembled monolayers-based immunosensor for detection of Escherichia coli using electrochemical impedance spectroscopy[END_REF]. In order to detect Gram-negative bacteria, a Brazilian team chose to work on bio-sensors with gold electrodes, functionalized with gold nanoparticles modified with cysteine, then they add clavanin A antimicrobial peptides. The modifications at the electrodes were evaluated by electrochemical impedance microscopy and cyclic voltammetry analysis [START_REF] De Miranda | A simple nanostructured biosensor based on clavanin A antimicrobial peptide for gram-negative bacteria detection[END_REF] (figure 3). The sensor successfully differentiates concentrations of E. coli ATCC 2522 between 10 1 to 10 4 CFU. Therefore, the proposed biosensor is a usable and simple alternative for bacterial detection with a possible use for quantitative detection. b)Electrochemical capacitance spectroscopy detection (ECS) Among electrochemical biosensors, capacitive biosensors show interesting properties due to changes in the dielectric properties of the electrode surface. Isil et al have shown that molecular printing is a well-suited technique with real-time detection of E. coli in the range of 10 2 -10 7 CFU/mL [START_REF] Idil | Whole cell based microcontact imprinted capacitive biosensor for the detection of Escherichia coli[END_REF]. The biosensors presented in this section demonstrated performance in terms of sensitivity, selectivity and reproducibility. Using different types of materials, researchers in this field have been able to demonstrate their optical, electrical or chemical capability for the detection of microorganisms.

III. NANOMATERIALS AND NANO-OBJECTS FOR THE DETECTION OF E-COLI BACTERIA

The use of nanoparticles and nano-objects increases in health and wellness, particularly for the detection or study of their effects on pathogenic bacteria. Due to their small size (<100 nm), nanoparticles have a larger specific surface area than the same materials on a larger scale. Quantum effects are important and can benefit for biosensing. In addition, below a size of about 20 nm, an increased surface reactivity appears due to surface energy [START_REF] Auffan | Towards a definition of inorganic nanoparticles from an environmental, health and safety perspective[END_REF]. It is this feature that gives them chemical reactivity and new physical properties (optical, mechanical, electrical and magnetic) that are of interest for interacting with biological systems, particularly in the case of biological species recognition.

Nanoparticles are used either for their ability to amplify the detection signal or for their biological actions on detection. In addition, the use of nanoparticles is compatible with the fabrication of ultra-sensitive, rapid, small and portable biosensors (e.g. usable as patches). In this way, nanoparticles contribute to the emergence of new detection techniques and also to new transduction methods. With regard to the detection of pathogens, efforts are focused on some properties of nanoparticles [START_REF] Sanvicens | Nanoparticlebased biosensors for detection of pathogenic bacteria[END_REF], [START_REF] Wang | Nanomaterial-based amplified transduction of biomolecular interactions[END_REF]. There are several methods for the detection of biological species, including E. coli, using the specific properties of nanoparticles, of which the most common and most promising are electrochemical, optical, magnetic and gravimetric based detection methods.

On the other hand, recent advances in micronanotechnologies have led to the emergence of innovative devices with the integration of nano-objects such as nanowires (NWs), carbon nanotubes (CNTs) and nanomaterials in biosensors. The use of biosensors combining a biological recognition mechanism with a physical transduction technique is very promising because they are able to detect pathogens quickly and accurately. Due to their high surface-to-volume ratio, nanostructures are potential functional probes for detecting filler species as their dimensions are comparable to those of chemical and biological species. These technologies can be combined with specific sensitive layers to enable selective chemical detection. The miniaturization of biosensors offers several advantages, including reduced sample and reagent consumption and increased sensitivity with cost-effective disposable chips through mass production. These nanoobjects as biochemical nano-sensors are promising for marker-free, real-time, biomolecule-sensitive detection. Their functionalization based on the chemical modification of the surfaces allows a more specific detection. They have shown great interest in the many biorecognition studies for biological analysis such as DNA, proteins, bacteria or viruses.

A. Electrochemical nanoparticles-based biosensors

The combined effects of the electronic properties and electrocatalytic activity of some nanoparticles can be a benefit for electrochemical detection (voltammetry, impedancemetry, amperometry...). Generally, these methods combine tools for the enhancement of the electrochemical signal, and recognition of biological elements.

For instance, anode material combined with nanoparticles binded to biological agent to target bacteria or acting as markers can be used in biosensors, and different detection signals can be obtained according to the detection method. In particular, recent work [START_REF] Zou | Gold nanoparticles-based multifunctional nanoconjugates for highly sensitive and enzyme-free detection of E.coli K12[END_REF] reported the detection of E. coli using differential pulse voltammetry, integrating gold nanoparticles on the surface of a polypyrrole-reduced graphene oxide (PPy-rGO) composite for immobilization of a capture antibody (Ab1), and conjugated onto ferrocene doped polypyrrole-gold nanoparticles composite for immobilisation of detection antibody (Ab2) (figure 4). In this case, the attachment of Ab2 allows the generation of electrons by the oxidation of ferrocene. Due to a high surface area, good conductivity and biocompatibility of the nanocomposite, enzyme-free detection of E. coli K12 was shown, with a linear range from 1.0 × 10 1 to 1.0 × 10 7 CFU/mL and a low detection limit of 10 CFU/mL. Another approach could be the use of anode material combined with nanoparticles to increase the surface area of the working electrode, and to improve the electron transfer, leading to higher electrochemical signal.

A recent study reported the photo-electrochemical detection of bacteria with zinc oxide (ZnO) nanowires modified with gold nanoparticles (AuNPs) and CdS quantum dots (CdS-QDs) as photo anode material [START_REF] Dong | CdS quantum dots/Au nanoparticles/ZnO nanowire array for self-powered photoelectrochemical detection of Escherichia coli O157:H7[END_REF]. Zinc oxide is a good semiconductor for photochemical detection biosensor [START_REF] Chen | Overall photoelectrochemical water splitting at low applied potential over ZnO quantum dots/nanorods homojunction[END_REF], and its form as nanowires increases the surface of the electrode. The quantum dots promote the light absorption from UV to visible region and gold nanoparticles improve the electrons transfer. In the study reported by Dong et al. [START_REF] Dong | CdS quantum dots/Au nanoparticles/ZnO nanowire array for self-powered photoelectrochemical detection of Escherichia coli O157:H7[END_REF], E. coli O 157:H7 detection was ensured after assembling aptamer biological recognition element to the ZnO/AuNPs/QDs-CdS complex (figure 5). The results showed linear detection over a range from 10 to 10 7 CFU/mL with a detection limit of 1.125 CFU/mL. Such an approach is a good alternative for low noise, fast, and ultra-high sensitive detection of bacteria in comparison with the classical electrochemical detection.

B. Optical nanoparticle-based biosensors

The main optical detection methods of E. coli based on nanoparticles are colorimetry, fluorescence and surface plasmon resonance (SPR). The use of nanoparticles is particularly interesting for the detection of biological elements by the SPR technique, thanks to the presence of intense absorption peaks leading to an amplification of the detection, or to the change in refractive index at the sensor surface depending on analyte concentration and allowing to investigate the binding events of molecular interactions in real time and label-free. For example, a study has shown a reduction of the detection limit of E. coli by a factor of 5500 compared to a sensor without the use of gold nanoparticles [START_REF] Joung | High sensitivity detection of 16s rRNA using peptide nucleic acid probes and a surface plasmon resonance biosensor[END_REF]. More recently, highly selective E. coli SPR detection at low detection limit 0.57 CFU/mL in urine was reported using silver nanoparticles entrapped into imprinted polymer mixture to increase the sensitivity of the sensor (figure 6) [START_REF] Özgür | Surface plasmon resonance based biomimetic sensor for urinary tract infections[END_REF]. For fluorescence detection semiconductor quantum dots (CdSe/Zn,...) combined with biological agent are used, due to their fluorescence properties of higher intensity than classical fluorophores [START_REF] Michalet | Quantum Dots for Live Cells[END_REF] [START_REF] Ahmadian-Fard-Fini | Hydrothermal green synthesis of magnetic Fe3O4-carbon dots by lemon and grape fruit extracts and as a photoluminescence sensor for detecting of E. coli bacteria[END_REF]. In the case of quantum dots, it is possible to adapt their fluorescence emission spectra according to their size and composition [START_REF] Edgar | High-sensitivity bacterial detection using biotintagged phage and quantum-dot nanocomplexes[END_REF] [START_REF] Mechery | Using Bioconjugated Nanoparticles To MonitorE. coli in a Flow Channel[END_REF], which makes them very attractive for the detection of bacteria. Moreover, thanks to the high selectivity of bioreceptors, multiple detection within the same sample could be possible [START_REF] Maalouf | Comparison of two innovatives approaches for bacterial detection: paramagnetic nanoparticles and self-assembled multilayer processes[END_REF], [START_REF] Wang | Fluorescent nanoparticles for multiplexed bacteria monitoring[END_REF]. Recently, an original study reported E. coli detection on smartphone-based electrogenerated chemiluminescence (ECL) system using graphene quantum dots/silver nanoparticles complexes (GQDs/AgNPs) [START_REF] Li | Electrogenerated chemiluminescence on smartphone with graphene quantum dots nanocomposites for Escherichia Coli detection[END_REF] (figure 7). In this case, graphene quantum dots are used to amplify and stabilize the luminescent signal, functional modules on smartphone for voltage excitation and luminescence capture, and luminescence image analysis. Based on this system, E. coli detection was demonstrated in a linear response in the concentration range 10-10 7 CFU/mL with a limit of detection of 5 CFU/mL.

C. Magnetic nanoparticle-based biosensors

Magnetic nanoparticles have specific properties (biocompatibility, non-toxic, and presence of functional groups) and are of interest for ultrasensitive biosensors and multiple pathogen detection. In addition, some hybrid magnetic nanoparticles provide various and effective detection methods thanks to the coupling of their magnetic properties to their electrical [START_REF] Wang | A homogeneous magnetic bead-based impedance immunosensor for highly sensitive detection of Escherichia coli O157:H7[END_REF], [START_REF] You | Gold Nanoparticle-Coated Starch Magnetic Beads for the Separation, Concentration, and SERS-Based Detection of E. coli O157:H7[END_REF] or their optical properties [START_REF] Hu | Green one-step synthesis of carbon quantum dots from orange peel for fluorescent detection of Escherichia coli in milk[END_REF]- [START_REF] Wang | The Escherichia coli O157:H7 DNA detection on a gold nanoparticle-enhanced piezoelectric biosensor[END_REF]. In particular, combination of magnetic nanoparticles with biological agents is an efficient strategy for identification, pre-concentration and separation of a bacterial strain in a given analyte through the application of a magnetic field. Magnetic nanoparticles can be used as homogeneous carriers for the capture of targets and the electrode as sensing element. For example, combination of double antibody sandwich complex (magnetic bead-E. coli O157:H7 monoclonal antibody)-(E. coli O157:H7)-(enzyme-labeled anti-E.coli O157:H7 antibody) has been used for the detection of E. coli O157:H7, using interdigited gold electrodes impedance system [START_REF] Wang | Fluorescent nanoparticles for multiplexed bacteria monitoring[END_REF] (figure 8). The immobilization process has been performed with magnetic beads instead of electrode functionalization and sample absorption to improve reproducibility and sensitivity. In this case, detection limits ranging linearly from 10 2 to 10 6 CFU/mL has been achieved. Another work, reported E. coli detection in the milk in the range 5-10 8 CFU/mL with a LOD of 10 CFU/mL by fluorescence method combining carbon quantum dots synthesized from orange peel with magnetic nanoparticles (Fe304) prepared by simple, non-polluting, time saving and low cost method [START_REF] Li | Designing an aptamer based magnetic and upconversion nanoparticles conjugated fluorescence sensor for screening Escherichia coli in food[END_REF].

D. Gravimetric nanoparticle-based biosensors

Gravimetric detection method is based on a piezoelectric material (quartz, lead zirconium titanate-PZT…) microbalance whose principle is the measurement of the mass variation, related to the variation of the resonance frequency of the piezoelectric material correlated to the mass of the analyte deposited on the electrode of the microbalance. Thanks to the benefits of nanoparticles positioned on one of the balance electrodes, a better immobilization of the bioreceptors is possible, and thus an amplification of the detection signal and a better sensitivity of the sensor can be reached. Several studies have demonstrated the benefits using nanoparticles for the detection of E. coli, but few studies reported on their detection by gravimetric detection method. However, previous work has shown that it is possible to improve the performance of a biosensor for the detection of E. coli O157:H7 compared to a conventional sensor. It was due to the high and specific biotin-binding affinity of streptavidin to attach gold or iron oxide nanoparticles to the complementary DNA strand leading to detection limits of 2.0x10 3 CFU/mL and 2.67x10 2 CFU/mL, respectively (figure 9) [START_REF] Mao | A nanoparticle amplification based quartz crystal microbalance DNA sensor for detection of Escherichia coli O157:H7[END_REF]. More recent work showed higher detection limit, below 5 x 10 5 CFU/mL and above 10 6 CFU/mL, by direct loading on PZT material electrode (i.e. without any modification of the bacteria) [START_REF] Tsai | Resonant efficiency improvement design of piezoelectric biosensor for bacteria gravimetric sensing[END_REF].

E. Carbon nanotubes based biosensors

Carbon nanotubes (CNTs) are considered interesting materials for biosensing applications because these nanoobjects can be functionalized with antibodies or aptamers for the detection of bacterial species. They can be synthesized as single-walled carbon nanotubes (SWNTs) composed of a single layer of carbon atoms, with very high aspect ratios (diameters of the order of 1 nm). As an example, Andrade et al. [START_REF] Andrade | Nanostructured sensor based on carbon nanotubes and clavanin A for bacterial detection[END_REF] propose a new approach for the development of nanostructured biosensors based on carbon nanotubes (CNTs) and antimicrobial peptides for the detection of bacteria (figure 10). The gold electrodes were coated with a self-assembled monolayer of cysteine. The terminal carboxyl groups of CNTs activated by EDC/NHS (1-ethyl-3-(3dimethylaminopropyl) carbodiimide /N-hydroxysuccinimide) allowed the covalent immobilization of CNTs on the surface, and then, of the antimicrobial peptide clavanine A (ClavA) on the NTCs. The surface-immobilized ClavA was then used to detect different concentrations of bacterial cells by electrochemical impedance spectroscopy (EIS) measurements. The infectious agents tested were E. coli, Klesiella pneumoniae, Enterococcus faecalis, and Bacillus subtilis. This biosensor was capable of distinguishing bacterial concentrations in the range of 10 2 -10 6 CFU /mL for E. coli, and the authors showed the ability of the sensor to differentiate Gram-negative bacteria (E. coli and K. pneumoniae) from Gram-positive bacteria thanks to electrostatic and hydrophobic interactions with ClavA.

In addition, carbon nanotube (CNT)-based field-effect transistors (FETs) are recent sensing device configurations. When the biological material is adsorbed on the surface of CNTs, electron transfer occurs, which changes their electrical conductivity.

This property has been used for biosensors by functionalizing single valley carbon nanotube based FETs (SWCNT-FETs) with aptamer for E. coli detection by electrical measurements [START_REF] So | Detection and Titer Estimation ofEscherichia coli Using Aptamer-Functionalized Single-Walled Carbon-Nanotube Field-Effect Transistors[END_REF]. In this case, the SWNT-FETs showed a decrease of the electrical conductance of more than 50% after binding with 10 7 CFU/mL of E. coli to the aptamer-functionalized SWNT-FET. Other work reported [START_REF] Zhang | Electronic detection of Escherichia coli O157 H7 using single-walled carbon nanotubes field-effect transistor biosensor[END_REF] the use of SWCNT-FET functionalized with a solution Fig. 8. Illustration of the magnetic bead-based impedance method for the detection of E. coli O157:H7, using interdigited gold electrodes. From Wang et al. [START_REF] Wang | A homogeneous magnetic bead-based impedance immunosensor for highly sensitive detection of Escherichia coli O157:H7[END_REF], reprinted with permission. of PASE (1-pyrenebutanoic acid succinimidyl ester) as linker molecule and anti-E-coli antibodies (figure 11). The changes of the electrical resistance of the CNTFET indicate the possibility of this device as biosensor to detect E. coli O157:H7, and concentration as low as 8.2 × 10 2 CFU/mL was detected.

F. Graphene based biosensors

Graphene is a remarkable two-dimensional (2D) material that is of great interest due to its exceptional electrical and thermal conductivity and good mechanical behaviour, with promising applications in electronic devices, especially sensors. The most common methods of graphene synthesis include chemical vapour deposition (CVD) and mechanical exfoliation. Graphene, which is a single atom-thick flat sheet of carbon atoms perfectly arranged in a honeycomb network, has great potential for biosensing. Huang et al. have demonstrated a graphene-based biosensor for electrically detecting E. coli with high sensitivity and specificity [START_REF] Huang | Graphene-based biosensors for detection of bacteria and their metabolic activities[END_REF]. To specifically detect E. coli, anti-E. coli antibodies were first immobilized on the graphene film via 1-pyrene-butanoic acid succinimidyl ester linked with graphene by  stacking. A significant increase in the conductance of the graphene device was observed after exposure to E. coli K12 at a concentration as low as 10 CFU/mL, while no significant response was triggered by a high concentration of another bacterial strain. In addition, graphene-based FET devices functionalized with antibodies or antimicrobial peptides (AMPs) have been used for electronic detection of E. coli. Thakur et al. developed a field effect transistor based on thermally reduced graphene oxide (rGO FET) passivated with an ultrathin layer of Al2O3 [START_REF] Thakur | Rapid detection of single E. coli bacteria using a graphene-based field-effect transistor device[END_REF].

Self-assembly of thiol was used to anchor the GO to the interdigital Au electrodes with fingers spaced by 2 μm. To anchor a high amount of anti-E. coli, Au nanoparticles (NP) functionalized by glutathione were sprayed onto the FET. The anti-E. coli anchored to the AuNP surface selectively captures the E. coli cells, and this lead to a change in the electrical conductivity of the rGO channel. The electrical conductivity of the sensor was recorded by monitoring the change in drain current for different concentrations of E. coli solutions, since the transistor is back-gate biased. The high negative charge density on the surface of E. coli cells strongly modulated the concentration of the majority charge carriers in the monolayer, allowing real-time monitoring of the E. coli concentration in a given sample. With a low single cell detection limit, the FET sensor had a linear range of 1 to 100 CFU in 1 μL sample volume (i.e. 10 3 to 10 5 CFU/mL) [START_REF] Thakur | Rapid detection of single E. coli bacteria using a graphene-based field-effect transistor device[END_REF] (figure 12). Techniques have also been used to produce graphene nanoribbons less than 10 nm wide, which are of interest due to quantum confinement of electrons and edge effects. As an example, Chen et al. highlighted a new graphene derivative, the holey reduced graphene oxide (hRGO), functionalized with positively charged Magainin I to link Gram-negative bacteria by electrostatic interaction [START_REF] Chen | Electronic Detection of Bacteria Using Holey Reduced Graphene Oxide[END_REF]. Even after functionalization, this material preserved its electronic properties. Fan et al. demonstrated a graphene-based biosensor to electrically detect E. coli with a high sensitivity and specificity [START_REF] Fan | Hybrid graphene oxide based ultrasensitive SERS probe for label-free biosensing[END_REF]. Kaur et al. used bacterial-DNA interactions using a nanostructured electrode-based sensor, using CNTs in transverse and longitudinal directions and the subsequent bridging with terephthalaldehyde (TPA), to form 3D bridged rebar graphene. This material exhibited enhanced electrical properties and facilitated chemical functionality. It was functionalized with a specific anti-E. coli DNA aptamer. The developed nanostructured aptasensor demonstrated a low detection limit and a sensitivity of ~ 10 1 CFU/mL towards E. coli [START_REF] Kaur | Bridged Rebar Graphene functionalized aptasensor for pathogenic E. coli O78:K80:H11 detection[END_REF].

G. Silicon nanowire-based biosensors

Silicon nanowires (SiNW) are also potential functional probes for pathogen detection due to their unique physical properties. Their one-dimensional structure provides the smallest confinement for electron transport in the longitudinal direction, their large surface area promotes interaction between target cells and nanomaterials, thus leading to a higher sensitivity. They offer unique advantages for the development of pathogen sensors in several respects. The advantages of this smaller scale approach are the possibility of mass production and reduced unit costs thanks to recent advances in micro and nano-manufacturing technologies compatible with CMOS (Complementary Metal Oxide Semiconductor) silicon technology. The lab-on-a-chip system for the rapid detection of bacteria could revolutionize health prevention. Some work has been reported on the use of SiNWs as sensitive units for chemical and biochemical species, but few works reported on the relevance of using the SiNW network, in particular for bacteria detection. The interest of sensors using SiNWs as sensitive units is based on a large contact surface (high surface/volume ratio for each nanowire) allowing potential significant interactions with bacteria, inducing a change in electrical conduction through the network of silicon nanowires after their binding. Indeed, when the device with functionalized nanowires is exposed to macromolecules with a net positive or negative electrical charge, this leads to a change in the charge density at the surface of the nanowire, and thus a change of the global electrical resistance. In particular, Le Borgne et al [START_REF] Borgne | Bacteria electrical detection using 3D silicon nanowires based resistor[END_REF] demonstrated E. coli ATCC 35218 E. coli detection in the range 8×10 3 -8×10 8 CFU/mL using SiNWs array based electrical resistor. Some other works showed that nanostructures promote adhesion of cells including bacteria.

In particular, Wang et al. [START_REF] Wang | High antibacterial efficiency of pDMAEMA modified silicon nanowire arrays[END_REF] reported that the graft density on the modified nanowire arrays was much higher than on analogous smooth silicon, leading to higher bacterial adhesion on the nanowire arrays. Moreover, some studies have shown that nanostructures and their specific physicochemical properties can be used to regulate the responses and movements of bacterial cells. Jeong et al. [START_REF] Jeong | Bacterial recognition of silicon nanowire arrays[END_REF], used a welldefined silicon nanowire array platform and single cell imaging, and showed that the attachment locations of bacteria are strongly influenced by the presence of nanowires on a surface, with preferential attachment to nanowires (figure 13). These results demonstrated that nanoscale topography could affect movement and attachment of bacteria. According to Kim et al. [START_REF] Kim | Cellular nano-transistor: An electronic-interface between nanoscale semiconductors and biological cells[END_REF], the coupling between microcellular species and semiconductor nanomaterials in a cellular nanotransistor enables the charge carriers to be confined to one or two dimensions and to the bio/nano interface, which leads to a very sensitive modification of the density of the carriers. In other work, Susarrey-Arce et al.

[81] investigated the interaction and the viability of bacteria on SiNWs surfaces, enhanced with different chemical functionalities, using model microorganisms E. coli and S. aureus. Recently, Li et al [START_REF] Li | Capture and detection of urine bacteria using a microchannel silicon nanowire microfluidic chip coupled with MALDI-TOF MS[END_REF] have shown fast bacterial identification in urine samples by capturing bacteria on a microchannel silicon nanowire microfluidic chip, followed by matrix-assisted laser desorption ionization-time of flight mass spectroscopy (MALDI-TOF MS) detection. Bacteria with a concentration of 10 6 CFU/mL in urine samples have been detected.

IV. CONCLUSION

Rapid and sensitive detection of pathogenic bacteria is extremely important for medical diagnosis or for monitoring pathogens in various food products. The limitations of most conventional diagnostic methods are a lack of ultrasensitivity or a delay in obtaining results. This review draws a state of the art of biosensors based on nano-objects measuring the presence or activity of E. coli, highlighting the technological challenges in terms of sensitivity, specificity, reproducibility, and the limitations and advantages of simplicity and transportability (Table 1). Detection becomes more sensitive and more flexible using nanoparticles as markers, and real-time electrical detection methods are dominant in comparison with optical ones. The lowest detection limit can be achieved for sensors based on metal with optical detection techniques in contrast with electrical detection methods (and more specifically using measurement conductance). Biosensors based on nano-objects for E-coli detection are of particular interest for label-free, real-time, ultra-high sensitive, and required selective detection to effectively counteract food-borne infections for example. Further improvements are needed for the development of such sensors because of their potential integration in largescale addressable network compatible with the fabrication of biological lab-on-chip, that distinguish them from other sensor technologies available today.

Recent

  Developments for the Detection of Escherichia coli biosensors based on nanoobjects -A review Yousra Benserhir, Anne-Claire Salaün, Florence Geneste, Laurent Pichon and Anne Jolivet-Gougeon ember, IEEE I II. ESTABLISHED METHODS FOR THE DETECTION OF PATHOGENS

Fig. 2 .Fig. 3 .

 23 Fig. 2. Schematic picture of flow cell with two-electrode configuration and polydimethylsiloxane (PDMS) flow chamber. From Krupinska et al. [43], reprinted with permission.

Fig. 1 .

 1 Fig. 1. Schematic representation of the NF-LAPS sensor comprising a three-electrode system. From Shaibani et al. [41], reprinted with permission.

Fig. 4 .

 4 Fig. 4. Schematic illustration of: preparation of the PPy-rGO/AuNPs nanocomposite followed by Ab1 attachment and detection of E. coli K12. From Zou et al. [50], reprinted with permission.

Fig. 5 .

 5 Fig. 5. Assembly process of the photoelectrochemical-biosensing for E. coli O157:H7. The process I states the PEC response based on aptamer/CdS QDs/Au NPNWs as a control. From Dong et al. [51], reprinted with permission.

Fig. 6 .

 6 Fig. 6. Schematic representation of E. coli imprinted polymeric film synthesis via microcontact imprinting technique. From Özgür et al. [52], reprinted with permission.

Fig. 7 .

 7 Fig. 7. Nanocomposites and system. (a) Preparation of the GQDs nanocomposites. (b) Smartphone-based ECL system with ITO electrode and platinum electrode. (c) ECL principle for E.coli detection. From Li et al. [61], reprinted with permission.

Fig. 9 .

 9 Fig. 9. Quartz crystal microbalance (QCM) sensor fabrication, and procedure for E. coli O157:H7 detection. From Mao et al [67], reprinted with permission.

Fig. 10 .

 10 Fig. 10. Schematic representation of the nanostructured sensor based on carbon nanotubes: the peptide is bound on the CNTs surface and the interaction of the sensor with the bacteria is evaluated. From Andrade et al. [69], reprinted with permission.

Fig. 11 .

 11 Fig. 11. Schematic illustration for fabrication of CNT-FET E-coli biosensor. From Zhang et al. [71], reprinted with permission.

Fig. 12 .

 12 Fig. 12. Schematic depicting various steps involved in the fabrication of FET sensor (1) Self-assembly of rGO (2) deposition of atomic layer of Al2O3 (3) deposition of AuNPs (4) immobilization of anti-E. coli and (5) E. coli sensing From Thakur et al. [73], reprinted with permission.

Fig. 13 .

 13 Fig. 13. SEM images showing bacteria attached on the Si patterned nanowires and its corresponding merged image of fluorescence micrographs Si nanowire. From Jeong et al. [79], reprinted with permission.

TABLE I SUMMARY

 I OF RECENT BIOSENSORS BASED ON NANO-OBJECTS FOR E. COLI DETECTION

				Range	Detection
	Nano-material	Bacteria	Detection technique	concentration	limit	Characteristics
				(CFU/mL) a	(CFU/mL) a
	ZnO nanorods array	E. coli	Light-addressable potentiometry	10 -10 5	10 2	Label-free, sensitive and effective
		O 157:H7 (DNA)				detection of bacterial DNA [40]
	Cysteine modified	E. coli	Cyclic voltammetry and impedance	1-10 4	-
	gold NPs	ATCC 25922	spectroscopy			Viable, simple and specific detection
						[45]
	Polypyrrole-reduced	E. coli	Pulse difference voltammetry	10 -10 7	10	Enzyme free, good stability, high
	graphene oxide	K12				selectivity and excellent
	modified gold					reproducibility detection [50]
	nanoparticles
	ZnONWS/AuNPs/Q	E. coli	Photo-electrochemical (amperometry)	10 -10 7	1.125	Low noise, fast, and ultrasensitive
	D-Cds composites	O 157:H7				detection [51]
	Silver nanoparticles	E. coli	Surface plasmon resonance	1.5×10 1 -	0.576	Real-time, label-free, selective, and
		ATCC 25922		1.5×10 6		low detection limit [54]
	Graphene quantum	E. coli	Electrogenerated chemiluminescence	10 -10 7	5	Low-cost, real-time, smartphone
	dots/SilverNPs					based detection [61]
	nanocomposites
	Carbon quantum	E. coli	Fluorescence	500 -10 6	487	Sensitive, specific detection with
	dots/magnetic	0 157 :H7				low-cost, simple method for
	nanoparticles(Fe3O4)					preparation of nanocomposites [64]
	Single walley	E. coli	Impedance spectroscopy	10 2 -10 6	10 2	Differentiation of gram-negative and
	carbon nanotubes					gram-positive bacteria, good
						reproducibility, very sensitive [65]
	Single walley	E. coli	Conductance measurements	8.2×10 2 -	-	Real-time detection, and compatible
	carbon nanotubes	O157:H7		8.2×10 8		with electronic platform [71]
	Graphene	E. coli	Conductance measurements			Real-time detection, and compatible
		K12		10 -10 5	-	with electronic platform [72]
	Graphene	E. coli	Conductance measurements			Real-time detection, and compatible
		ATCC 25922		10 3 -10 5	-	with electronic platform [73]
	Graphene		Cyclic voltammetry and impedance			Real-time, sensitive and specific
		E. coli	spectroscopy	10 -10 6	-	detection [76]
		O78:K80:H11
						Real-time detection, and compatible
	Silicon nanowires	E. coli	Conductance measurements	8×10 3 -8×10 8	-	with CMOS electronic platform [76]
	array	ATCC 35218


A Belme

M Martinelli

L Hascoët

V Pascual

A Dervieux
email: alain.dervieux@sophia.inria.fr

AD-based perturbation methods for uncertainties and errors

Keywords: Automatic Differentiation, second-order derivatives, uncertainty, error correction, design, CFD

The progress of Automatic Differentiation (AD) and its impact on perturbation methods is the object of this paper. AD studies show an important activity for developing methods addressing the management of modern CFD kernels, taking into account the language evolution, and intensive parallel computing. The evaluation of a posteriori error analysis and of resulting correctors will be addressed. Recents works in the AD-based contruction of second-derivatives for building reducedorder models based on a Taylor formula will be presented on the test case of a steady compressible flow around an aircraft.

Introduction

While high fidelity models are mainly used for deterministic design, which assumes a perfect knowledge of the environmental and operational parameters, uncertainty can arise in many aspects of the entire designproduction-operational process: from the assumptions done in the mathematical model describing the underlying physical process, to the manufacturing tolerances, and to the operational parameters and conditions that could be affected by unpredictable factors (e.g. atmospheric conditions). Exact and approximate techniques for propagating these uncertainties require additional computational effort but are progressively well established. The proposed study takes place in NODESIM-CFD FP6 project [START_REF] Martinelli | Tangent-on-Tangent vs. Tangent-on-Reverse for Second Differentiation of Constrained Functionals[END_REF][START_REF]Non-deterministic simulation for CFD-based design technologies[END_REF]. The AD tool TAPENADE [START_REF] Hascoët | TAPENADE 2.1 user's guide[END_REF] has been developed for a large range of applications where the code to code direct and reverse differentiation is needed. Direct and reverse Automatic Differentiation are used for addressing numerical error reduction since they help building correctors. This discussed in Section 3. Uncertainty propagation is addressed by a perturbation technique using the first terms of Taylor series of the high-fidelity model (Method of Moments). This is built as an application of Automatic Differentiation (AD). Efficient first and second derivatives softwares are produced by TAPENADE, thanks to recent improvements that we shortly present in first section. Then the development of

1 44th AAAF, Nantes (F) 2009
a Method of Moments using second derivatives obtained with TAPENADE is presented in Section 2.

Automated Differentiation improvements

Our AD tool TAPENADE has been extended to deal with Fortran95 and with ANSI C ( [START_REF] Pascual | Extension of TAPENADE towards Fortran 95[END_REF][START_REF] Pascual | TAPENADE for C[END_REF]). Figure 1 shows the architecture of tapenade. It is implemented mostly in java (115 000 lines) except for the separate front-ends which can be written in their own languages. Front-and

Differentiation Engine

Imperative Language Analyzer back-ends communicate with the kernel via an intermediate abstract language ("IL") that makes the union of the constructs of individual imperative languages. Notice also the clear separation between the general-purpose program analysis and the differentiation engine itself. Thanks to the language-independent internal representation of programs, this still makes a single and only tool, and every development benefits to differentiation of each input language. One of these developments concerned the pointer analysis. The reverse mode now accepts most uses of pointers and allocation. Another development concerned declarations. The differentiated program respects the order of declarations, uses the include files and keeps the comments from the original program. Generated codes are more readable and often smaller. We also investigated extensions to TAPENADE to successive differentiations, in particular to efficiently handle tangent differentiation of the stack primitives present in the reverse differentiated codes. We implemented user directives for the reverse differentiation of a frequent class of parallel loops (directive II-LOOP) and for opti-mal checkpointing in reverse differentiation ( [START_REF] Naumann | A Framework for Proving Correctness of Adjoint Message-Passing Programs[END_REF], [START_REF] Hascoët | Cheaper Adjoints by Reversing Address Computations[END_REF], [START_REF] Tber | Building the Tangent and Adjoint codes of the Ocean General Circulation Model OPA with the Automatic Differentiation tool TAPENADE[END_REF]). TAPENADE lets the user specify finely which procedure calls must be checkpointed or not with the directive NOCHECKPOINT.

Numerical errors reduction

Error estimates and corectors

Let us recall first how linearised -direct or adjoint-states can be useful for improving numerical accuracy issues.

Numerical error involves the deviation between the solution W = W (x, y, z) of mathematical model i.e. of the nonlinear PDE symbolized by: Ψ(W ) = 0 .

(

) 1
and the output data produced by the computations, i.e. the more or less perfect numerical solution of the discrete system:

Ψ h (W h ) = 0 ∈ R N (2)
The discrete unknown W h is the N-dimensional array of degrees of freedom:

W h ∈ R N , W h = [(W h ) i ] .
The output data produced by the computation do not involve a function W h but, instead the array W h which needs to be transformed via an interpolation: Let V ∈ L 2 (Ω) a space of rather smooth function. In practice, V ⊂ C 0 ( Ω). Let R h be a linear interpolation operator transforming an array of N degrees of freedom into a continuous function:

R h : R N → V v h → R h v h . (3)
Let:

W h (x, y, z) = (R h W h )(x, y, z).
Similarly, we need an operator from continuous functions to arrays. Let T h be an operator transforming a continuous function into an array of N degrees of freedom:

T h : V → R N v → T h v. (4)
It is useful to take the adjoint of R h :

T h = R * h . (5)
The deviation between the PDE solution and the numerical one can be defined as W -W h . It consists mainly of approximation errors, of algorithmic errors, arising typically because iterative algorithmes are not iterated infinitely, and of round-off errors due to the fact that the programm is run in floating point arithmetics. We discuss here mainly of approximation errors, although the other ones may be also addressed in part by the method studied here. Another way to postprocess a computation is to use it for evaluating a scalar functional: Let j a smooth linear functional applying W into the scalar number:

j(u) = (g, W ) L 2 (Ω)
where g is a given L 2 (Ω) function. This allows to define:

g h = T h g g h = R h g h = R h T h g (6)
The continuous adjoint writes:

( ∂Ψ ∂u ) * p = g .
The discrete adjoint equation is then defined by:

∂Ψ h ∂u h T p h = T h g . (7)
And we can then consider:

p h = R h p h .
A fundamental assumption of the present analysis is that this discrete adjoint is a good enough approximation of continuous adjoint p for allowing to replace p by p h the calculations which follow. In order to evaluate the approximation error, two kinds of estimates can be applied:

A posteriori estimate:

Ψ(W ) -Ψ(W h ) = -Ψ(W h ) (8)
where Ψ(W h ) is the continuous residual applied to discrete solution. Then:

W -W h ≈ -∂Ψ ∂W -1 Ψ(W h ). (9)
A priori estimate:

Ψ h (T h W ) -Ψ h (W h ) = -Ψ h (T h W ) ( 10
)
where Ψ h (T h W ) is the discrete residual applied to discretised continuous solution.Then:

T h W -W h ≈ -∂Ψ h ∂W h -1 Ψ h (T h W ). ( 11
)
We observe that these estimates involve unavailable continuous functions. In the a posteriori estimate, the solution of the continuous linearised system can be approximated thanks to the discrete Jacobian. For the a priori estimate, we can also solve this issue in some particular case see [START_REF] Loseille | Adaptation de maillage anosotrope 3D multi-echelles et ciblée à une fonctionnelle pour la Mécanique des Fluides, application à la prédiction du bang sonique[END_REF], by replacing Ψ h (T h W ) by an expression T h Θ h (W h ) depending only of W h . Corresponding to these estimates, we have the following field correctors:

δW h = -R h ∂Ψ h ∂W h -1 T h Ψ(W h ) ( 12
)
δR h W h = -R h ∂Ψ h ∂W h -1 T h Θ h (W h ) (13)
and the following direct-linearized goal-oriented correctors:

δ 1 j = -g , R h ∂Ψ h ∂W h -1 T h Ψ(W h ) L 2 (Ω) (14)
δ 2 j = -g , R h ∂Ψ h ∂W h -1 T h Θ h (W h ) L 2 (Ω) . ( 15
)
Also follows the adjoint-based goal-oriented correctors:

δ 1 j = -(p h , T h Ψ(W h )) L 2 (Ω) (16
)

δ 2 j = -(p h , T h Θ h (W h )) L 2 (Ω) . (17)
We recognize in ([?]) the superconvergent corrector of [?]. We observe that, thanks to the choice of T h as the adjoint operator of R h , the linearised-based and adjointbased formulation are perfectly equivalent. At the contrary, the effort to compute them are very different, particularly in the case of unsteady PDE, since the adjoint system has to be solved reverse in time, while using the state solution at all time levels. This remark leads to the following recommendations:

-use the direct linearised formulation in any case you only need a corrector for the field as well as a corrector for one or several output functionals.

-the adjoint formulation is compulsory when you wish to derive an goal-oriented optimal mesh.

The second recommendation is motivated by the fact that an optimal mesh will be derived from minimisation of the error term in which we need to put in evidence the dependance of error with respect to mesh. Since the adjoint is an approximation of a continuous function, it does not much depend of mesh. At the contrary, the continuous residual T h Ψ(W h ) or the truncation error T h Θ h (W h ) are proportional to a power of the mesh size. In [START_REF] Loseille | Adaptation de maillage anosotrope 3D multi-echelles et ciblée à une fonctionnelle pour la Mécanique des Fluides, application à la prédiction du bang sonique[END_REF], the truncation error is expressed in terms of second derivatives of solution field and allows the derivation of an optimal mesh.

An example

To end this discussion, we give an numerical example of corrector evaluation built on a finite-element approximation. we can write Euler equations under the form:

W ∈ V = H 1 (Ω) 5 , ∀φ ∈ V, Ω F(W )∇φ dΩ - ∂Ω φ F(W ).n dΓ = 0 . ( 18
)
where F(W ) accounts for the different boundary conditions. Let us introduce a discretization of the previous EDP. Let τ h a tetrahedrization of Ω with N vertices. It will rely on a discrete space of functions:

V h = {φ h ∈ H 1 (Ω) 5 , ∀T ∈ τ h , φ h |T ∈ P 1 }
the canonical basis of which is denoted:

V h = span[N i ], N i (x j ) = δ ij ∀i, j, vertices of τ h ,
and on the interpolation operator:

Π h : V → V h , Π h φ(x i ) = φ(x i ), ∀i, vertex of τ h .
Comparing with the previous abstract theory, we get:

R h : R 5N → V h , f h → R h f h = Σ i [f h ] i N i , T h : V → R 5N , φ → T h φ = [φ(x i )] .
The discretization is set into the discrete space, but also it differs from the continuous statement in two features, a discrete flux F h instead of F:

F h : V → V ′
and an extra term of artificial diffusion D h :

W h ∈ V h , ∀φ h ∈ V h , (Ψ h (W h ), φ h ) V ′ ×V = 0, with (Ψ h (W h ), φ h ) V ′ ×V = Ω φ h ∇.F h (W h ) dΩ - Γ φ h Fh (W h ).n dΓ + Ω φ h D h (W h )dΩ . (19)
The discrete fluxes are chosen as follows:

F h (W ) = F h (Π h W ) = Π h F(Π h W ) . Fh (W ) = Fh (Π h W ) = Π h F(Π h W ). ( 20
)
After some calculations and simplifications, the main error term appears as follows:

( ∂Ψ h ∂W h δW h , φ h ) = - Ω ∇φ h (F(W ) -Π h F(W ))dΩ + Γ φ h ( Fout (W ) -Π h Fout (W )).n dΓ (21)
with F(W ).n = F(W ).n-F(W ).n. A Gauss quadrature is applied for the evaluation of the right hand side. We have applied this to a steady subsonic flow and give some preliminary results. Figure 2 compares the entropy generation in the flow computed directly and the same flow corrected by formula (21). Entropy level is one order of magnitude smaller.

Uncertainty propagation techniques

In optimization problems, uncertainty propagation analysis may concern the study of the cost functional

j : γ → j(γ) := J(γ, W ) ∈ R (22)
where all varying parameters are represented by the uncertain (i.e. not-deterministic) control variables γ ∈ R n , and where the state variables W = W (γ) ∈ R N are solution of the (nonlinear) state equation

Ψ(γ, W ) = 0. ( 23
)
It is important to note that the state equation (23) contains the governing PDE of the mathematical model of the physical system of interest (for example the stationary part of the Euler or Navier-Stokes equations) and it can be viewed as an equality constraint for the functional (22). The basic probabilistic approaches for analyzing the propagation of uncertainties are Monte-Carlo methods.

A full nonlinear Monte-Carlo method gives us a complete and exact information about uncertainty propagation in the form of its PDF, but with a prohibitively expensive cost in terms of CPU time. In NODESIM-CFD, several other probabilistic approaches for analyzing the propagation of uncertainties are considered such as Latine Hypercubes and Polynomial Chaos. We contribute on perturbative methods based on the Taylor expansion.

Perturbation methods

To reduce the computational cost, we may think to use only some (derivate) quantities characterizing the distribution of the input variables instead of an entire sample drawn from a population with a given PDF. Therefore, the idea behind the Method of Moments is based on the Taylor series expansion of the original nonlinear functional (22) around the mean value of the input control (µ γ = E[γ]), and then computing some statistical moments of the output (usually mean and variance). In this way, we are assuming that the input control γ can be decomposed as sum of a fully deterministic quantity µ γ with a stochastic perturbation δγ u with the property E[δγ u ] = 0. With these definitions, the Taylor series expansion of the functional j(γ) around the mean value µ γ is:

j(γ) = j(µ γ + δγ u ) = j(µ γ ) + Gδγ u + + 1 2 δγ * u Hδγ u + O(||δγ u || 3 ) (24)
where G = ∂j ∂γ u µγ is the gradient of the functional respect to the uncertain variables and H = ∂ 2 j ∂γ u 2 µγ is the Hessian matrix, both evaluated at the mean of the input variables µ γ .

By considering various orders of the Taylor expansion (24) and taking the first and the second statistical moment, we can approximate the mean µ j and the variance σ 2 j of the functional j(γ) in terms of its derivatives evaluated at µ γ and in terms of statistical moments of the control γ.

First order moment methods:

     µ j = j(µ γ ) + O E δγ 2 u σ 2 j = E Gδγ u 2 + O E δγ 3 u ( 25
)
Second order moment methods:

µ j = j(µ γ ) + 1 2 E δγ * u Hδγ u +O E δγ 3 u σ 2 j = E Gδγ u 2 + E Gδγ u δγ * u Hδγ u - 1 4 E δγ * u Hδγ u 2 + 1 4 E δγ * u Hδγ u 2 +O E δγ 4 u
With this method it is clear that we are using only some partial informations about the input uncertainties, in fact we are using only some statistical moments of the control variable instead of full information available with its PDF, and we will not have anymore the PDF of the propagated uncertainty, but only its approximate mean and variance. Another important point is that the Method of Moments is applicable only for small uncertainties, due to the local nature of Taylor expansion approximation. Two things should be noted here: the first one is that for the Method of Moments we need the derivatives of the functional respect to the control variables affected by uncertainties: in particular we need the gradient for the first order method, and gradient and Hessian for the second order method. Due to the fact that j(γ) = J(γ, W ), where W = W (γ) is solution of the state equation ( 23) we have for the derivative:

∂j ∂γu = ∂J ∂γu + ∂J ∂W ∂W ∂γu
Since we know the solution W (γ) by its numerical values as result of a program (implementing an appropriate method, e.g. fixed point method), it is intersting to use of Automatic Differentiation tools (like TAPENADE) in order to obtain the needed derivatives. The same remarks apply to the computation of the Hessian matrix.

In particular we note that the derivatives are computed at the mean value of the control µ γ , so they are fully deterministic and can be picked out from the expectations in the equations (25) or (26). In other words we can write

E Gδγ u 2 = i,k G i G k E δγ (i) u δγ (k) u = i,k G i G k C ik E δγ * u Hδγ u = i,k H ik E δγ (i) u δγ (k) u = i,k H ik C ik E Gδγ u δγ * u Hδγ u = i,k,l G l H ik E δγ (i) u δγ (k) u δγ (l) u E δγ * u Hδγ u 2 = i,k,l,m H ik H lm E δγ (i) u δγ (k) u δγ (l) u δγ (m) u ( 26
)
where

G i = ∂j ∂γ (i) u µγ
are the elements of the gradient,

H ik = ∂ 2 j ∂γ (i) u ∂γ (k) u µγ
are the elements of the Hessian matrix and C ik = E δγ

(i) u δγ (k) u = cov(γ (i) u , γ (k)
u ) are the elements of the covariance matrix. Every expectaction term E[. . . ] in the equations (4.1), is defined by the statistical model of the uncertainties and could be computed in a preprocessing phase.

For example, for the important case where the uncertainties are random and normally distribuited, we have:

E δγ (i) u δγ (k) u δγ (l) u = 0 E δγ (i) u δγ (k) u δγ (l) u δγ (m) u = C ik C lm + C il C km + C im C kl
and if these (normal) uncertainties are idependents, then holds the relation

C ik = σ 2 i δ ij where σ 2 i = E δγ (i) u δγ (i) u
and the equations (4.1) become

E Gδγ u 2 = i G 2 i σ 2 i E δγ * u Hδγ u = i H ii σ 2 i E Gδγ u δγ * u Hδγ u = 0 E δγ * u Hδγ u 2 = i,k H ii H kk + 2H 2 ik σ 2 i σ 2 k (27) Since we have the term E δγ * u Hδγ u 2 /4
, the error is still of the order of E δγ 4 u . Computing the other terms of same order require the knowledge of order of derivatives higher than the second. From the previous discussion, it is clear that in order to apply the Method of Moments we need to solve only one (expensive) nonlinear system with derivatives (at the mean µ γ ) and then apply the (inexpensive) equations ( 25) or (26) where, for the fully nonlinear Monte-Carlo approach of the previous section, we need to solve N ≫ 1 nonlinear systems (23).

First and second-order derivatives of a functional

We are interested by obtaining the first and second derivatives of a functional j depending of γ ∈ R n , and expressed in terms of a state W ∈ R N as follows:

ψ(γ) = Ψ(γ, W (γ)) = 0 j(c) = J(γ, W (γ)) ( 28
)
Our problem can be viewed from two different point of view: the first one is consider the solution algorithm for state equation as part of j itself, i.e. considering j as a function of the control variables γ only. The second one is consider the system made by two different routines: one of them is the routine that solves the nonlinear system Ψ(γ, W (γ)) = 0 (and contains the evaluation the residual Ψ(γ, W )), and the other is the routine J(γ, W ) that computes the value of the functional from the state variables W and (eventually) the control variables γ.

The first approach lead to a straightforward algorithm for first order derivatives, in fact we just need to differentiate the entire routine j with tangent or reverse mode . In this context, the routine j contains the iterative solver method for the state equation, and the differentiated routines will also contains this loop in differentiated form. If we need n iter loop iterations in order to obtain the nonlinear solution, and we assume for each iteration an unitary cost, we can analyze the cost for the gradient of the functional.

Using tangent mode, the cost for the entire gradient will be n(n iter α T ) where n is the number of components of the gradient and 1 < α T < 4 is the overhead associated with the differentiated code respect to the original one. For this stategy, the memory requirements will be of the same order of the undifferentiated code.

With reverse mode we are able to obtain the entire gradient with a single evaluation of the differentiated routine, but the total cost (in terms of CPU time and memory) will depends on the strategy used by the AD tool to solve the problem of inverse order differentiation for the original routine. For the case of a Store-All (SA) strategy, the CPU cost will be (n iter α R ) with 1 < α R <, i.e. α R times the undifferentiated code, but the required memory will be n times greater. For a Recompute-All (RA) strategy the CPU cost will be (n 2 iter α R ), i.e. (n iter α R ) the nonlinear solution, but the memory will be the same of the undifferentiated routine. For real large programs, neither SA or RA strategy can work, so we need a special storage/recomputation trade-off in order to be efficient using checkpoints. Obviously, with checkpointing the CPU cost will be greater than the cost of SA strategy and can be shown that the cost for the differentiated code will be of the order of s √ n iter (where s is the number of snapshots available).

It is clear that for gradient computation with n ≫ 1, the reverse mode is faster than tangent mode, but for a program containing an iterative algorithm, the reverse mode is not always applicable. The problem relies on the fact that the Reverse mode computation is performed in the opposite way of the original code (backward sweep) after a forward sweep needed to store the variable needed in the successive phases.

For the previous arguments, we prefer differentiate not the entire program (solution of the state equation + functional evaluation), but the two main component in a separate way, using the fact that at the solution, the residuals will be zero (i.e. we don't differentiate the routine containing the main loop, but only the quantities involved after the last iteration). For this second approach, we have to analyse the influence of state equation and the functional evaluation in more details. This is the purpose of the next sections.

First derivative

Using the chain rule, the gradient of the functional j(γ) = J(γ, W (γ)) is given by

dj dγ = ∂J ∂γ + ∂J ∂W dW dγ
where the derivatives of the state variables W (c) are obtained solving the linear system

dψ dγ = ∂Ψ ∂γ + ∂Ψ ∂W dW dγ = 0
Therefore, two strategies can be applied: Direct differentiation :

It consists in computing the Gateaux-derivatives with respect to each component direction (e i = (0, . . . 0, 1, 0, . . . , 0)

This has to be applied to each component of γ, i.e. n times and the cost is n linearised N -dimensional systems to solve. If we choose to solve the single system (30) with an iterative matrix-free method, and the solution is obtained after n iter step, the total cost will be of the order of α T n iter,T , i.e. n iter,T evaluation of the matrix-by-vector operation ∂Ψ ∂W x, where each evaluation costs α T times the evaluation of the state residual Ψ(γ, W ) (and the cost of the state residual is taken as reference equal to 1). Therfore, the cost of the full gradient will be nα T n iter,T .

Inverse differentiation (Reverse mode)

The complete gradient is given by the equation

dj dγ * = ∂J ∂γ * - ∂Ψ ∂γ * Π 0 (31)
where Π 0 is the solution of the linear system

∂Ψ ∂W * Π = ∂J ∂W * (32)
This computation needs only one extra linearised Ndimensional system, the adjoint system (some methods for calculation of the adjoint solutions are described in ). If we choose to solve the adjoint system (32) with an iterative matrix-free method, we can apply the same estimate done as in the case of the Tangent mode differentiation, but this time the overhead associated with the evaluation of the matrix-by-vector operation ∂Ψ ∂W * x respect to the state residual evaluation will be α R (and usually α R > α T , and the number of iteration n iter,R for the convercence of the solution could be different from n iter,T of the previous case (but the asymptotical rate of convergence will be the same of the original linear system ∂Ψ ∂W x = b, see ). Therefore the cost for the gradient will be α R n iter,R , and the Reverse mode differentiation for the gradient computation is cheaper than the Tangent mode if n ≫ 1.

Second derivative

For second derivatives we have different possibilities:

Direct-direct option

This methods was initially investigated by along with various other algorithms, but the publication does not go into the implementation details for a generic fluid dynamic code. Here we present the mathematical background behind the idea and the efficient AD implementation of Ghate and Giles but with a different analysis of the computational cost.

Starting from the derivative (29), we perform another differentiation respect to the variable γ k obtaining

d 2 j dγ i dγ k = D 2 i,k J + ∂J ∂W d 2 W dγ i dγ k (33)
where

D 2 i,k J = ∂ ∂γ
d 2 j dγ i dγ k = D 2 i,k J - ∂J ∂W ∂Ψ ∂W -1 D 2 i,k Ψ = D 2 i,k J -Π * 0 D 2 i,k Ψ (35)
where Π 0 is the solution of the adjoint system (32) evaluated at the point (γ, W

Inverse-direct

This consists in the direct derivation in any direction e i , i = 1, n of the (non-scalar) function:

Inverse-inverse

If we are interested in a (scalar!) functional depending on the gradient, then it can be interesting to apply a second inverse differentiation. We do not focus on this direction at the moment.

Numerical experiments

The testcase considered here corresponds to the optimization of the wing shape of a business aircraft (courtesy of Piaggio Aero Ind.), for a transonic regime (see the shape and the mesh in Fig. 3). The nominal operational conditions are defined by the free-stream Mach number M ∞ = 0.83 and the incidence α = 2 • . We suppose that only these two quantities are subject to random fluctuations. For simplicity, we assume that their PDF are Gaussian with given mean and variance. The mean values correspond to the nominal values. The section of the initial wing shape corresponds to the NACA 0012 airfoil.

For the present work, due to the fact that we consider only two uncertain variables, we used a ToT approach for the Hessian evaluation. A comparision of the resulting approximate drag coefficient with respect to the real value is given in Fig. 5. We want to point out the fact that the second-order approximation requires to solve only one nonlinear state equation Ψ = 0 plus 4 linear systems using ToT. The real values of the functional plotted in

Figure 1 :

 1 Figure 1: Overall Architecture of tapenade

Figure 2 :

 2 Figure 2: Entropy spurious generation for a direct computation of a steady flow and for a corrected one.

  , W (c)) where W (c) and Π(c, W (c)) are solutions of the above two state systems. With some algebra we obtain The derivation needs the solution of the adjoint systems ∂Ψ ∂W * Π 0 = ∂J ∂W * (36) and 2n perturbed N -dimensional linear systems (for the full Hessian): i where all the functions in the equations (36)-(37) are evalued at the final state (in order to verify Ψ(c, W (c)) = 0).

  Fig 4 require 21×21 nonlinear simulations.

Figure 3 :

 3 Figure 3: Wing shape and mesh in the symmetry plane.

Figure 4 :

 4 Figure 4: Drag coefficient vs. Mach number and angle of attack (first-order spatial accuracy) for the transonic wing: nonlinear simulations (first image); percentage relative difference between the nonlinear simulations and the second-order Taylor approximation (second image). For the top plot we have solved 21×21 nonlinear systems Ψ = 0.

  T , where 1 is at the i-esim compo-

	nent):
	dj dγ i	=	dj dγ	e i =	∂J ∂γ i	+	∂J ∂W	dW dγ i	(29)
	with:
		∂Ψ ∂W	dW dγ i	= -	∂Ψ ∂γ	e i

  Ψ). Therfore the full Hessian costs nα T [n iter,T + (n + 1)α T /2]. With similar arguments, if we want only the diagonal part of the Hessian, the cost is nα T [n iter,T + α T ].

	γ)) solution of the state
	equation Ψ(γ, W ) = 0. The n derivatives be computed (and stored) using tangent mode dif-dW should dγ i
	ferentiation of the nonlinear solver algorithm, and
	each derivatives costs n iter α T . If we need the full
	Hessian matrix we have to evaluate the quantity (35)
	n(n + 1)/2 times, i.e. we have to evaluate the terms
	D 2 i,k Ψ and D 2 i,k J for i = 1, . . . , n and j = i, . . . , n due
	to the simmetry of the Hessian, and each evaluation of
	(D 2 i,k Ψ costs α 2 T (the evaluation of D 2 i,k J) is negligible
	respect to (D 2 i,k

Acknowlegments: This studies are supported in part by NODESIM-CFP FP-6 European project. Figure 5: First-order (first image) and second-order (second image) Taylor approximation around α = 2 • and M = 0.83; For the 1st and 2nd-order Taylor approximation, we have solved only one nonlinear system.