Numerical solutions of elliptic inverse problems via the equation error method

To estimate a parameter in an elliptic boundary value problem, the method of equation error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement). The estimated parameter converges to the exact value as the measured data converge to the exact value, provided Tikhonov regularization is used to control the instability inherent in the problem. The error in the estimated solution can be bounded in an appropriate quotient norm; estimates can be derived for both the underlying (infinite-dimensional) problem and a finite-element discretization that can be implemented in a practical algorithm. Numerical experiments demonstrate the efficacy and limitations of the method.

Inverse thermodilution with conventional pulmonary artery catheters for the assessment of cerebral, hepatic, renal, and femoral blood flow

Assessment of regional blood flow changes is difficult in the clinical setting. We tested whether conventional pulmonary artery catheters (PACs) can be used to measure regional venous blood flows by inverse thermodilution (ITD). Inverse thermodilution was tested in vitro and in vivo using perivascular ultrasound Doppler (USD) flow probes as a reference. In anesthetized pigs, PACs were inserted in jugular, hepatic, renal, and femoral veins, and their measurements were compared with simultaneous USD flow measurements from carotid, hepatic, renal, and femoral arteries and from portal vein. Fluid boluses were injected through the PAC's distal port, and temperature changes were recorded from the proximally located thermistor. Injectates of 2 and 5 mL at 22 degrees C and 4 degrees C were used. Flows were altered by using a roller pump (in vitro), and infusion of dobutamine and induction of cardiac tamponade, respectively. In vitro: At blood flows between 400 mL . min-1 and 700 mL . min-1 (n = 50), ITD and USD correlated well (r = 0.86, P < 0.0001), with bias and limits of agreement of 3 +/- 101 mL . min-1. In vivo: 514 pairs of measurements had to be excluded from analysis for technical reasons, and 976 were analyzed. Best correlations were r = 0.87 (P < 0.0001) for renal flow and r = 0.46 (P < 0.0001) for hepatic flow. No significant correlation was found for cerebral and femoral flows. Inverse thermodilution using conventional PAC compared moderately well with USD for renal but not for other flows despite good in vitro correlation in various conditions. In addition, this method has significant technical limitations.

SPATIAL UNDERSTANDING OF MATRIX INVERSION FOR INVERSE FORCE ESTIMATION

A basic approach to study a NVH problem is to break down the system in three basic elements – source, path and receiver. While the receiver (response) and the transfer path can be measured, it is difficult to measure the source (forces) acting on the system. It becomes necessary to predict these forces to know how they influence the responses. This requires inverting the transfer path. Singular Value Decomposition (SVD) method is used to decompose the transfer path matrix into its principle components which is required for the inversion. The usual approach to force prediction requires rejecting the small singular values obtained during SVD by setting a threshold, as these small values dominate the inverse matrix. This assumption of the threshold may be subjected to rejecting important singular values severely affecting force prediction. The new approach discussed in this report looks at the column space of the transfer path matrix which is the basis for the predicted response. The response participation is an indication of how the small singular values influence the force participation. The ability to accurately reconstruct the response vector is important to establish a confidence in force vector prediction. The goal of this report is to suggest a solution that is mathematically feasible, physically meaningful, and numerically more efficient through examples. This understanding adds new insight to the effects of current code and how to apply algorithms and understanding to new codes.

Real-Time Joint Coupling of the Spine for Inverse Kinematics

In this paper we propose a simple model for the coupling behavior of the human spine for an inverse kinematics framework. Our spine model exhibits anatomically correct motions of the vertebrae of virtual mannequins by coupling standard swing and revolute joint models. The adjustement of the joints is made with several simple (in)equality constraints, resulting in a reduction of the solution space dimensionality for the inverse kinematics solver. By reducing the solution space dimensionality to feasible spine shapes, we prevent the inverse kinematics algorithm from providing infeasible postures for the spine.In this paper, we exploit how to apply these simple constraints to the human spine by a strict decoupling of the swing and torsion motion of the vertebrae. We demonstrate the validity of our approach on various experiments.

Distance-based kriging relying on proxy simulations for inverse conditioning

Let us consider a large set of candidate parameter fields, such as hydraulic conductivity maps, on which we can run an accurate forward flow and transport simulation. We address the issue of rapidly identifying a subset of candidates whose response best match a reference response curve. In order to keep the number of calls to the accurate flow simulator computationally tractable, a recent distance-based approach relying on fast proxy simulations is revisited, and turned into a non-stationary kriging method where the covariance kernel is obtained by combining a classical kernel with the proxy. Once the accurate simulator has been run for an initial subset of parameter fields and a kriging metamodel has been inferred, the predictive distributions of misfits for the remaining parameter fields can be used as a guide to select candidate parameter fields in a sequential way. The proposed algorithm, Proxy-based Kriging for Sequential Inversion (ProKSI), relies on a variant of the Expected Improvement, a popular criterion for kriging-based global optimization. A statistical benchmark of ProKSI’s performances illustrates the efficiency and the robustness of the approach when using different kinds of proxies.

Inverse relationship between fractionated electrograms and atrial fibrosis in persistent atrial fibrillation: combined magnetic resonance imaging and high-density mapping

OBJECTIVES This study sought to evaluate the relationship between fibrosis imaged by delayed-enhancement (DE) magnetic resonance imaging (MRI) and atrial electrograms (Egms) in persistent atrial fibrillation (AF). BACKGROUND Atrial fractionated Egms are strongly related to slow anisotropic conduction. Their relationship to atrial fibrosis has not yet been investigated. METHODS Atrial high-resolution MRI of 18 patients with persistent AF (11 long-lasting persistent AF) was registered with mapping geometry (NavX electro-anatomical system (version 8.0, St. Jude Medical, St. Paul, Minnesota)). DE areas were categorized as dense or patchy, depending on their DE content. Left atrial Egms during AF were acquired using a high-density, 20-pole catheter (514 ± 77 sites/map). Fractionation, organization/regularity, local mean cycle length (CL), and voltage were analyzed with regard to DE. RESULTS Patients with long-lasting persistent versus persistent AF had larger left atrial (LA) surface area (134 ± 38 cm(2) vs. 98 ± 9 cm(2), p = 0.02), a higher amount of atrial DE (70 ± 16 cm(2) vs. 49 ± 10 cm(2), p = 0.01), more complex fractionated atrial Egm (CFAE) extent (54 ± 16 cm(2) vs. 28 ± 15 cm(2), p = 0.02), and a shorter baseline AF CL (147 ± 10 ms vs. 182 ± 14 ms, p = 0.01). Continuous CFAE (CFEmean [NavX algorithm that quantifies Egm fractionation] <80 ms) occupied 38 ± 19% of total LA surface area. Dense DE was detected at the left posterior left atrium. In contrast, the right posterior left atrium contained predominantly patchy DE. Most CFAE (48 ± 14%) occurred at non-DE LA sites, followed by 41 ± 12% CFAE at patchy DE and 11 ± 6% at dense DE regions (p = 0.005 and p = 0.008, respectively); 19 ± 6% CFAE sites occurred at border zones of dense DE. Egms were less fractionated, with longer CL and lower voltage at dense DE versus non-DE regions: CFEmean: 97 ms versus 76 ms, p < 0.0001; local CL: 153 ms versus 143 ms, p < 0.0001; mean voltage: 0.63 mV versus 0.86 mV, p < 0.0001. CONCLUSIONS Atrial fibrosis as defined by DE MRI is associated with slower and more organized electrical activity but with lower voltage than healthy atrial areas. Ninety percent of continuous CFAE sites occur at non-DE and patchy DE LA sites. These findings are important when choosing the ablation strategy in persistent AF.

Prediction of Cranio-Maxilofacial Surgical Planning using an Inverse Soft Tissue Modelling Approach

In cranio-maxillofacial surgery, the determination of a proper surgical plan is an important step to attain a desired aesthetic facial profile and a complete denture closure. In the present paper, we propose an efficient modeling approach to predict the surgical planning on the basis of the desired facial appearance and optimal occlusion. To evaluate the proposed planning approach, the predicted osteotomy plan of six clinical cases that underwent CMF surgery were compared to the real clinical plan. Thereafter, simulated soft-tissue outcomes were compared using the predicted and real clinical plan. This preliminary retrospective comparison of both osteotomy planning and facial outlook shows a good agreement and thereby demonstrates the potential application of the proposed approach in cranio-maxillofacial surgical planning prediction.

Solute transport in crystalline rocks at Äspö — II: Blind predictions, inverse modelling and lessons learnt from test STT1

Based on the results from detailed structural and petrological characterisation and on up-scaled laboratory values for sorption and diffusion, blind predictions were made for the STT1 dipole tracer test performed in the Swedish A¨ spo¨ Hard Rock Laboratory. The tracers used were nonsorbing, such as uranine and tritiated water, weakly sorbing 22Na+, 85Sr2 +, 47Ca2 +and more strongly sorbing 86Rb+, 133Ba2 +, 137Cs+. Our model consists of two parts: (1) a flow part based on a 2D-streamtube formalism accounting for the natural background flow field and with an underlying homogeneous and isotropic transmissivity field and (2) a transport part in terms of the dual porosity medium approach which is linked to the flow part by the flow porosity. The calibration of the model was done using the data from one single uranine breakthrough (PDT3). The study clearly showed that matrix diffusion into a highly porous material, fault gouge, had to be included in our model evidenced by the characteristic shape of the breakthrough curve and in line with geological observations. After the disclosure of the measurements, it turned out that, in spite of the simplicity of our model, the prediction for the nonsorbing and weakly sorbing tracers was fairly good. The blind prediction for the more strongly sorbing tracers was in general less accurate. The reason for the good predictions is deemed to be the result of the choice of a model structure strongly based on geological observation. The breakthrough curves were inversely modelled to determine in situ values for the transport parameters and to draw consequences on the model structure applied. For good fits, only one additional fracture family in contact with cataclasite had to be taken into account, but no new transport mechanisms had to be invoked. The in situ values for the effective diffusion coefficient for fault gouge are a factor of 2–15 larger than the laboratory data. For cataclasite, both data sets have values comparable to laboratory data. The extracted Kd values for the weakly sorbing tracers are larger than Swedish laboratory data by a factor of 25–60, but agree within a factor of 3–5 for the more strongly sorbing nuclides. The reason for the inconsistency concerning Kds is the use of fresh granite in the laboratory studies, whereas tracers in the field experiments interact only with fracture fault gouge and to a lesser extent with cataclasite both being mineralogically very different (e.g. clay-bearing) from the intact wall rock.

Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.

Symmetric inverse consistent nonlinear registration driven by mutual information.

A nonlinear viscoelastic image registration algorithm based on the demons paradigm and incorporating inverse consistent constraint (ICC) is implemented. An inverse consistent and symmetric cost function using mutual information (MI) as a similarity measure is employed. The cost function also includes regularization of transformation and inverse consistent error (ICE). The uncertainties in balancing various terms in the cost function are avoided by alternatively minimizing the similarity measure, the regularization of the transformation, and the ICE terms. The diffeomorphism of registration for preventing folding and/or tearing in the deformation is achieved by the composition scheme. The quality of image registration is first demonstrated by constructing brain atlas from 20 adult brains (age range 30-60). It is shown that with this registration technique: (1) the Jacobian determinant is positive for all voxels and (2) the average ICE is around 0.004 voxels with a maximum value below 0.1 voxels. Further, the deformation-based segmentation on Internet Brain Segmentation Repository, a publicly available dataset, has yielded high Dice similarity index (DSI) of 94.7% for the cerebellum and 74.7% for the hippocampus, attesting to the quality of our registration method.

Inverse modelling of the 14C bomb pulse in stalagmites to constrain the dynamics of soil carbon cycling at selected European cave sites

The decomposition of soil organic matter (SOM) is temperature dependent, but its response to a future warmer climate remains equivocal. Enhanced rates of decomposition of SOM under increased global temperatures might cause higher CO2 emissions to the atmosphere, and could therefore constitute a strong positive feedback. The magnitude of this feedback however remains poorly understood, primarily because of the difficulty in quantifying the temperature sensitivity of stored, recalcitrant carbon that comprises the bulk (>90%) of SOM in most soils. In this study we investigated the effects of climatic conditions on soil carbon dynamics using the attenuation of the 14C ‘bomb’ pulse as recorded in selected modern European speleothems. These new data were combined with published results to further examine soil carbon dynamics, and to explore the sensitivity of labile and recalcitrant organic matter decomposition to different climatic conditions. Temporal changes in 14C activity inferred from each speleothem was modelled using a three pool soil carbon inverse model (applying a Monte Carlo method) to constrain soil carbon turnover rates at each site. Speleothems from sites that are characterised by semi-arid conditions, sparse vegetation, thin soil cover and high mean annual air temperatures (MAATs), exhibit weak attenuation of atmospheric 14C ‘bomb’ peak (a low damping effect, D in the range: 55–77%) and low modelled mean respired carbon ages (MRCA), indicating that decomposition is dominated by young, recently fixed soil carbon. By contrast, humid and high MAAT sites that are characterised by a thick soil cover and dense, well developed vegetation, display the highest damping effect (D = c. 90%), and the highest MRCA values (in the range from 350 ± 126 years to 571 ± 128 years). This suggests that carbon incorporated into these stalagmites originates predominantly from decomposition of old, recalcitrant organic matter. SOM turnover rates cannot be ascribed to a single climate variable, e.g. (MAAT) but instead reflect a complex interplay of climate (e.g. MAAT and moisture budget) and vegetation development.

Modulation of orientation-selective neurons by motion: when additive, when multiplicative?

The recurrent interaction among orientation-selective neurons in the primary visual cortex (V1) is suited to enhance contours in a noisy visual scene. Motion is known to have a strong pop-up effect in perceiving contours, but how motion-sensitive neurons in V1 support contour detection remains vastly elusive. Here we suggest how the various types of motion-sensitive neurons observed in V1 should be wired together in a micro-circuitry to optimally extract contours in the visual scene. Motion-sensitive neurons can be selective about the direction of motion occurring at some spot or respond equally to all directions (pandirectional). We show that, in the light of figure-ground segregation, direction-selective motion neurons should additively modulate the corresponding orientation-selective neurons with preferred orientation orthogonal to the motion direction. In turn, to maximally enhance contours, pandirectional motion neurons should multiplicatively modulate all orientation-selective neurons with co-localized receptive fields. This multiplicative modulation amplifies the local V1-circuitry among co-aligned orientation-selective neurons for detecting elongated contours. We suggest that the additive modulation by direction-specific motion neurons is achieved through synaptic projections to the somatic region, and the multiplicative modulation by pandirectional motion neurons through projections to the apical region of orientation-specific pyramidal neurons. For the purpose of contour detection, the V1-intrinsic integration of motion information is advantageous over a downstream integration as it exploits the recurrent V1-circuitry designed for that task.