A fully discrete framework for the adaptive solution of inverse problems

We investigate and contrast the differences between the discretize-then-differentiate and differentiate-then-discretize approaches to the numerical solution of parameter estimation problems. The former approach is attractive in practice due to the use of automatic differentiation for the generation of the dual and optimality equations in the first-order KKT system. The latter strategy is more versatile, in that it allows one to formulate efficient mesh-independent algorithms over suitably chosen function spaces. However, it is significantly more difficult to implement, since automatic code generation is no longer an option. The starting point is a classical elliptic inverse problem. An a priori error analysis for the discrete optimality equation shows consistency and stability are not inherited automatically from the primal discretization. Similar to the concept of dual consistency, We introduce the concept of optimality consistency. However, the convergence properties can be restored through suitable consistent modifications of the target functional. Numerical tests confirm the theoretical convergence order for the optimal solution. We then derive a posteriori error estimates for the infinite dimensional optimal solution error, through a suitably chosen error functional. This estimates are constructed using second order derivative information for the target functional. For computational efficiency, the Hessian is replaced by a low order BFGS approximation. The efficiency of the error estimator is confirmed by a numerical experiment with multigrid optimization

Space-time adaptive solution of inverse problems with the discrete adjoint method

Adaptivity in both space and time has become the norm for solving problems modeled by partial differential equations. The size of the discretized problem makes uniformly refined grids computationally prohibitive. Adaptive refinement of meshes and time steps allows to capture the phenomena of interest while keeping the cost of a simulation tractable on the current hardware. Many fields in science and engineering require the solution of inverse problems where parameters for a given model are estimated based on available measurement information. In contrast to forward (regular) simulations, inverse problems have not extensively benefited from the adaptive solver technology. Previous research in inverse problems has focused mainly on the continuous approach to calculate sensitivities, and has typically employed fixed time and space meshes in the solution process. Inverse problem solvers that make exclusive use of uniform or static meshes avoid complications such as the differentiation of mesh motion equations, or inconsistencies in the sensitivity equations between subdomains with different refinement levels. However, this comes at the cost of low computational efficiency. More efficient computations are possible through judicious use of adaptive mesh refinement, adaptive time steps, and the discrete adjoint method. This paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent, adaptive grid, adaptive step models. Discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation. However, this approach brings several important challenges. The adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations. A reduction in accuracy of the discrete adjoint sensitivities may appear due to the intergrid transfer operators. Moreover, the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations. This work shows that several of these potential issues can be avoided for the discontinuous Galerkin (DG) method. The adjoint model development is considerably simplified by decoupling the adaptive mesh refinement mechanism from the forward model solver, and by selectively applying automatic differentiation on individual algorithms. In forward models discontinuous Galerkin discretizations can efficiently handle high orders of accuracy, $h/p$-refinement, and parallel computation. The analysis reveals that this approach, paired with Runge Kutta time stepping, is well suited for the adaptive solutions of inverse problems. The usefulness of discrete discontinuous Galerkin adjoints is illustrated on a two-dimensional adaptive data assimilation problem

Transbronchic lung biopsy

Este o metoda semiinvazivă de prelevare endobronșică, utilă pentru diagnosticul afectâțiunilor pulmonare, având sediul dincolo de zonele de vizualizare directă endoscopică (din periferia câmpurilor pulmonare). Constă în recoltarea de țesut pulmonar situat între două bronhii distale. Indicații dpdv radiologic - Opacități interstițiale – difuze sau localizate; Opacități periferice (dincolo de vizualizarea directă endoscopică). Indicații dpdv etiologic - Patologie tumorală malignă; Boli interstițiale pulmonare; Boli infecțioase pulmonare; Boli fungice. Contraindicații ale BPT sunt generale ale bronhoscopiei - cardiovasculare - cardiopatie ischemică, pulmonare; neurologice, sau contraindicații speciale - deficit de coagulare, plămân unic, HTA. Materiale necesare și dotări - nu este necesară o dotare specială față de dotarea standard a unei săli de bronhoscopie – fibrobronhoscop și pensa de biopsie. Este prudent să se efectueze în vecinatatea unei secții ATI sau chirurgie toracică (datorită posibilelor incidente). Tehnica - pensa de biopsie este impinsa închisă în periferie în bronhia corespunzătoare existând apoi două variante de continuare: 1. miscări “du-te vino” pe o distanță de cca. 1cm și se închide pensa; și 2. pacientul este pus să inspire adânc, pensa este deschisă și împinsă către distal cu apnee. Se recoltează 4 - 6 fragmente. Incidente – accidente: Durere; Sângerări 10 - 20% din cazuri; Pneumotorax - 5 -10 % din cazuri; Sincopa - f. rar. Pentru creșterea siguranței tehnice și a randamentului se utilizează urmărirea sub ecran fluoroscopic a BPT cu un echipament Rontgen complex, cu brat în C și monitor Tv. Crioterapia este folosită în bronhoscopie de mulți ani, iar utilizarea sa principală este tratamentul și excizia leziunilor endobronsice, în special în cazurile de obstrucție bronșică. Obținerea unui diagnostic histologic precis cu BPT convențională este limitată de dimensiunea redusă a fragmentului de biopsie. În cazul probelor de criobiopsie dimensiunile sunt duble, iar structurile sunt păstrate. BPT este soluția de elecție recomandată înainte de metodele chirurgicale de prelevare bioptică – toracoscopie sau toracotomie. BPT scade indicația manevrelor chirurgicale de prelevare cu 50 – 60%. Asocierea BPT – LBA duce la creșterea randamentului și rapiditatea diagnosticului NBP cu localizare periferică, chiar în faza precoce.It is a semi-invasive method of endobronchial sampling useful for the diagnosis of lung diseases based beyond the areas of direct endoscopic visualization (from the periphery of the lung fields). It consists in the collection of lung tissue located between two distal bronchi Radiological indications - Interstitial opacities - diffuse or localized; Peripheral opacities (beyond direct endoscopic visualization). Etiological indications - Malignant tumor pathology; Interstitial lung diseases; Infectious pulmonary diseases; Fungal diseases Contraindications of BPT are: general bronchoscopy - cardiovascular - ischemic cardiopathy, pulmonary; neurological. or special contraindications - coagulation deficiency, single lung, hypertension Necessary materials and equipment - no special equipment is required compared to the standard equipment of a bronchoscopy room - fibrobronchoscope and biopsy forceps. It is prudent to perform it in the vicinity of an ATI or thoracic surgery department (due to possible incidents). Technique - the biopsy forceps is pushed closed in the periphery into the corresponding bronchus, and there are then two options for continuation: 1. "go and come" movements over a distance of approx. 1cm and close the clamp; and 2. - the patient is made to inhale deeply, the clamp is opened and pushed distally with apnea. 4 - 6 fragments are collected. Incidents - accidents: Pain; Bleeding 10 - 20% of cases; Pneumothorax - 5-10% of cases; Syncope - rarely. In order to increase technical safety and efficiency, BPT monitoring under a fluoroscopic screen is used with a complex X-ray equipment, with C-arm and TV monitor Cryotherapy has been used in bronchoscopy for many years, and its main use is the treatment and excision of endobronchial lesions, especially in cases of bronchial obstruction. Obtaining a precise histological diagnosis with conventional BPT is limited by the small size of the biopsy fragment. In the case of cryobiopsy samples, the sizes are double, and the structures are preserved BPT is the solution of choice recommended before the surgical methods of biopsy sampling - thoracoscopy or thoracotomy. BPT reduces the indication of surgical sampling maneuvers by 50-60%. The association of BPT - LBA leads to an increase in the yield and speed of diagnosis of NBP with peripheral localization, even in the early phase

EBUS-TBNA practical aspects

În ultimele două decenii, EBUS - TBNA a apărut ca o tehnică extrem de eficientă și minim invazivă pentru prelevarea de mase peribronșice, mediastinale și pulmonare pentru examenul histopatologic. EBUS - TBNA poate oferi rezultate rapide la fața locului, cu o expertiza relativ redusă și are un profil foarte sigur. S-a dovedit a fi semnificativ rentabil în comparație cu tehnicile considerate “gold-standard” anterior. Indicații principale ale EBUS-TBNA sunt: Diagnosticul, stadializarea și restadializarea cancerului pulmonar; Diagnosticul limfadenopatiei mediastinale si hilare; Diagnosticul limfoamelor; Leziuni bronsice . Limitele EBUS-TBNA - EBUS este limitat la mediastinul anterosuperior, iar EUS este adesea utilizat pentru a preleva de la nivelul mediastinul posteroinferior. EBUS și EUS pot fi adesea efectuate în aceeași sesiune consecutiv. EBUS este dificil de realizat din punct de vedere tehnic în anumite locații anatomice, cum ar fi lobii superiori, deoarece este necesară angularea extremă a ecobronhoscopului. Pentru pacienții supusi bronhoscopiei sub sedare conștientă, o tuse semnificativă poate limita succesul procedurii. Contraindicațiile pentru EBUS sunt similare cu cele ale bronhoscopiei în general. Contraindicațiile specifice EBUS-TBNA sunt legate de coagulopatii (induse medicamentos sau inerente). Sângerarea endoganglionară / endotumorală este în general redusă fiind limitată de rezistența extrinsecă a capsulei ganglionare sau intrinsecă endotumorală. Localizări practice și imagistice ale adenopatiilor mediastinale.In the last two decades, EBUS - TBNA has emerged as an extremely efficient and minimally invasive technique for sampling peribronchial, mediastinal and pulmonary masses for histopathological examination. EBUS - TBNA can provide quick results on the spot, with relatively little expertise and has a very safe profile. It has proven to be significantly cost-effective compared to the previously considered "gold-standard" techniques. The main indications of EBUS-TBNA are: Diagnosis, staging and restaging of lung cancer; Diagnosis of mediastinal and hilar lymphadenopathy; Diagnosis of lymphomas; Bronchial lesions. Limitations of EBUS-TBNA - EBUS is limited to the anterosuperior mediastinum, and EUS is often used to sample the posteroinferior mediastinum. EBUS and EUS can often be performed in the same session consecutively. EBUS is technically difficult to perform in certain anatomical locations, such as the upper lobes, because extreme angulation of the echo-bronchoscope is required. For patients undergoing bronchoscopy under conscious sedation, a significant cough can limit the success of the procedure. Contraindications for EBUS are similar to those of bronchoscopy in general. The specific contraindications of EBUS-TBNA are related to coagulopathy (drug-induced or inherent). Endonodal / endotumoral bleeding is generally reduced, being limited by the extrinsic resistance of the nodal capsule or intrinsic endotumoral. Practical and imaging locations of mediastinal adenopathies

Second order adjoints for solving PDE-constrained optimization problems

Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as PDE-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems requires the computation of derivatives of the model output with respect to model parameters. The first order derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of matrix-vector products between the Hessian of the cost functional and user defined vectors. Traditionally, the construction of second order derivatives for large scale models has been considered too costly. Consequently, data assimilation applications employ optimization algorithms that use only first order derivative information, like nonlinear conjugate gradients and quasi-Newton methods. In this paper we discuss the mathematical foundations of second order adjoint sensitivity analysis and show that it provides an efficient approach to obtain Hessian-vector products. We study the benefits of using of second order information in the numerical optimization process for data assimilation applications. The numerical studies are performed in a twin experiment setting with a two-dimensional shallow water model. Different scenarios are considered with different discretization approaches, observation sets, and noise levels. Optimization algorithms that employ second order derivatives are tested against widely used methods that require only first order derivatives. Conclusions are drawn regarding the potential benefits and the limitations of using high-order information in large scale data assimilation problems

DENSERKS: Fortran sensitivity solvers using continuous, explicit Runge-Kutta schemes

DENSERKS is a Fortran sensitivity equation solver package designed for integrating models whose evolution can be described by ordinary differential equations (ODEs). A salient feature of DENSERKS is its support for both forward and adjoint sensitivity analyses, with built-in integrators for both first and second order continuous adjoint models. The software implements explicit Runge-Kutta methods with adaptive timestepping and high-order dense output schemes for the forward and the tangent linear model trajectory interpolation. Implementations of six Runge-Kutta methods are provided, with orders of accuracy ranging from two to eight. This makes DENSERKS suitable for a wide range of practical applications. The use of dense output, a novel approach in adjoint sensitivity analysis solvers, allows for a high-order cost-effective interpolation. This is a necessary feature when solving adjoints of nonlinear systems using highly accurate Runge-Kutta methods (order five and above). To minimize memory requirements and make long-time integrations computationally efficient, DENSERKS implements a two-level checkpointing mechanism. The code is tested on a selection of problems illustrating first and second order sensitivity analysis with respect to initial model conditions. The resulting derivative information is also used in a gradient-based optimization algorithm to minimize cost functionals dependent on a given set of model parameters

Alcohol is a possible endoscopic method for resolving bronchial bump dehiscences

Dehiscența bontului de rezecție pulmonară reprezintă un moment critic în evoluția pacientului postchirurgical, necesitând reintervenție chirurgicală, ceea ce reprezintă o nouă agresiune asupra pacientului. Ideea de a folosi etanolul pur a aparut în urma observării apariției unor granulații mucoase după inhalarea accidentală de alcool de către unii pacienți. Studii efectuate pe plămânul de iepure care menționau reacție granulomatoasă semnificativă urmată de epitelizare și parțială fibroză ca urmare a injectării în cantități mici a etanolului în peretele bronșic. Injectarea cu alcool a produs o zonă de necroză și ulterior regenerare epitelială și fibroza septală alveolară. La acești pacienți intervenția chirurgicală a fost imposibil de aplicat din cauze variate - contraindicații cardiace; stare generală afectată; spectrul bacteriologic – MDR-Tuberculoza; refuzul pacientului.Procedura a fost efectuata după epuizarea altor tratamente - drenaj pleural prelungit; terapie cu antibiotice (3-4 săptămâni); intervenție Eleosser Procedeul în sine este relativ simplu de aplicat necesitand doar fibrobronhoscop, ac de puncție transbronșică și etanol, putându-se efectua în orice sală de endoscopie bronșică echipată standard. Tehnica - acul de puncție transbronșică se introduce sub control vizual, strict intramucos. Se injectează strict intramucos alcool etanolic pur, fracționat (câte 0,5 ml) la marginea orificiului dehiscent. Se toaletează imediat arborele bronșic. Se evaluează după cca. 3-4 săptămâni, necesitatea repetării manevrei. Faze – 1. degradarea proteinei mucoasei cu îngustarea moderată a orificiului dehiscent; 2. modificarea granulatiei; 3. regenerarea epitelială; 4. remaniere fibrotică. Prezentare 5 cazuri. Discuții - rezultate net favorabile s-au obţinut în 8 cazuri dintr-un total de 11, necesitând mai multe instilaţii bronşice intramucoase. Eșecurile posibile se pot datora unor situații speciale: hiperglicemie; supurații preexistente; instilarea intramucoasă incorectă. Nu au fost observate efecte secundare locale sau sistemice notabile după instilare cu etanol. Monitorizarea pacientului este necesară la fiecare 3-6 luni în primul an. În loc de concluzii - această procedură este ieftină și usor de aplicat pentru pacient. Procesul în sine este relativ simplu de implementat și poate fi efectuat în orice cameră de endoscopie bronșică echipată standard.The dehiscence of the lung resection abutment represents a critical moment in the evolution of the post-surgical patient, requiring surgical reintervention, which represents a new aggression on the patient. The idea of using pure ethanol arose after observing the appearance of some mucous granulations after the accidental inhalation of alcohol by some patients. Studies performed on the rabbit lung that mentioned a significant granulomatous reaction followed by epithelization and partial fibrosis as a result of the injection of small amounts of ethanol into the bronchial wall. Alcohol injection produced an area of necrosis and subsequent epithelial regeneration and alveolar septal fibrosis In these patients, surgical intervention was impossible for various reasons - cardiac contraindications; impaired general condition; bacteriological spectrum – MDR-Tuberculosis; the patient's refusal. The procedure was performed after the exhaustion of other treatments - prolonged pleural drainage; antibiotic therapy (3-4 weeks); Eleosser intervention The procedure itself is relatively simple to apply, requiring only a fibrobronchoscope, a transbronchial puncture needle and ethanol, and can be performed in any standard equipped bronchial endoscopy room Technique - the transbronchial puncture needle is inserted under visual control, strictly intramucosally. Pure, fractionated ethanolic alcohol (0.5 ml each) is injected strictly intramucosally at the edge of the dehiscent hole. Clean the bronchial tree immediately. It is evaluated after approx. 3-4 weeks, the need to repeat the maneuver. Phases – 1. degradation of the mucosal protein with moderate narrowing of the dehiscent hole; 2. granulation modification; 3. epithelial regeneration; 4. fibrotic remodeling. Presentation of 5 cases. Discussions - clearly favorable results were obtained in 8 cases out of a total of 11, requiring several intramucosal bronchial instillations. Possible failures may be due to special situations: hyperglycemia; pre-existing suppurations; incorrect intramucosal instillation. No notable local or systemic side effects were observed after ethanol instillation. Patient monitoring is required every 3-6 months during the first year. Instead of conclusions - this procedure is cheap and easy to apply for the patient. The process itself is relatively simple to implement and can be performed in any standard equipped bronchial endoscopy room

A Hybrid Approach to Estimating Error Covariances in Variational Data Assimilation

Data Assimilation (DA) involves the combination of observational data with the underlying dynamical principles governing the system under observation. In this work we combine the advantages of the two prominent advanced data assimilation systems, the 4D-Var and the ensemble methods. The proposed method consists of identifying the subspace spanned by the major 4D-Var error reduction directions. These directions are then removed from the background covariance through a Galerkin-type projection. This generates an updated error covariance information at both end points of an assimilation window. The error covariance information is updated between assimilation windows to capture the ``error of the day''. Numerical results using our new hybrid approach on a nonlinear model demonstrate how the background covariance matrix leads to an error covariance update that improves the 4D-Var DA results

On the discrete adjoints of adaptive time stepping algorithms

We investigate the behavior of adaptive time stepping numerical algorithms under the reverse mode of automatic differentiation (AD). By differentiating the time step controller and the error estimator of the original algorithm, reverse mode AD generates spurious adjoint derivatives of the time steps. The resulting discrete adjoint models become inconsistent with the adjoint ODE, and yield incorrect derivatives. To regain consistency, one has to cancel out the contributions of the non-physical derivatives in the discrete adjoint model. We demonstrate that the discrete adjoint models of one-step, explicit adaptive algorithms, such as the Runge--Kutta schemes, can be made consistent with their continuous analogs using simple code modifications. Furthermore, we extend the analysis to cover second order adjoint models derived through an extra forward-mode differentiation of the discrete adjoint code. Two numerical examples support the mathematical derivations

Decline in running performance in highest-level soccer: analysis of the UEFA champions league matches

It is widely recognized that there is a decline in match running performance (MRP) towards the end of matches. To clarify whether it is primarily a consequence of fatigue, pacing or situational influences, this study aimed to examine MRP across 15-min match periods for players on different playing positions. Players’ MRP (n = 244) were examined from the UEFA Champions League matches (n = 20) using a semiautomatic optical tracking system. Linear mixed models for repeated measures were adjusted to analyze MRP over the six 15-min match periods while controlling the influence of situational factors. No effects of match outcome, match location, team, and opponent quality on total distance (TD) and high-intensity running (HIR) for players in all playing positions were found (F = 0.03–2.75; all p > 0.05). Significant differences in TD (F = 17.57–53.01; η2 = 0.39–0.52, all large effect sizes) and HIR (F = 3.67–7.64; η2 = 0.05–0.19, small to medium effect sizes) among six 15-minute match periods were found for players in all playing positions. In addition, players in all playing positions covered less TD (d = 1.41–2.15, large to very large effect sizes) and HIR (d = 0.16–0.6, trivial to medium effect sizes) in the last compared to the first 15-min match period. No differences in TD and HIR between the last two match periods in the second half were observed. This study confirmed that soccer players reduce MRP towards the end of matches, and suggest that the decline of MPR in highest-level soccer may be a consequence of pacing strategies