Mathematical Constraints Research Articles

SOME CONSIDERATIONS ON PHYSICAL ANALYSIS OF DATA

In this paper, we present some general considerations about data analysis from the perspective of a physical scientist and advocate the physical, instead of mathematical, analysis of data. These considerations have been accompanying our development of novel adaptive, local analysis methods, especially the empirical mode decomposition and its major variation, the ensemble empirical mode decomposition, and its preliminary mathematical explanations. A particular emphasis will be on the advantages and disadvantages of mathematical and physical constraints associated with various analysis methods. We argue that, using data analysis in a given temporal domain of observation as an example, the mathematical constraints imposed on data may lead to difficulties in understanding the physics behind the data. With such difficulties in mind, we promote adaptive, local analysis method, which satisfies fundamental physical principle of consequent evolution of a system being not able to change the past evolution of the system. We also argue, using the ensemble empirical mode decomposition as an example, that noise can be helpful to extract physically meaningful signals hidden in noisy data.

Research and Design of Mechanical System Virtual Prototyping

The effectiveness and actualities of mechanical system virtual prototyping software were summarized. The process from prototype to utility, commercialization and industrialization were analyzed in software engineering. The basic concept of practical improvement for a mechanical system virtual prototyping software was presented. The key points of the mathematical model, topology structure, constraint type, kinematic and dynamic simulation and data exchange were discussed by the theory of multibody system dynamics and the technique of object-oriented.

A new look at the statistical assessment of approximate and rigorous methods for the estimation of stabilized formation temperatures in geothermal and petroleum wells

Approximate and rigorous solutions of seven heat transfer models were statistically examined, for the first time, to estimate stabilized formation temperatures (SFT) of geothermal and petroleum boreholes. Constant linear and cylindrical heat source models were used to describe the heat flow (either conductive or conductive/convective) involved during a borehole drilling. A comprehensive statistical assessment of the major error sources associated with the use of these models was carried out. The mathematical methods (based on approximate and rigorous solutions of heat transfer models) were thoroughly examined by using four statistical analyses: (i) the use of linear and quadratic regression models to infer the SFT; (ii) the application of statistical tests of linearity to evaluate the actual relationship between bottom-hole temperatures and time function data for each selected method; (iii) the comparative analysis of SFT estimates between the approximate and rigorous predictions of each analytical method using a β ratio parameter to evaluate the similarity of both solutions, and (iv) the evaluation of accuracy in each method using statistical tests of significance, and deviation percentages between ‘true’ formation temperatures and SFT estimates (predicted from approximate and rigorous solutions). The present study also enabled us to determine the sensitivity parameters that should be considered for a reliable calculation of SFT, as well as to define the main physical and mathematical constraints where the approximate and rigorous methods could provide consistent SFT estimates.

Life around the scallop theorem

Locomotion on small scales is dominated by the effects of viscous forces and, as a result, is subject to strong physical and mathematical constraints. Following Purcell's statement of the scallop theorem which delimitates the types of swimmer designs which are not effective on small scales, we review the different ways the constraints of the theorem can be escaped for locomotion purposes.

A trust-region filter-SQP method for mathematical programs with linear complementarity constraints

A trust-region filter-SQP method for mathematical programs withlinear complementarity constraints (MPLCCs) is presented. The methodis similar to that proposed by Liu, Perakis and Sun[Computational Optimization and Applications, 34, 5-33, 2006]but it solves the trust-region quadratic programming subproblems at eachiteration and uses the filter technique to promote the global convergence. As a result,the method here is more robust since it admits the use of LagrangianHessian information and its performance is not affected by any penalty parameter.The preliminary numerical results on test problems generated by the QPECgen generatorshow that the presented method is effective.

An arcless controlled switch

A new current controlled switch is introduced in this paper. The switch has inherent control on the current and operates based on the motion of a conducting medium through a set of transition contacts. Basic theory concerning controlled motion of the conducting medium through a liquid metal (LM) interface is exposed followed by experimental study. The structure of the device and the design basis are described and the Lorentz force on the moving conducting medium is calculated using FEM software. Acting major role in the device operation, the forces on the conducting medium have been characterized and the motion equations of this conducting medium are solved using a numerical method. Accordingly, the design criteria, which are converted to a set of mathematical and physical constraints, are checked and the structure or quantities are modified where needed. Finally, a specimen is made and the switch operation is studied.

Compact inter-domain routing under real-world constraints

Due to the “natural” growth of the Internet, the scaling properties of today’s inter-domain routing system worsen at a steep rate. Certain operational practices and a number of limitations of the routing protocol itself further exacerbate the scalability problem. In order to address this threat, this paper introduces 2SIDR, a two-step inter-domain routing approach. 2SIDR aims at significantly reducing the state requirements of routers while minimizing the incurred path inflation due to the lack of full routing state. 2SIDR leverages insights from theoretical approaches termed compact routing. But instead of adhering to mathematical constraints 2SIDR takes practical constraints from Internet operations into consideration, i.e., data that is available in practice and business relationships. We call this practical compactness as it deliberately gives up mathematical bounds in order to satisfy real-world requirements. Two variants of 2SIDR were analyzed extensively based on multiple sources of data gathered from the Internet to construct an Internet-scale AS-level topology. Various aspects were analyzed such as the state/stretch trade-off and the effect of observed routing policies.

Low-dose x-ray phase-contrast and absorption CT using equally sloped tomography

Tomographic reconstruction from undersampled and noisy projections is often desirable in transmission CT modalities for purposes of low-dose tomography and fast acquisition imaging. However under such conditions, due to the violation of the Nyquist sampling criteria and the presence of noise, reconstructions with acceptable accuracy may not be possible. Recent experiments in transmission electron tomography and coherent diffraction microscopy have shown that the technique of equally sloped tomography (EST), an exact tomographic method utilizing an oversampling iterative Fourier-based reconstruction, provides more accurate image reconstructions when the number of projections is significantly undersampled relative to filtered back projection and algebraic iterative methods. Here we extend this technique by developing new reconstruction algorithms which allow for the incorporation of advanced mathematical regularization constraints, such as the nonlocal means total variational model, in a manner that is consistent with experimental projections. We then evaluate the resulting image quality of the developed algorithm through simulations and experiments at the European Synchrotron Radiation Facility on image quality phantoms using the x-ray absorption and phase contrast CT modalities. Both our simulation and experimental results have indicated that the method can reduce the number of projections by 60–75% in parallel beam modalities, while achieving comparable or better image quality than the conventional reconstructions. As large-scale and compact synchrotron radiation facilities are currently under rapid development worldwide, the implementation of low-dose x-ray absorption and phase-contrast CT can find broad applications in biology and medicine using these advanced x-ray sources.

Pressure–Strain Correlation Modeling: Towards Achieving Consistency with Rapid Distortion Theory

In turbulence closure modeling, it is widely accepted that the rapid pressure-strain correlation (RPSC) model be consistent with the rapid distortion theory (RDT). It is desirable to achieve this consistency with a closure model that is computationally tractable and satisfies the requisite mathematical constraints of realizability and linearity in the appropriate variables. In this investigation, starting from a detailed modal analysis of two-dimensional mean flows, we identify important flow features to be incorporated into the model. However, the dynamical system analysis shows that the suggested physics cannot be embodied in a model with all desired computational and mathematical attributes. To resolve this conflict, we propose a slight compromise in the physical requirement and ease one of the linearity constraints leading to a "best possible" tractable model. Overall, the present work provides important insight into RPSC closure modeling challenges—arising from the interplay among physical fidelity, computational viability and mathematical constraints—and proposes avenues for future improvement.

New method to measure and improve consistency of baroreflex sensitivity values

Baroreflex sensitivity (BRS) is an important prognostic index in cardiovascular diseases, however, its use is complicated by different methods difficult to compare and standardize, often providing conflicting results. We tested whether the simple ratio of RR interval to systolic blood pressure global variabilities (assessed by standard deviations) is a reliable measure of BRS, by measuring the agreement with six established methods. In addition, we tested whether high-pass filtering of data, by removing slow non-baroreflex-mediated fluctuations, could improve the agreement between different BRS methods. In 1,409 subjects, we compared 6 established methods (derived by cross-spectral and sequence analysis) and the new method, supine and in response to tilting (1,175 subjects). Data were analyzed after linear detrending, high-pass filtering at 0.025 and 0.05Hz. Although all seven methods showed a general agreement, the new method consistently showed the lowest distance from the median of the remaining methods (0.04±0.06ms/mmHg over 2,584 files, p<0.05 with respect to the second best method). High-pass filtering improved (p<0.001) the agreement between methods without reducing the sensitivity to changes induced by tilting. Only the new method could provide estimates in all 2,584 files tested. The new method intercepts the mean information of all other methods better than any other method, hence providing a simple, easy to standardize (no mathematical constraints) and yet robust and reliable BRS estimate. High-pass filtering markedly improves the agreement of all methods, without loss of sensitivity, and could be routinely used in clinical trials, to provide comparable BRS estimates.

SETI and SEH (Statistical Equation for Habitables)

The statistics of habitable planets may be based on a set of ten (and possibly more) astrobiological requirements first pointed out by Stephen H. Dole in his book “Habitable planets for man” (1964). In this paper, we first provide the statistical generalization of the original and by now too simplistic Dole equation. In other words, a product of ten positive numbers is now turned into the product of ten positive random variables. This we call the SEH, an acronym standing for “Statistical Equation for Habitables”. The mathematical structure of the SEH is then derived. The proof is based on the central limit theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be arbitrarily distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov form of the CLT, or the Lindeberg form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that (1) The new random variable N Hab , yielding the number of habitables (i.e. habitable planets) in the Galaxy, follows the lognormal distribution. By construction, the mean value of this lognormal distribution is the total number of habitable planets as given by the statistical Dole equation. But now we also derive the standard deviation, the mode, the median and all the moments of this new lognormal N Hab random variable. (2) The ten (or more) astrobiological factors are now positive random variables. The probability distribution of each random variable may be arbitrary . The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT “translates” into our SEH by allowing an arbitrary probability distribution for each factor. This is both astrobiologically realistic and useful for any further investigations. (3) An application of our SEH then follows. The (average) distance between any two nearby habitable planets in the Galaxy may be shown to be inversely proportional to the cubic root of N Hab . Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function, apparently previously unknown and dubbed “Maccone distribution” by Paul Davies in 2008. (4) Data Enrichment Principle. It should be noticed that ANY positive number of random variables in the SEH is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the SEH we call the “Data Enrichment Principle”, and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. (5) A practical example is then given of how our SEH works numerically. We work out in detail the case where each of the ten random variables is uniformly distributed around its own mean value as given by Dole back in 1964 and has an assumed standard deviation of 10%. The conclusion is that the average number of habitable planets in the Galaxy should be around 100 million±200 million, and the average distance in between any couple of nearby habitable planets should be about 88 light years±40 light years. (6) Finally, we match our SEH results against the results of the Statistical Drake Equation that we introduced in our 2008 IAC presentation. As expected, the number of currently communicating ET civilizations in the Galaxy turns out to be much smaller than the number of habitable planets (about 10,000 against 100 million, i.e. one ET civilization out of 10,000 habitable planets). And the average distance between any two nearby habitable planets turns out to be much smaller than the average distance between any two neighboring ET civilizations: 88 light years vs. 2000 light years, respectively. This means an ET average distance about 20 times higher than the average distance between any couple of adjacent habitable planets.

Probabilistic models for stochastic elliptic partial differential equations

Mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively. Yet, additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities, referred to as physical requirements, have not been examined systematically. It is shown that current models for random coefficients constructed solely by mathematical considerations can violate physical constraints and, consequently, be of limited practical use. We develop alternative models for the random coefficients of stochastic differential equations that satisfy both mathematical and physical constraints. Theoretical arguments are presented to show potential limitations of current models and establish properties of the models developed in this study. Numerical examples are used to illustrate the construction of the proposed models, assess the performance of these models, and demonstrate the sensitivity of the solutions of stochastic differential equations to probabilistic characteristics of their random coefficients.

Constraints on the turnout gap between high and low knowledge (or income) voters: Combining the Duncan-Davis method of bounds with the Taagepera method of bounds

Countries differ quite substantially in mean turnout levels, and it is equally well known that there may be substantial within-country variation as well, for example, between high income and low-income groupings or between high political knowledge and low political knowledge groupings. It has been hypothesized that the size of such between-group gaps will fall as turnout rises, and conversely ( Franklin, 2004. Blais, 2000). However, as Franklin (2004) also noted, there are mathematical constraints on the size of the turnout gap that are related to the level of turnout. For example, in the limit, if turnout is 100%, then all groups must have identical turnout. Here we build on this insight by adapting the classic work on boundary conditions done by two sociologists ( Duncan and Davis, 1953) to show precisely what the boundary constraints look like over the entire range of turnout values. Then we show how these constraints can help make sense of the strong relationship found between overall turnout and the size of the gap between voters above and below the median in political knowledge in the Fisher et al. (2008) cross-national study. To do so we draw on ideas in Rein Taagepera (2007, 2008) about how to use boundary condition information to develop better theoretical models.

Hybrid optimization with improved tabu search

Finding a global optimum of an unknown system has attracted a great deal of interest in many engineering problems. In this settings, meta-heuristics are very common as efficient approaches for solving complex real-world problems in global continuous optimization problems (GCOPs) as they can approximate solutions without considering mathematical constraints such as differentiability. In this study, we propose a method based on tabu search (TS) and Nelder–Mead (NM) search strategy in application to GCOPs. To increase the robustness of the proposed method, we add a new phase, referred to as partitioning phase, before diversification which is realized by the TS. The TS is improved and then followed by the NM search strategy. The partitioning phase aims at distributing initial random solutions in the search space. By doing this, we increase the robustness of the method. The TS has an interesting ability of covering a wide solution space by promoting the search far away from the current solution and consecutively decreasing the possibility of trapping in local minima. The neighbour search-strategy of the TS is improved to accelerate the speed of finding the near optimum solution. Instead of just generating random neighbours around the current solution, we generate some neighbours in the direction of the previous move as well as some neighbours in the previous best crown. When certain criteria are reached for the diversification of the search space, the NM search strategy is carried out with the focus on the intensification of the solution found in the diversification phase. We assess the performance of the algorithm for a range of standard test functions available in the literature and compare the obtained results with those available in the literature. There are two main advantages of the proposed method; first, it can apply to any GCOP without considering any constraints and secondly, it shows better performance (in terms of function evaluation, success rate, and average error) for the functions with less than four input variables and relatively small or medium input domains. In other cases, the method still has acceptable perfomance and produces the results that are comparable with the results produced by other methods.

The Statistical Drake Equation

We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this “the Statistical Drake Equation”. The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: (1) The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. (2) The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT “translates” into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. (3) An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density function, apparently previously unknown and dubbed “Maccone distribution” by Paul Davies. (4) DATA ENRICHMENT PRINCIPLE. It should be noticed that ANY positive number of random variables in the Statistical Drake Equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the statistical Drake equation, we call the “Data Enrichment Principle,” and we regard it as the key to more profound future results in the fields of Astrobiology and SETI. Finally, a practical example is given of how our statistical Drake equation works numerically. We work out in detail the case, where each of the seven random variables is uniformly distributed around its own mean value and has a given standard deviation. For instance, the number of stars in the Galaxy is assumed to be uniformly distributed around (say) 350 billions with a standard deviation of (say) 1 billion. Then, the resulting lognormal distribution of N is computed numerically by virtue of a MathCad file that the author has written. This shows that the mean value of the lognormal random variable N is actually of the same order as the classical N given by the ordinary Drake equation, as one might expect from a good statistical generalization.

Physical Phenomena and Length Scales Governing the Behaviour of Wildfires: A Case for Physical Modelling

This paper is an overview of the physical mechanisms and length scales governing the propagation of wildfires. One of the objectives is to identify the physical and mathematical constraints in the modelling of wildfires when using a “fully” physical approach. The literature highlights two regimes in the propagation of surface fires, i.e. wind-driven fires and plume-dominated fires, which are governed by radiation and convective heat transfer, respectively. This division leads to the identification of two governing length scales: the extinction length characterising the absorption of radiation by vegetation, and the integral turbulent length scale characterising the interaction between wind and canopy. Some numerical results published during the last decade using a fully physical approach are presented and discussed with a focus on the models FIRESTAR, FIRELES, FIRETEC and WFDS. Numerical simulations were compared with experimental data obtained at various scales, from laboratory to field fires in grassland and in Mediterranean shrubland. Some perspectives are presented concerning the potential coupling between physical fire models with mesoscale atmospheric models to study the impacts of wildfires at larger scale. Some of the topics on wildfire physical modelling that need further research are identified in the conclusions.

TU-B-201B-01: Radiation Dose Reduction in Medical X-Ray CT Using Equally-Sloped Tomography

Purpose: An exact Fourier‐based iterative reconstruction technique termed Equally‐Sloped Tomography (EST) has recently been developed to allow a reduction of radiation dose while maintaining image quality. Here we optimize this technique by incorporating a Gradient Descent algorithm with mathematical regularization constraints, including the Non‐Local Means Total Variation model. Using this technique, we conducted a series of comparative experiments to quantify the image quality, and explore radiation dose reduction in CT through reduction of X‐ray flux. Method and Materials: A standard quality control phantom containing multiple contrastvisibility and resolution inserts was imaged using a Siemens Somatom Sensation 64 scanner. The X‐ray flux was systematically lowered from 583mAs to 39mAs. All scans were performed under axial mode with constant tube current. Comparison reconstructions were produced by Filtered‐Back Projection (FBP) with a standard uncropped ramp filter. The resolution, signal‐to‐noise ratio (SNR) and contrast‐to‐noise ratio (CNR) were measured and compared as a function of flux. Results: Compared to the FBP reconstruction of 500mAs, the optimized EST reconstruction of 50mAs has comparable SNRs and CNRs. The low‐contrast regions of the phantom exhibit similar visibility. In addition, no resolution degradation is observed in the low‐dose EST reconstruction.Conclusion: As radiation dose is linearly proportional to the X‐ray flux, our results indicate that the optimized EST technique may potentially reduce the CT dose up to 90%, while producing comparable image resolution, quality and contrast as the full‐dose FBP reconstruction. The reasons that EST allows for significant reduction of the CT dose are because (i) an exact Fourier‐based iterative algorithm is used in the reconstruction where both physical and mathematical constraints can be naturally incorporated, (ii) the algorithm is guided towards a lowest‐possible noisy state, which is also strictly consistent with the measured data. Conflict of Interest: Research sponsored by UC Discovery / TomoSoft Technologies Grant #ITL07‐10166.

The living Drake equation of the Tau Zero Foundation

The living Drake equation is our statistical generalization of the Drake equation such that it can take into account any number of factors. This new result opens up the possibility to enrich the equation by inserting more new factors as long as the scientific learning increases. The adjective “Living” refers just to this continuous enrichment of the Drake equation and is the goal of a new research project that the Tau Zero Foundation has entrusted to this author as the discoverer of the statistical Drake equation described hereafter. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this “the Statistical Drake Equation”. The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be arbitrarily distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov form of the CLT, or the Lindeberg form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: (1) The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the lognormal distribution. Then, the mean value, standard deviation, mode, median and all the moments of this lognormal N can be derived from the means and standard deviations of the seven input random variables. (2) In fact, the seven factors in the ordinary Drake equation now become seven independent positive random variables. The probability distribution of each random variable may be arbitrary. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT “translates” into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. (3) An application of our statistical Drake equation then follows. The (average) distance between any two neighbouring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, this distance now becomes a new random variable. We derive the relevant probability density function, apparently previously unknown (dubbed “Maccone distribution” by Paul Davies). (4) Data Enrichment Principle. It should be noticed that any positive number of random variables in the statistical Drake equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the statistical Drake equation we call the “Data Enrichment Principle”, and regard as the key to more profound, future results in Astrobiology and SETI.

MRI Coil Design Using Boundary-Element Method With Regularization Technique: A Numerical Calculation Study

A new boundary element method (BEM) based method is described for the design of coils for magnetic resonance imaging (MRI) systems. BEM is an effective approach for solving the electromagnetic forward problem and has been used in the design of MRI gradient coils. However, BEM-based gradient coil design faces an ill-posed mathematical problem, which is conventionally handled by means of a Lagrange multiplication method. This work attempts to improve the BEM method for MRI coil designs by applying the Tikhonov regularization scheme to solve the ill-posed matrix system formulated by the BEM forward model. The objective of the study is not to design some specific gradient or radio-frequency (RF) coils for MRI system, but to discuss the design scheme with practical regularization and constraints. The proposed approach was explained in the design of MRI coils including biplanar transverse gradient coils and RF phased array coils. With the consideration of the practical engineering requirements, physical constraints such as wire intervals are transformed into mathematical constraints and formulated into BEM equations. The examples demonstrate that the proposed method is efficient and flexible for the design of MRI coils with arbitrary geometries and engineering constraints.

Mathematical formulation and analysis of the nonlinear system reconstruction of the online image-guided adaptive control of hyperthermia.

A nonlinear system reconstruction can theoretically provide timely system reconstruction when designing a real-time image-guided adaptive control for multisource heating for hyperthermia. This clinical need motivates an analysis of the essential mathematical characteristics and constraints of such an approach. The implicit function theorem (IFT), the Karush-Kuhn-Tucker (KKT) necessary condition of optimality, and the Tikhonov-Phillips regularization (TPR) were used to analyze and determine the requirements of the optimal system reconstruction. Two mutually exclusive generic approaches were analyzed to reconstruct the physical system: The traditional full reconstruction and the recently suggested partial reconstruction. Rigorous mathematical analysis based on IFT, KKT, and TPR was provided for all four possible nonlinear reconstructions: (1) Nonlinear noiseless full reconstruction, (2) nonlinear noisy full reconstruction, (3) nonlinear noiseless partial reconstruction, and (4) nonlinear noisy partial reconstruction, when a class of nonlinear formulations of system reconstruction is employed. Effective numerical algorithms for solving each of the aforementioned four nonlinear reconstructions were introduced and formal derivations and analyses were provided. The analyses revealed the necessity of adding regularization when partial reconstruction is used. Regularization provides the theoretical support for one to uniquely reconstruct the optimal system. It also helps alleviate the negative influences of unavoidable measurement noise. Both theoretical analysis and numerical examples showed the importance of having a good initial guess for accomplishing nonlinear system reconstruction. Regularization is mandatory for partial reconstruction to make it well posed. The Tikhonov-Phillips regularized Gauss-Newton algorithm has nice theoretical performance for partial reconstruction of systems with and without noise. The Levenberg-Marquardt algorithm is a more robust algorithmic option compared to the Gauss-Newton algorithm for nonlinear full reconstruction. A severe limitation of nonlinear reconstruction is the time consuming calculations required for the derivatives of temperatures to unknowns. Developing a method of model reduction or implementing a parallel algorithm can resolve this. The results provided herein are applicable to hyperthermia with blood perfusion nonlinearly depending on temperature and in the presence of thermally significant blood vessels.