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error chooses the value that minimizes the error in the PDE and boundary condition (the solution of the BVP having been replaced by a measurement).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.