The electrical engineering handbook
Miller, E.K. “Computational Electromagnetics” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 © 2000 by CRC Press LLC 45 Computational Electromagnetics 45.1 Introduction 45.2 Background Discussion Modeling as a Transfer Function • Some Issues Involved in Developing a Computer Model 45.3 Analytical Issues in Developing a Computer Model Selection of Solution Domain • Selection of Field Propagator 45.4 Numerical Issues in Developing a Computer Model Sampling Functions • The Method of Moments 45.5 Some Practical Considerations Integral Equation Modeling • Differential Equation Modeling • Discussion • Sampling Requirements 45.6 Ways of Decreasing Computer Time 45.7 Validation, Error Checking, and Error Analysis Modeling Uncertainties • Validation and Error Checking 45.8 Concluding Remarks 45.1 Introduction The continuing growth of computing resources is changing how we think about, formulate, solve, and interpret problems. In electromagnetics as elsewhere, computational techniques are complementing the more traditional approaches of measurement and analysis to vastly broaden the breadth and depth of problems that are now quantifiable. Computational electromagnetics (CEM) may be broadly defined as that branch of electromagnetics that intrinsically and routinely involves using a digital computer to obtain numerical results. With the evolu- tionary development of CEM during the past 20-plus years, the third tool of computational methods has been added to the two classical tools of experimental observation and mathematical analysis. This discussion reviews some of the basic issues involved in CEM and includes only the detail needed to illustrate the central ideas involved. The underlying principles that unify the various modeling approaches used in electromagnetics are emphasized while avoiding most of the specifics that make them different. Listed throughout are representative, but not exhaustive, numbers of references that deal with various specialty aspects of CEM. For readers interested in broader, more general expositions, the well-known book on the moment method by Harrington [1968]; the books edited by Mittra [1973, 1975], Uslenghi [1978], and Strait [1980]; the monographs by Stutzman and Thiele [1981], Popovic, et al. [1982], Moore and Pizer [1984], and Wang [1991]; and an IEEE Press reprint volume on the topic edited by Miller et al. [1991] are recommended, as is the article by Miller [1988] from which this material is excerpted. This chapter is excerpted from E. K. Miller, “A selective survey of computational electromagnetics,” IEEE Trans. Antennas Propagat. , vol. AP-36, pp. 1281–1305, ©1988 IEEE. E.K. Miller Los Alamos National Laboratory © 2000 by CRC Press LLC 45.2 Background Discussion Electromagnetics is the scientific discipline that deals with electric and magnetic sources and the fields these sources produce in specified environments. Maxwell’s equations provide the starting point for the study of electromagnetic problems, together with certain principles and theorems such as superposition, reciprocity, equivalence, induction, duality, linearity, and uniqueness, derived therefrom [Stratton, 1941; Harrington, 1961]. While a variety of specialized problems can be identified, a common ingredient of essentially all of them is that of establishing a quantitative relationship between a cause (forcing function or input) and its effect (the response or output), a relationship which we refer to as a field propagator , the computational characteristics of which are determined by the mathematical form used to describe it. Modeling as a Transfer Function The foregoing relationship may be viewed as a gener- alized transfer function (see Fig. 45.1) in which two basic problem types become apparent. For the anal- ysis or the direct problem, the input is known and the transfer function is derivable from the problem spec- ification, with the output or response to be deter- mined. For the case of the synthesis or inverse problem, two problem classes may be identified. The easier synthesis problem involves finding the input, given the output and transfer function, an example of which is that of determining the source voltages that produce an observed pattern for a known antenna array. The more difficult synthesis problem itself separates into two problems. One is that of finding the transfer function, given the input and output, an example of which is that of finding a source distribution that produces a given far field. The other and still more difficult is that of finding the object geometry that produces an observed scattered field from a known exciting field. The latter problem is the most difficult of the three synthesis problems to solve because it is intrinsically transcendental and nonlinear. Electromagnetic propagators are derived from a particular solution of Maxwell’s equations, as the cause mentioned above normally involves some specified or known excitation whose effect is to induce some to-be- determined response (e.g., a radar cross section, antenna radiation pattern). It therefore follows that the essence of electromagnetics is the study and determination of field propagators to thereby obtain an input–output transfer function for the problem of interest, and it follows that this is also the goal of CEM. Some Issues Involved in Developing a Computer Model We briefly consider here a classification of model types, the steps involved in developing a computer model , the desirable attributes of a computer model, and finally the role of approximation throughout the modeling process. Classification of Model Types It is convenient to classify solution techniques for electromagnetic modeling in terms of the field propagator that might be used, the anticipated application, and the problem type for which the model is intended to be used, as is outlined in Table 45.1. Selection of a field propagator in the form, for example, of the Maxwell curl equations, a Green’s function, modal or spectral expansions, or an optical description is a necessary first step in developing a solution to any electromagnetic problem. Development of a Computer Model Development of a computer model in electromagnetics or literally any other disciplinary activity can be decomposed into a small number of basic, generic steps. These steps might be described by different names but FIGURE 45.1The electromagnetic transfer function relates the input, output, and problem. © 2000 by CRC Press LLC would include at a minimum those outlined in Table 45.2. Note that by its nature, validation is an open-ended process that cumulatively can absorb more effort than all the other steps together. The primary focus of the following discussion is on the issue of numerical implementation. Desirable Attributes of a Computer Model A computer model must have some minimum set of basic properties to be useful. From the long list of attributes that might be desired, we consider: (1) accuracy, (2) efficiency, and (3) utility the three most important as summarized in Table 45.3. Accuracy is put foremost because results of insufficient or unknown accuracy have uncertain value and may even be harmful. On the other hand, a code that produces accurate results but at unacceptable cost will have hardly any more value. Finally, a code’s applicability in terms of the depth and breadth of the problems for which it can be used determines its utility. The Role of Approximation As approximation is an intrinsic part of each step involved in developing a computer model, we summarize some of the more commonly used approximations in Table 45.4. We note that the distinction between an approximation at the conceptualization step and during the formulation is somewhat arbitrary, but choose to use the former category for those approximations that occur before the formulation itself. TABLE 45.1 Classification of Model Types in CEM Field Propagator Description Based on Integral operator Green’s function for infinite medium or special boundaries Differential operator Maxwell curl equations or their integral counterparts Modal expansions Solutions of Maxwell’s equations in a particular coordinate system and expansion Optical description Rays and diffraction coefficients Application Requires Radiation Determining the originating sources of a field and patterns they produce Propagation Obtaining the fields distant from a known source Scattering Determining the perturbing effects of medium inhomogeneities Problem type Characterized by Solution domain Time or frequency Solution space Configuration or wave number Dimensionality 1D, 2D, 3D Electrical properties of medium and/or boundary Dielectric, lossy, perfectly conducting, anisotropic, inhomogeneous, nonlinear, bianisotropic Boundary geometry Linear, curved, segmented, compound, arbitrary TABLE 45.2 Steps in Developing a Computer Model Step Activity Conceptualization Encapsulating observation and analysis in terms of elementary physical principles and their mathematical descriptions Formulation Fleshing out of the elementary description into a more complete, formally solved, mathematical representation Numerical implementation Transforming into a computer algorithm using various numerical techniques Computation Obtaining quantitative results Validation Determining the numerical and physical credibility of the computed results © 2000 by CRC Press LLC 45.3 Analytical Issues in Developing a Computer Model Attention here is limited primarily to propagators that use either the Maxwell curl equations or source integrals which employ a Green’s function, although for completeness we briefly discuss modal and optical techniques as well. Selection of Solution Domain Either the integral equation (IE) or differential equation (DE) propagator can be formulated in the time domain, where time is treated as an independent variable, or in the frequency domain, where the harmonic TABLE 45.3 Desirable Attributes in a Computer Model Attribute Description Accuracy The quantitative degree to which the computed results conform to the mathematical and physical reality being modeled. Accuracy, preferably of known and, better yet, selectable value, is the single most important model attribute. It is determined by the physical modeling error ( e P ) and numerical modeling error ( e N ). Efficiency The relative cost of obtaining the needed results. It is determined by the human effort required to develop the computer input and interpret the output and by the associated computer cost of running the model. Utility The applicability of the computer model in terms of problem size and complexity. Utility also relates to ease of use, reliability of results obtained, etc. TABLE 45.4 Representative Approximations that Arise in Model Development Approximation Implementation/Implications Conceptualization Physical optics Surface sources given by tangential components of incident field, with fields subsequently propagated via a Green’s function. Best for backscatter and main-lobe region of reflector antennas, from resonance region ( ka > 1) and up in frequency. Physical theory of diffraction Combines aspects of physical optics and geometrical theory of diffraction, primarily via use of edge-current corrections to utilize best features of each. Geometrical theory diffraction Fields propagated via a divergence factor with amplitude obtained from diffraction coefficient. Generally applicable for ka > 2–5. Can involve complicated ray tracing. Geometrical optics Ray tracing without diffraction. Improves with increasing frequency. Compensation theorem Solution obtained in terms of perturbation from a reference, known solution. Born–Rytov Approach used for low-contrast, penetrable objects where sources are estimated from incident field. Rayleigh Fields at surface of object represented in terms of only outward propagating components in a modal expansion. Formulation Surface impedance Reduces number of field quantities by assuming an impedance relation between tangential E and H at surface of penetrable object. May be used in connection with physical optics. Thin-wire Reduces surface integral on thin, wirelike object to a line integral by ignoring circumferential current and circumferential variation of longitudinal current, which is represented as a filament. Generally limited to ka < 1 where a is the wire radius. Numerical Implementation ¶ f / ¶ x ® ( f + – f – )/( x + – x – ) ò f ( x ) dx ® å f ( x i ) D x i Differentiation and integration of continuous functions represented in terms of analytic operations on sampled approximations, for which polynomial or trigonometric functions are often used. Inherently a discretizing operation, for which typically D x < l /2 p for acceptable accuracy. Computation Deviation of numerical model from physical reality Affects solution accuracy and relatability to physical problem in ways that are difficult to predict and quantify. Nonconverged solution Discretized solutions usually converge globally in proportion to exp(– AN x ) with A determined by the problem. At least two solutions using different numbers of unknowns N x are needed to estimate A . © 2000 by CRC Press LLC time variation exp( j w t ) is assumed. Whatever propagator and domain are chosen, the analytically formal solution can be numerically quantified via the method of moments (MoM) [Harrington, 1968], leading ultimately to a linear system of equations as a result of developing a discretized and sampled approximation to the continuous (generally) physical reality being modeled. Developing the approach that may be best suited to a particular problem involves making trade-offs among a variety of choices throughout the analytical formulation and numerical implementation, some aspects of which are now considered. Selection of Field Propagator We briefly discuss and compare the characteristics of the various propagator-based models in terms of their development and applicability. Integral Equation Model The basic starting point for developing an IE model in electromagnetics is selection of a Green’s function appropriate for the problem class of interest. While there are a variety of Green’s functions from which to choose, a typical starting point for most IE MoM models is that for an infinite medium. One of the more straightforward is based on the scalar Green’s function and Green’s theorem. This leads to the Kirchhoff integrals [Stratton, 1941, p. 464 et seq .], from which the fields in a given contiguous volume of space can be written in terms of integrals over the surfaces that bound it and volume integrals over those sources located within it. Analytical manipulation of a source integral that incorporates the selected Green’s function as part of its kernel function then follows, with the specific details depending on the particular formulation being used. Perhaps the simplest is that of boundary-condition matching wherein the behavior required of the electric and/or magnetic fields at specified surfaces that define the problem geometry is explicitly imposed. Alternative formulations, for example, the Rayleigh–Ritz variational method and Rumsey’s reaction concept, might be used instead, but as pointed out by Harrington [in Miller et al., 1991], from the viewpoint of a numerical imple- mentation any of these approaches lead to formally equivalent models. This analytical formulation leads to an integral operator, whose kernel can include differential operators as well, which acts on the unknown source or field. Although it would be more accurate to refer to this as an integrodifferential equation, it is usually called simply an integral equation. Two general kinds of integral equations are obtained. In the frequency domain, representative forms for a perfect electric conductor are (45.1a) (45.1b) where E and H are the electric and magnetic fields, respectively, r , r ¢ are the spatial coordinate of the observation and source points, the superscript inc denotes incident-field quantities, and j ( r , r ¢ ) = exp[– jk Έ r – r ¢ Έ ]/ Έ r – r ¢ Έ is the free-space Green’s function. These equations are known respectively as Fredholm integral equations of the first and second kinds, differing by whether the unknown appears only under the integral or outside it as well [Poggio and Miller in Mittra, 1973]. Differential-Equation Model A DE MoM model, being based on the defining Maxwell’s equations, requires intrinsically less analytical manipulation than does derivation of an IE model. Numerical implementation of a DE model, however, can differ significantly from that used for an IE formulation in a number of ways for several reasons: nEr n nHrr,r n Er,r r,r r ´=´ ¢ ´ ¢¢ - ¢ × ¢¢ Ñ ¢¢ Î ò inc () { [ ()]( ) [ ( ) ( )} ; 1 4p wm j j j ds S S nHrnHr nnHr r,r r´=´+´ ¢ ´ ¢ ´ ¢ Ñ ¢¢ Î ò () () [ ()] ( )};2 1 2 inc ds p j S S © 2000 by CRC Press LLC 1.The differential operator is a local rather than global one in contrast to the Green’s function upon which the integral operator is based. This means that the spatial variation of the fields must be developed from sampling in as many dimensions as possessed by the problem, rather than one less as the IE model permits if an appropriate Green’s function is available. 2.The integral operator includes an explicit radiation condition, whereas the DE does not. 3.The differential operator includes a capability to treat medium inhomogeneities, non-linearities, and time variations in a more straightforward manner than does the integral operator, for which an appro- priate Green’s function may not be available. These and other differences between development of IE and DE models are summarized in Table 45.5, with their modeling applicability compared in Table 45.6. Modal-Expansion Model Modal expansions are useful for propagating electromagnetic fields because the source-field relationship can be expressed in terms of well-known analytical functions as an alternate way of writing a Green’s function for special distributions of point sources. In two dimensions, for example, the propagator can be written in terms of circular harmonics and cylindrical Hankel functions. Corresponding expressions in three dimensions might involve spherical harmonics, spherical Hankel functions, and Legendre polynomials. Expansion in terms of analytical solutions to the wave equation in other coordinate systems can also be used but requires computation TABLE 45.5 Comparison of IE- and DE-Field Propagators and Their Numerical Treatment Differential Form Integral Form Field propagator Maxwell curl equations Green’s function Boundary treatment At infinity (radiation condition) Local or global “lookback” to approximate outward propagating wave Green’s function On object Appropriate field values specified on mesh boundaries to obtain stairstep, piecewise linear, or other approximation to the boundary Appropriate field values specified on object contour which can in principle be a general, curvilinear surface, although this possibility seems to be seldom used Sampling requirements No. of space samples N x µ ( L / D L ) D N x µ ( L / D L ) D –1 No. of time steps N t µ ( L / D L ) » cT / d tN t µ ( L / D L ) » cT / d t No. of excitations N rhs µ (L/DL) N rhs µ (L/DL) (right-hand sides) Linear system L is problem size D is no. of problem dimensions (1, 2, 3) T is observation time DL is spatial resolution dt is time resolution Sparse, but larger Dense, but smaller. In this comparison, note that we assume the IE permits a sampling of order one less than the problem dimension, i.e., inhomogeneous problems are excluded. Dependence of solution time on highest-order term in (L/DL) Frequency domain T w µ N x 2(D–1)/D+1 = (L/DL) 3D–2 T w µ N x 3 = (L/DL) 3(D–1) Time domain Explicit T t µ N x N t N rhs = (L/DL) D+1+r T t µ N x 2 N t N rhs = (L/DL) 2D–1+r ; 0 £ r £ 1 Implicit T t µ N x 2(D–1)/D+1 = (L/DL) 3D–2 , D = 2, 3; T t µ N x 3 = (L/DL) 3(D–1) µ N x N t N rhs = (L/DL) 2+r , D = 1; 0 £ r £ 1 Note that D is the number of spatial dimensions in the problem and is not necessarily the sampling dimensionality d. The distinction is important because when an appropriate Green’s function is available, the source integrals are usually one dimension less than the problem dimension, i.e., d = D – 1. An exception is an inhomogeneous, penetrable body where d = D when using an IE. We also assume for simplicity that matrix solution is achieved via factorization rather than iteration but that banded matrices are exploited for the DE approach where feasible. The solution-time dependencies given can thus be regarded as upper-bound estimates. See Table 45.10 for further discussion of linear-system solutions. © 2000 by CRC Press LLC of special functions that are generally less easily evaluated, such as Mathieu functions for the two-dimensional solution in elliptical coordinates and spheroidal functions for the three-dimensional solution in oblate or prolate spheroidal coordinates. One implementation of modal propagators for numerical modeling is that due to Waterman [in Mittra, 1973], whose approach uses the extended boundary condition (EBC) whereby the required field behavior is satisfied away from the boundary surface on which the sources are located. This procedure, widely known as the T-matrix approach, has evidently been more widely used in optics and acoustics than in electromagnetics. In what amounts to a reciprocal application of EBC, the sources can be removed from the boundary surface on which the field-boundary conditions are applied. These modal techniques seem to offer some computational advantages for certain kinds of problems and might be regarded as using entire-domain basis and testing functions but nevertheless lead to linear systems of equations whose numerical solution is required. Fourier transform solution techniques might also be included in this category since they do involve modal expansions, but that is a specialized area that we do not pursue further here. Modal expansions are receiving increasing attention under the general name “fast multipole method,” which is motivated by the goal of systematically exploiting the reduced complexity of the source-field interactions as their separation increases. The objective is to reduce the significant interactions of a Green’s-function based matrix from being proportional to (N x ) 2 to of order N x log (N x ), thus offering the possibility of decreasing the operation count of iterative solutions. TABLE 45.6 Relative Applicability of IE- and DE-Based Computer Models Time Domain Frequency Domain DE IE Issue DE IE Medium ÖÖLinear ÖÖ ~ x Dispersive ÖÖ Ö x Lossy ÖÖ Ö ~ Anisotropic ÖÖ Ö x Inhomogeneous Ö x Ö x Nonlinear x x Ö x Time-varying x x Object ~ Ö Wire ~ Ö ÖÖClosed surface ÖÖ ÖÖPenetrable volume ÖÖ ~ Ö Open surface ~ Ö Boundary Conditions ÖÖInterior problem ÖÖ ~ Ö Exterior problem ~ Ö ÖÖLinear ÖÖ ÖÖNonlinear x x ÖÖTime-varying x x ~ x Halfspace ~ Ö Other Aspects ~ ~ Symmetry exploitation ÖÖ ~ Ö Far-field evaluation ~ Ö x ~ Number of unknowns ~ Ö Ö ~ Length of code ~ x Suitability for Hybridizing with Other: ~ Ö Numerical procedures ÖÖ x ~ Analytical procedures ~ Ö x ~ GTD x Ö Ö signifies highly suited or most advantageous. ~ signifies moderately suited or neutral. x signifies unsuited or least advantageous. © 2000 by CRC Press LLC Geometrical-Optics Model Geometrical optics and the geometrical theory of diffraction (GTD) are high-frequency asymptotic techniques wherein the fields are propagated using such optical concepts as shadowing, ray tubes, and refraction and diffraction. Although conceptually straightforward, optical techniques are limited analytically by the unavail- ability of diffraction coefficients for various geometries and material bodies and numerically by the need to trace rays over complex surfaces. There is a vast literature on geometrical optics and GTD, as may be ascertained by examining the yearly and cumulative indexes of such publications as the Transactions of the IEEE Antennas and Propagation Society. 45.4 Numerical Issues in Developing a Computer Model Sampling Functions At the core of numerical analysis is the idea of polynomial approximation, an observation made by Arden and Astill [1970] in facetiously using the subtitle “Numerical Analysis or 1001 Applications of Taylor’s Series.” The basic idea is to approximate quantities of interest in terms of sampling functions, often polynomials, that are then substituted for these quantities in various analytical operations. Thus, integral operators are replaced by finite sums, and differential operators are similarly replaced by generalized finite differences. For example, use of a first-order difference to approximate a derivative of the function F(x) in terms of samples F(x + ) and F(x – ) leads to (45.2a) and implies a linear variation for F(x) between x + and x – as does use of the trapezoidal rule (45.2b) to approximate the integral of F(x), where h = x + – x – . The central-difference approximation for the second derivative, (45.2c) similarly implies a quadratic variation for F(x) around x 0 = x + – h/2 = x – + h/2, as does use of Simpson’s rule (45.2d) to approximate the integral. Other kinds of polynomials and function sampling can be used, as discussed in a large volume of literature, some examples of which are Abramowitz and Stegun [1964], Acton [1970], and Press et al. [1986]. It is interesting to see that numerical differentiation and integration can be accomplished using the same set of function samples and spacings, differing only in the signs and values of some of the associated weights. Note also that the added degrees of freedom that arise when the function samples can be unevenly spaced, as in Gaussian quadrature, produce a generally more accurate result (for well-behaved functions) for dF x dx Fx Fx h xxx () ( ) ( ) » - ££ +- -+ ; Fxdx h Fx Fx x x () [( ) ( )]»+ - + ò +- 2 dFx dx Fx Fx Fx h 2 2 0 2 2() [( ) ( ) ( ) » -+ +- Fxdx h Fx Fx Fx x x () [( ) ( ) ( )]»++ - + ò +- 6 4 0 © 2000 by CRC Press LLC a given number of samples. This suggests the benefits that might be derived from using unequal sample sizes in MoM modeling should a systematic way of determining the best nonuniform sampling scheme be developed. The Method of Moments Numerical implementation of the moment method is a relatively straightforward, and an intuitively logical, extension of these basic elements of numerical analysis, as described in the book by Harrington [1968] and discussed and used extensively in CEM [see, for example, Mittra, 1973, 1975; Strait, 1980; Poggio and Miller, 1988]. Whether it is an integral equation, a differential equation, or another approach that is being used for the numerical model, three essential sampling operations are involved in reducing the analytical formulation via the moment method to a computer algorithm as outlined in Table 45.7. We note that operator sampling can ultimately determine the sampling density needed to achieve a desired accuracy in the source–field rela- tionships involving integral operators, especially at and near the “self term,” where the observation and source points become coincident or nearly so and the integral becomes nearly singular. Whatever the method used for these sampling operations, they lead to a linear system of equations or matrix approximation of the original integral or differential operators. Because the operations and choices involved in developing this matrix descrip- tion are common to all moment-method models, we shall discuss them in somewhat more detail. When using IE techniques, the coefficient matrix in the linear system of equations that results is most often referred to as an impedance matrix because in the case of the E-field form, its multiplication of the vector of unknown currents equals a vector of electric fields or voltages. The inverse matrix similarly is often called an admittance matrix because its multiplication of the electric-field or voltage vector yields the unknown-current vector. In this discussion we instead use the terms direct matrix and solution matrix because they are more generic descriptions whatever the forms of the originating integral or differential equations. As illustrated in the following, development of the direct matrix and solution matrix dominates both the computer time and storage requirements of numerical modeling. In the particular case of an IE model, the coefficients of the direct or original matrix are the mutual impedances of the multiport representation which approximates the problem being modeled, and the coeffi- cients of its solution matrix (or equivalent thereof) are the mutual admittances. Depending on whether a subdomain or entire-domain basis has been used (see Basic Function Selection), these impedances and admittances represent either spatial or modal interactions among the N ports of the numerical model. In either case, these TABLE 45.7Sampling Operations Involved in MoM Modeling DE Model IE Model Equation L(s¢)f(s¢) = g(s¢) L(s,s¢)f(s¢) = g(s) Sampling of: Unknown via basis- functions b j (s¢) using f(s¢) » åa j b j (s¢) Subdomain bases usually of low order are used. Known as FD procedure when pulse basis is used, and as FE approach when bases are linear. Can use either subdomain or entire-domain bases. Use of latter is generally confined to bodies of rotation. Former is usually of low order, with piecewise linear or sinusoidal being the maximum variation used. Equation via weight functions w i (s) <w i (s),L(s,s¢)åa j b j (s¢)> = <w i (s),g(s) > to get Z ij a j = g i Pointwise matching is commonly employed, using a delta function. Pulse and linear matching are also used. Pointwise matching is commonly employed, using a delta function. For wires, pulse, linear, and sinusoidal testing is also used. Linear and sinusoidal testing is also used for surfaces. Operator Operator sampling for DE models is entwined with sampling the unknown in terms of the difference operators used. Sampling needed depends on the nature of the integral operator L(s,s¢). An important consideration whenever the field integrals cannot be evaluated in closed form. Solution of: Z ij a j = g i for the a j Interaction matrix is sparse. Time-domain approach may be explicit or implicit. In frequency domain, banded-matrix technique usually used. Interaction matrix is full. Solution via factorization or iteration. . For the case of the synthesis or inverse problem, two problem classes may be identified. The easier synthesis problem involves finding the input, given the. cumulatively can absorb more effort than all the other steps together. The primary focus of the following discussion is on the issue of numerical implementation.