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The Renormalization Group and Singular Perturbations: Multiple-Scales, Boundary Layers and Reductive Perturbation Theory

Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither {\it ad hoc\/} assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially-extended systems near bifurcation points, deriving both amplitude equations and the center manifold.


Introduction
Asymptotic and perturbative analysis has played a significant role in applied mathematics and theoretical physics. In many cases, regular perturbation methods are not applicable, and various singular perturbation techniques must be used. 1−6 Examples of widely-used techniques for ordinary differential equations (ODEs) include 1,2 the methods of multiple scales, boundary layers or asymptotic matching, WKB, stretched coordinates, averaging, the method of reconstitution, 4 and center manifold theory. 6 Although these methods are well known, each has its own drawbacks, preventing mechanical (or algorithmic) application. Indeed, it is probably fair to say that the practice of asymptotic analysis is something of an art.
Multiple-scales analysis has proven to be a particularly useful tool for constructing uniform or global approximate solutions for both small and large values of independent variables. In this method a set of scaled variables, which are regarded as independent variables (although they are ultimately related to one another), is introduced to remove all secular terms. The choice of the set is, in some cases, nontrivial, and may only be justified post hoc. Nevertheless, this method is usually considered the most general, subsuming the others mentioned below.
Differential equations whose highest order derivatives are multiplied by a small parameter ǫ often yield solutions with narrow regions of rapid variation, known as boundary layers. Boundary layer techniques can be applied if the thickness of these regions tends to zero as ǫ → 0; otherwise, WKB must be used. The limitation of WKB is that it applies to linear equations only. Although boundary layer methods apply to nonlinear as well as to linear problems, the determination of the expansion parameter can be subtle. Furthermore, matching of outer and inner expansions via intermediate expansions is required, sometimes involving delicate arguments that are difficult to perform mechanically.
Another class of related problems concerns partial differential equations (PDEs) describing nonequilibrium, spatially-extended systems near bifurcation points. Such systems often exhibit spatial-temporal patterns modulated by an envelope function (or amplitude) which varies slowly compared with the pattern itself. Extracting the long wavelength, slow timescale behavior of such systems is the task of reductive perturbation methods, 7 which are themselves related to multiple-scales analysis.
The purpose of this paper is to present a unified, and physically-motivated approach to these classes of problems, based upon the renormalization group (RG).
The essence of the renormalization group method is to extract structurally stable features of a system which are insensitive to details. 8−11 For example, field theories, critical phenomena, polymers and other statistical mechanical systems exhibit universal scaling functions and critical exponents in the limit Λ/ξ → 0, where Λ is some ultra-violet cut-off and ξ is the (temperature-dependent) correlation length. The renormalization group is the principal tool with which to elucidate this universal behavior and is properly regarded as a means of asymptotic analysis.
The usefulness of this point of view has been amply demonstrated 12,13 by the relationship between the renormalization group and intermediate asymptotics. 14 In particular, the large-time asymptotic behavior of certain initial-value problems is given by a similarity solution of the governing PDE, where the similarity variable contains anomalous exponents which may not be determined a priori by elementary dimensional considerations. Nevertheless, renormalized perturbation theory combined with the renormalization group, gives an expansion for the anomalous exponents and the solution. 15 The similarities between renormalization group and singular perturbation methods extend also to technical details: both perturbative renormalization group and conventional singular perturbation methods remove secular or divergent terms from perturbation series. These formal similarities invite a natural question: what is the relation, if any, between conventional asymptotic methods and the renormalization group?
In this paper, which is an extended version of our preliminary report, 16 we demonstrate that singular perturbation methods may be naturally understood as renormalized perturbation theory, and that amplitude equations obtainable by reductive perturbation methods may be derived as renormalization group equations.
Our studies indicate that the renormalization group method may have several advantages compared with conventional methods. Although we recognize that our analysis is at the formal, heuristic level, we suggest that a more careful mathematical analysis would be worthwhile, given the potential usefulness of our central claim.
One advantage of the renormalization group method is that the starting point is a straightforward naive perturbation expansion, for which very little a priori knowledge is required. That is, one does not need to guess or otherwise introduce unexpected fractional power laws or logarithmic functions of ǫ in an ad hoc manner. It seems that these ǫ-dependent space/time scales arise naturally during the analysis.
We will show that the renormalization group approach sometimes seems to be more efficient and accurate in practice than standard methods in extracting global information from the perturbation expansion. Standard methods often attempt to represent an asymptotic solution in terms of asymptotic sequences of a few simple functions of the expansion parameter, such as exp, log, powers and so on. The renormalization group can generate its own problem-adapted asymptotic sequence without matching: in the examples given in section 4, these turn out to be complicated functions conveniently defined by an integral representation. For small ǫ, this asymptotic sequence can be expanded to reproduce the solutions conventionally obtained by asymptotic matching, although in the examples that we have studied so far, the conventional approximant is practically inferior to the one obtained by the RG. In switchback problems, the RG perturbation series may need to be carried out to higher than lowest order, then expanded in ǫ, in order to reproduce the (inferior) conventional result.
A related advantage of the renormalization group seems to be the lack of necessity to perform asymptotic matching. To illustrate this assertion, in section 3 we solve several ODEs with boundary layers, and in section 4 we address the difficult technical problem of switchback terms.
The renormalization group methods for partial differential equations such as the Barenblatt equation,12,13,14 and front propagation problems in reactiondiffusion equations, 11 are, in retrospect, examples of the general approach discussed in this paper. We emphasize that our renormalization group method has no connection with the so-called method of renormalization or uniformization 1 in the conventional perturbation literature; the latter is a mere variant of the method of stretched coordinates, and of narrow limited use.
Lastly, we wish to point out that recently, a method utilizing an invariance condition in the solution of multiple-scale singular perturbation problems was proposed independently by Woodruff, 17 based on ideas related to the renormalization group. In addition, Kunihiro 18 has shown that the renormalization group method that we proposed in ref. 16 may be interpreted geometrically in terms of the classical theory of envelopes.
The outline of this paper is as follows. In Section 2, we discuss the general relation between multiple-scale analysis and renormalization group. In Section 3, we show how boundary layer and WKB problems can be solved using the renormalization group. In Section 4, we demonstrate with several examples that the renormalization group approach has technical advantages to conventional asymptotic methods. In Section 5, the renormalization group is applied as a reductive perturbation tool to the derivation of global slow motion equations for partial differential equations. Center manifold theory is also briefly considered from the same point of view. We conclude in Section 6.

Multiple Scale Theory and RG
In this section, we show that multiple-scale analysis is equivalent to the RG, and that the solvability condition used in multiple scales to remove the secular divergences is equivalent to the physical assumption of renormalizability in RG theory.

Rayleigh Equation
The example we consider below is the Rayleigh equation, 19 closely related to the Van der Pol oscillator: It is known that the method of uniformization or renormalization 1 fails here, and this example is a textbook illustration of multiple scales analysis. We show here that from only the simple-minded straightforward expansion, not only is the RG capable of identifying automatically all different multiple scales required by multiple scales analysis, but also produces a uniformly valid asymptotic solution without encountering the ambiguity which often plagues higher order calculations in multiple scales analysis.
A naive expansion y = y 0 + ǫy 1 + ǫ 2 y 2 + · · · gives where R 0 , Θ 0 are constants determined by the initial conditions at arbitrary t = t 0 . This naive perturbation theory breaks down when ǫ(t − t 0 ) > 1 because of the secular terms. The arbitrary time t 0 may be interpreted as the (logarithm of the) ultraviolet cutoff in the usual field theory. 11 To regularize the perturbation series, we introduce an arbitrary time τ , split t − t 0 as t − τ + τ − t 0 , and absorb the terms containing τ − t 0 into the renormalized counterparts R and Θ of R 0 and Θ 0 , respectively. This is allowed because R 0 and Θ 0 are no longer constants of motion in the presence of the nonlinear perturbation.
We introduce a multiplicative renormalization constant Z 1 = 1 + ∞ 1 a n ǫ n and an additive one Z 2 = ∞ 1 b n ǫ n such that R 0 (t 0 ) = Z 1 (t 0 , τ )R(τ ) and Θ 0 (t 0 ) = Θ(τ ) + Z 2 (t 0 , τ ). The coefficients a n and b n (n ≥ 1) are chosen order by order in ǫ to eliminate the terms containing τ − t 0 as in the standard RG. 20−25 The choice a 1 = −(1/2)(1 − R 2 /4)(τ − t 0 ), b 1 = 0 removes the secular terms to order ǫ, and we obtain the following renormalized perturbation result 26 where R, Θ are now functions of τ . Since τ does not appear in the original problem, the solution should not depend on τ . Therefore, (∂y/∂τ ) t = 0 for any t. This is the RG equation, which in this case consists of two independent Solving (2.4) and equating τ and t eliminates the secular term, we get where R(0), Θ(0) are constants to be determined by the initial condition. Assuming the initial condition y(0) = 0, y ′ (0) = 2a, we find R(0) = 2a, Θ(0) = 0, and the final uniformly valid result reads which approaches a limit circle of radius 2 as t → ∞.
The second order RG calculation shows the assumption of perturbative renormalizability is consistent and no ambiguity arises at all. The corresponding amplitude and phase equation to order O(ǫ 3 ) are from which the multiple time scales T 1 = ǫt, T 2 = ǫ 2 t, · · · used in multiple scales analysis appear naturally (although the RG does not require such identifications).
When R = 2, (2.7) reduces to In this simple example, it was straightforward to determine the multiple time scales. However, it is well known that in many cases, within multiple scales analysis hidden intermediate scales must be included in the perturbation expansion so as to obtain the correct result. In the next example, will show that the RG method is a more straightforward but secure way to determine multiple slow time scales than the multiple scales method.

Mathieu Equation
The second illustrative example we examine using RG is the Mathieu equation 27 d 2 y dt 2 + (a + 2ǫ cos t)y = 0, (2.9) where a and ǫ are parameters.
The Floquet theory of linear periodic differential equations 1 predicts that in the (a, ǫ) plane there are some regions where the solutions to (2.9) remain bounded for all t and stable, and others where the solutions are unstable. Perturbative investigation shows that for sufficiently small ǫ, all solutions y(t) are stable for a > 0, a = n 2 /4, n = 0, 1, 2, · · ·. Without loss of generality, we investigate the stability of solutions near a = 1/4 and ǫ = 0 to find the stability boundary in the (a, ǫ) plane. We treat the boundary curve a as a function of ǫ and expand a in powers of ǫ: a(ǫ) = 1/4 + a 1 ǫ + a 2 ǫ 2 + · · ·. It is our goal to determine values of a 1 , a 2 , · · · perturbatively. Multiple-scale analysis can be applied to this problem, and the coefficients a 1 = 1, a 2 = −1/2 are determined. However, it turns out that the introduction of multiple time scales τ 1 = ǫt, τ 2 = ǫ 2 t, · · · is not sufficient to determine the second order coefficient a 2 even after the first order coefficient a 1 is set to 1. Through careful analysis, it is found that a new hidden time scale σ = ǫ 3/2 t must be introduced into the problem, and the perturbative expansion must be done in powers of ǫ 1/2 , rather than the original expansion in powers of ǫ. It is necessary to go to the fourth order in powers of ǫ 1/2 to determine a 2 . Thus, the procedure required to determine all necessary time scales is not mechanical: if any hidden scales are omitted or cannot be determined, correct results will not be guaranteed. This represents a typical shortcoming of multiple scales analysis. Now we demonstrate how the unexpected time scales such as σ = ǫ 3/2 t appear automatically from the RG equation, starting only with a straightforward perturbative expansion. Substituting a = 1/4 + a 1 ǫ + a 2 ǫ 2 + · · · in (2.9) and expanding in powers of ǫ (not ǫ 1/2 ) as y = y 0 + ǫy 1 + ǫ 2 y 2 + · · ·, we get and so on. First, let us determine the first order coefficient a 1 . The straightforward perturbation result, to O(ǫ), is given by where R 0 , Θ 0 are constants dependent on initial conditions given at some arbitrary time t 0 . Similarly, the secular divergences can be removed by regarding t 0 as a regularization parameter and renormalizing the bare amplitude A 0 and bare phase Θ 0 : R 0 (t 0 ) = Z 1 (t 0 , µ)R(µ), Θ 0 (t 0 ) = Z 2 (t 0 , µ) + Θ(µ), where µ is some arbitrary time scale, as was done in previous problems. The renormalized perturbation result is (2.14) The RG equation ∂y/∂µ = 0 for any t gives For convenience, we introduce the complex amplitude A = Re iΘ as A = B + iC, with its real and imaginary parts B = R cos Θ, C = R sin Θ. The equations for B(µ) and C(µ) are Thus, we have Solving this and setting µ = t, we get where K 1 is a constant, and the first slow time scale τ 1 = ǫt has appeared automatically. Obviously, for |a 1 | < 1, instability sets in, where the solution grows exponentially with time t, while for |a 1 | > 1, the solutions are bounded and stable. Therefore, near ǫ = 0, the stability boundary is a = 1/4 ± ǫ + O(ǫ 2 ).

(2.19)
Extending the renormalization procedure to the second order, we find all the secular divergences to this order can be removed completely, a sign of the consistency of perturbative renormalizability. Keeping only the two lowest harmonics with prime frequency and omitting other higher frequency terms which are not important for determining the stability boundary, we obtain the renormalized perturbation result, to order ǫ 2 , Accordingly, the equations for B(µ) and C(µ) become Thus, we get Keeping only the lowest order term of (2.23) gives which has the solution (setting µ = t) where K 2 is a constant, and the second and the third slow time scales σ = ǫ 3/2 t, τ 2 = ǫ 2 t appear naturally. We apparently have stable solutions for a 2 > −1/2 and unstable solutions for a 2 < −1/2. Therefore, to order ǫ 2 , the instability boundary is given by

Oscillator with Time Dependent Spring Constant
The third illustrative example is an oscillator governed by the equation The initial conditions are y(0) = 1 and y ′ (0) = 0. The regular perturbation theory breaks down for t → ∞, and multiple scales analysis can be applied to eliminate the secular behavior. However, it turns out that multiple time scales must be chosen as τ 0 = t, τ 1 = ǫ 1/2 t, τ 2 = ǫt, · · ·. Since the frequency of the oscillator is found to be time-dependent, the method of stretched coordinates or the so-called method of uniformization or renormalization (in the conventional applied mathematics sense) does not work here.
We will see how a uniformly valid solution can be constructed simply from the naive perturbation series with the aid of the RG. To solve (2.27), we assume a straightforward expansion in powers of ǫ (not ǫ 1/2 ), y = y 0 + ǫy 1 + ǫ 2 y 2 + · · ·. The bare perturbation result, to order ǫ, is given by As in the preceding examples, renormalizing the bare amplitude R 0 and phase Θ 0 removes the secular divergences. The renormalized perturbation result is where R, Θ are functions of arbitrary time scale µ, and a 1 , b 1 are arbitrary constants. The RG equation reads Solving (2.30) and setting µ = t in (2.29) give Thus, we obtain the uniformly valid result Imposing the boundary conditions y(0) = 1, y ′ (0) = 0 gives R(0) = 1, Θ(0) = 0, a 1 = 0, b 1 = −1. Therefore, the final result is where the frequency defined as ω = dΘ/dt becomes time-dependent: . Rewriting ǫt 2 as (ǫ 1/2 t) 2 , two slow time scales T 1 = ǫ 1/2 t, T 2 = ǫt are easily identified from the RG result (but these identifications are unnecessary in our approach).
The RG scheme given above is also applicable to quantum systems with discrete or continuous energy spectrums, especially those which involve resonance phenomena, e.g., the Rabi flopping, the Stark shift, the Bloch-Siegert shift. 29 The multiple time scale perturbation analysis has successfully given a unified framework for all quantum resonance. 30 In a similar way, the RG method simply recovers all resonance equations which turn out to be simply RG equations. The application of RG to the time-dependent Schrödinger equation also reproduces the Fermi's Golden Rule. 31 Here we will not give detailed calculations of these problems. In the next section, we will show that WKB problems can be easily solved using the RG method. Therefore, many quantum problems which are usually solved using WKB and/or multiple scales analysis can also be studied using the RG approach.
To summarize, it seems that the RG method is more efficient and mechanical than the multiple scales method in determining the multiple slow time scales. In the RG approach, the starting point is simply a straightforward naive perturbation series, and all necessary multiple scales arise naturally from RG equations.
The above examples reveal two important points, demonstrated more generally below: (1) the results of multiple scales analysis can be obtained from renormalized perturbation theory, and (2) the RG equation describes the long time scale motion of the amplitude and the phase.

Boundary-Layer Theory, WKB and RG
Another important class of singular problems is that for which the highest order derivative of the equation is multiplied by a small parameter ǫ, e.g., WKB and boundary layer problems.
Boundary-layer theory and asymptotic matching are a collection of singular perturbation methods for constructing a uniformly and globally valid solution by calculating the separated outer and inner solutions and then matching them across intermediate scale solutions. Quite often, the intermediate matching is very lengthy and only some particular matching method will work. WKB theory is well known to be a powerful tool for obtaining a global approximation to solutions of a linear differential equation whose highest derivative is multiplied by a small parameter ǫ. Many linear problems often solved by WKB theory can be solved by boundary layer theory; indeed, in these cases, boundary layer theory (thickness of the boundary layer goes to zero as ǫ → 0) is a special case of WKB (thickness of the boundary layer remains finite even as ǫ → 0). The limitation of the conventional WKB method is that it applies only to linear problems, while boundary layer theory works for linear as well as nonlinear problems.
In this section we will demonstrate explicitly that many boundary layer problems, linear or nonlinear, can be solved by the RG. The uniformly valid asymptotics of boundary layer problems can actually be constructed from the inner expansion alone, with the aid of the RG, without the need for intermediate matching.

Simple Linear Example
Consider the following simple example, which describes the motion of an overdamped linear oscillator: where ǫ is a small parameter. A standard dominant-balance argument shows that there exists a boundary layer of thickness δ = O(ǫ) at t = 0. Thus, we set t = ǫτ , and rewrite equation (3.1) as Naive expansion gives where the coefficients A 0 , B 0 are constants of integration and O(ǫ) refers to all the regular terms of order ǫ and higher, which are finite even in the limit τ − τ 0 → ∞. This naive perturbation theory breaks down due to the divergence of secular terms for large τ − τ 0 . However, this divergence can be removed by regarding τ 0 as a regularization parameter and renormalizing A 0 , B 0 as A 0 (τ 0 ) = Z 1 A(µ), and B 0 (τ 0 ) = Z 2 B(µ). Here µ is an arbitrary time, and A, B are the renormalized counterparts of A 0 , B 0 . The renormalization constants are chosen order by order in ǫ to eliminate the secular divergences. Split τ − τ 0 as (τ − µ) + (µ − τ 0 ), and then absorb the divergent part µ − τ 0 in the limit τ 0 → −∞ by redefining However, it is impossible that the actual solution y(τ ) can depend on the arbitrary time µ which is not present in the original problem. Thus we have the renormalization group equation ∂y/∂µ = 0 for any τ , which gives Extending the RG calculation to the second order gives, without any ambiguity, Solving them, setting µ = τ and setting back τ = t/ǫ in (3.4), we finally obtain the uniformly valid solution where C 1 , C 2 are constants to be determined by the initial conditions. Clearly, the RG result to order ǫ 2 recovers exactly that obtained by the standard singular methods. 1 Notice that the equations in (3.7) are nothing but the equations of motion for slow time scale: the amplitude equations. Thus, amplitude equations are renormalization group equations. We announced this result previously, and derived the Burgers equation as a renormalization group equation. 11 A much more complicated example illustrating this point will be given in Section 5.

Example with log ǫ
The second example we consider is 32 ǫy ′′ + xy ′ − xy = 0, y(0) = 0, y(1) = e. (3.9) A standard dominant-balance argument tells us that there exists a boundary layer of thickness of order ǫ 1/2 (but not ǫ) at x = 0. The complication in the conventional asymptotic matching stems from the fact that the inner expansion must contain not only powers of ǫ 1/2 but also those terms containing combinations of ǫ and log ǫ to make the intermediate matching successful. Here we explicitly show that the renormalized naive inner expansion in powers of ǫ 1/2 gives a uniformly valid asymptotic solution. This reveals that those unexpected terms containing log ǫ in the conventional approach are just an artifact of perturbative expansions of x −ǫ . (3.10) Thus the naive perturbation result to order ǫ is (3.12) where A 0 , B 0 are integration constants, and R.T. represents regular terms finite even in the limit X − X 0 → ∞ and log(X/X 0 ) → ∞. The divergence can be controlled by renormalizing ∞ 0 a n ǫ n/2 , a 0 = 1 and Z 2 (µ) = ∞ 0 b n ǫ n/2 , b 0 = 1, are renormalization constants and µ is some arbitrary position. The choice a 1 = X 0 − µ, log(X 0 /µ) successfully removes the divergences up to order ǫ, and the renormalized perturbation result is Solving these two equations, we obtain where C 1 , C 2 are constants to be determined by the given boundary conditions. Setting µ = X, we obtain Setting back X = x/ǫ 1/2 , we obtain the final uniformly valid asymptotic result to order ǫ, Thus the terms such as ǫ log ǫ, which are present in the inner expansion given in (e.g.,) ref. 1, are relics of the expansion of x −ǫ . It is worthwhile to note that the RG result is slightly different from the asymptotic matching result given by Bender and Orszag in their book. 32 To leading order, the former is while the latter is Comparing with the numerical result of the original equation (3.9), we find that in the boundary-layer region, the RG result (3.20) is a better approximant than the standard result (3.21).

Nonlinear Boundary Layer Problem
Boundary-layer analysis applies to nonlinear as well as to linear differential equations. In this section and in the following section, we will demonstrate that the RG method can be used to solve nonlinear boundary layer problems.
Let us consider the following illustrative nonlinear problem: There is only one boundary layer of thickness ǫ at Assuming an inner expansion Y = Y 0 + ǫY 1 + · · · gives the following asymptotic result as X → ∞, where A 0 , B 0 are integration constants and R.T. represents all regular terms in the expansion finite even in the limit X − X 0 → ∞. The renormalized perturbation result obtained as in the previous examples is The RG equation gives, to order ǫ, Solving (3.26) , we get where C 1 , C 2 are constants of integration to be determined by the given boundary conditions. Equating µ and X in (3.25) and restoring x = ǫX, we obtain the uniformly valid asymptotic result Imposing boundary conditions y(0) = 0, y(1) = 0 gives C 1 = 1, C 2 = − log 2 in the limit ǫ → 0 + . Therefore, the final result is This RG result recovers the leading order result from boundary layer analysis.

Nonlinear Problem of Carrier
In this section, we consider a first-order nonlinear model problem of Carrier, The exact solution can be obtained by integrating (3.30) once, It becomes, however, a nontrivial singular perturbation problem, if we pretend that we cannot obtain the exact solution. The method of strained coordinates or the method of asymptotic matching can be applied with a rather lengthy matching. We show here how to solve the problem using RG without matching, and give the exact result, starting only from the inner expansion.
First, we apply the usual dominant balance argument to make the structure of the equation clear. We introduce X ≡ η(ǫ)x and F = δ(ǫ)f ; the latter is needed because the equation is nonlinear. The original equation reads (X + (ǫη/δ)F )dF/dX + F = δ. ǫη/δ ≪ 1 corresponds to the outer limit, and ǫη/δ ∼ 1 is the only nontrivial alternative possibility. Hence, δ ∼ ǫ α and η ∼ ǫ α−1 with α ∈ (0, 1] are the useful scalings. The expansion parameter becomes δ ∼ ǫ α . It turns out that any choice of α is admissible in this case, and so we adopt the simplest choice α = 1. Accordingly, we rescale f as F = ǫf to convert the original equation (3.30) to (3.32) whose general positive solution is with A 0 a constant of integration determined by the initial condition given at some arbitrary x 0 . The first order equation is given by This linear equation has a general solution Thus, the straightforward perturbation result, to O(ǫ), is given by We see that this naive perturbation (3.37) breaks down formally for x ≫ x 0 . Actually, the domain of our problem is finite, and because x is not scaled, it is not possible that x ≫ x 0 can occur within the domain. A better argument is as follows. Since the boundary condition is F (1) = 2ǫ, near x = 1 the O(ǫ) term dominates; this is a singular perturbation, and indeed the perturbation term diverges relative to the zeroth order term.
The secular divergence can be removed by renormalizing A 0 by A 0 (x 0 ) = ZA(µ), and the renormalized perturbation result obtained is Setting µ = x and f = F/ǫ, we obtain the uniformly valid asymptotic result Imposing the boundary condition f (1) = 2 gives A(0) = 2ǫ + 4ǫ 2 . Therefore, the uniformly valid result to order ǫ, is given by This happens to be the exact solution to the problem. A further calculation demonstrates that all the higher order corrections vanish. The conventional methods can also recover the exact result, but clearly the RG is simpler.

Problem with Multiple Boundary Layers
In many situations, there exist multiple boundary layers at one side, for which multiple calculations of inner and outer solutions and their asymptotic matchings have to be made in different separated regions to obtain a uniformly valid solution. Again it turns out that the RG method manages to produce the solution without any matching needed. Let us consider the following initial-value problem 35 ǫ 3/2 y ′′′ + (ǫ 1/2 + ǫ + ǫ 3/2 )y ′′ + (1 + ǫ 1/2 + ǫ)y ′ + y = 0, (3.43) with initial conditions y(0) = 3, The exact solution is y(t) = e −x + e −x/ǫ 1/2 + e −x/ǫ . Pretending we do not know how to solve it exactly, we resort to conventional singular perturbation methods. It turns out that the conventional perturbation calculation is very tedious and rather challenging. By dominant balance, this problem is found to have two distinguished boundary layers at t = 0, of thickness of order ǫ 1/2 and ǫ, respectively. Therefore, one outer solution and two inner solutions must be calculated and two asymptotic matchings are necessary, if boundary layer theory is used. Starting only with the thinnest or innermost boundary layer by rescaling t by t = ǫT , and expanding y = Y (T ) (e.g.,) in ǫ 1/2 , the RG method successfully recovers the exact solution without any matching.

Linear Boundary-layer and WKB problems I: no turning points
To conclude this section, we show how linear boundary layer and WKB problems in general forms can be treated using RG in a unified fashion. This relationship between boundary layer theory and WKB is explained in Bender and Orszag's book. 1 The boundary-layer type problem we wish to study using RG has the following general form: converting (3.44) to This form is just the Schrödinger form, which can be solved by WKB methods. Consequently, we can treat both linear boundary layer and WKB problems in a unified way.
In the remainder of this section and in the following section, we will show how to solve Schrödinger equations using RG. Our strategy is to first introduce a natural change of the independent variable which allows one to obtain efficiently the non-perturbative part of the solution. The transformation is identical to the independent variable portion of the standard Liouville-Green transformation 36 or its natural generalization used by Langer, 37 but the crucial difference is that we do not introduce the new dependent variable. This is the analogue of the geometrical-optics approximation in WKB theory, 1 and is the starting point of a renormalized perturbation series, which reproduces the physical-optics and higher-order WKB approximations. Although it may be possible to derive even the geometric-optics approximation using RG, we have not succeeded in so doing. transformation is sensitive function Q(x). When Q(x) vanishes, its zeros lead to turning points in the standard WKB approach. The simplest WKB approximations break down there, and connection formulae are required in the conventional procedure in order to match approximations on either side. other hand, our procedure leads to a uniformly valid approximation. Langer 37 found that a suitable generalization of the Liouville-Green transformation can produce a uniformly valid approximation across the turning point. Again for the cases with turning points, we transform the independent variable only with a straightforward generalization of the no-turning point case. We emphasize that we are able to avoid the need to perform matching, and that the transformation of the dependent variable is produced naturally by RG. The use of RG is not responsible for the choice of the transformation of the independent variable, but our choice not to introduce the transformation of dependent variables in contrast to the approaches by Liouville and Green and Langer is motivated by RG. This allows us to choose a better transformation of the dependent variable, which agrees with the conventional result in the small ǫ limit. The corrections and prefactors which accompany the zeroth order Langer-type solution are calculated by RG, and do differ from and improve upon those obtained by the standard analysis. Furthermore, we can expand our asymptotic sequence in ǫ to reproduce the standard textbook results.
The remainder of this section concerns Schrödinger problems with no turning points. The following section discusses the case with one turning point, and gives an outline of how the methods can be generalized for higher numbers of turning points and for multiple-boundary-layer linear problems as well.
We will first rederive the well-known physical-optics approximation using the RG theory, valid when the function Q(x) has no zeroes in the interval of interest. Following Liouville and Green, we introduce a new independent variable t = f (x) implicitly determined as dt = √ Qdx/ǫ. The choice is natural from the perturbation point of view, because even when du/d(x/ǫ) is significant, dQ(x)/dx is not, so Q(x) can be regarded as a constant to order O(ǫ 0 ). Eq. Naively expanding u as u(t) = u 0 (t)+ǫu 1 (t)+· · ·, we get the bare perturbation result where O(ǫ) refers to all regular terms of order ǫ which remain finite even as The uniformly valid asymptotic result y(x) for the general linear boundary layer problem (3.44) is given by (3.45). For numerical evaluation of (3.33), we do not need any further expansion, because (3.45) is the uniformly valid result we want. To compare, however, with the conventional results due to asymptotic matching methods, let us make asymptotic expansions of Q(x).
As a simple check, let us assume that a(x), b(x) are some analytic functions, and a(x) > 0 for 0 ≤ x ≤ 1, and a(0) = 0. Obviously, in the whole region 0 ≤ x ≤ 1, as ǫ → 0, the term a 2 (x)/4ǫ 2 is the dominant term, compared to a ′ (x)/2 and b(x). Simply Taylor expanding as and imposing boundary conditions y(0) = A, y(1) = B, we obtain This expression can be simplified further, because the second term contributes appreciably only when x = O(ǫ 2 ) (ǫ → 0). Thus, This is exactly the same as the uniformly valid leading boundary layer or WKB result.
It is known that the case with a(0) = 0 is subtle. For simplicity, we consider only the cases a(x) = x α , b(x) = 1, where α > −1 so that there exists a boundary layer at x = 0.
When α > 1, the thickness of boundary layer is of order δ = O(ǫ). When x ≫ ǫ, the term a 2 (x)/4ǫ 2 dominates over other two terms, a ′ (x)/2 and b(x), in Q(x). However, when x ∼ O(ǫ), we have to be careful with the asymptotic expansion of Q(x), since the dominant term now is b(x) = 1. Thus, as ǫ → 0, the leading term of Q(x) is 1. The final uniformly valid approximation is When |α| ≤ 1, it is straightforward to check that the boundary layer is of thickness of order δ ∼ ǫ 2/(1+α) , and that the first and second term in Q(x) are of the same order, when x ∼ δ(ǫ). The uniformly valid expression turns out to be (3.58) Expanding the above leading uniformly valid result obtained with the aid of RG recovers the outer and inner solutions due to boundary layer theory and asymptotic matching. Note that the above results are obtained from the "inner expansion" alone without ever having to perform any asymptotic matching. This is practically important as we will see in the next section. However, the main message here is conceptually more important: conventional singular perturbation methods can be understood naturally as the standard renormalized perturbation procedure.

WKB analysis II: turning points
In order to complete this section, we begin by presenting a general discussion of Schrödinger equations and one-turning-point WKB problem, and at the end of this section, we generalize the case to multiple-turning-point and multipleboundary-layer problems.
The Schrödinger equation which we will consider in this section is When Q in (3.59) vanishes or changes its sign, the approach in the preceding subsection fails as can easily be seen from the presence of the factor Q −1/4 . If Q has an isolated zero at x = 0 of order α > 0, we can write locally Q(x) = x α ψ(x) with a positive definite function ψ without any loss of generality. A natural choice of the counterpart of the Liouville-Green transformation x → t is to remove the zeros from dt/dx: we introduce a new independent variable t = f (x) implicitly determined as dt = Q/t α dx/ǫ, and integrating it gives . (3.60) The original equation (3.46) is transformed into x as x → 0, S is a bounded function even near x = 0. Notice that in contrast to the conventional approaches due to Liouville and Green or Langer, we do not introduce the transformation for the dependent variable, which will be produced by the RG procedure. Here we work out the simplest case α = 1.
Expanding naively u in powers of ǫ as u = u 0 + ǫu 1 + ǫ 2 u 2 + · · ·, we obtain the bare perturbation result to order ǫ, where Ai, Bi are two linearly independent Airy functions, and the Bi(t) function is already discarded in the zeroth order solution, since it does not satisfy the physical condition u(+∞) = 0. In the limit t − t 0 → +∞, the second term of the first order perturbation Bi(t) t t0 dt ′ S(t ′ )Ai(t ′ )Ai ′ (t ′ ) remains finite. However, the term t t0 dt ′ S(t ′ )Ai ′ (t ′ )Bi(t ′ ) diverges and must be renormalized, giving the renormalized perturbation series where C(µ) is the counterpart of the bare amplitude C 0 (t 0 ), and O(ǫ) refers to all finite regular terms of order ǫ even in the limit t − t 0 → ∞. The RG equation du/dµ ≡ 0 gives Integrating (3.64) and setting µ = t, we get where C 0 is a constant of integration to be determined by boundary condition at t = 0. Thus we have arrived at the adiabatic invariant which differs from that usually obtained, leading to the the final uniformly valid solution The RG result (3.65) differs from the standard Langer formula, since (3.65) involves Airy functions Ai and Bi. Note that the new variable t given in (3.60) is a function of ǫ, and that as ǫ → 0 for fixed x, and t → ∞. In this limit, we can resort to the asymptotic properties of the Airy functions Ai(t) and Bi(t) for t → ∞, and find that Ai ′ (t)Bi(t) ∼ −1/2π, as t → ∞. Thus, (3.65) recovers the standard result However, the RG equation (3.64) is valid not only for relatively large µ, but also for small µ. For this reason, we expect that (3.67) is a better uniformly valid approximant than the standard Langer formula, for small and intermediate values of t, or for relatively large (or not small) ǫ cases. This is verified and can be clearly seen in Fig. 1, where we compare the RG result (3.67), the standard Langer formula, and the exact numerical solution of equation (3.46) for several values of ǫ. Thus, the RG results (3.67) without asymptotic matching improve upon those obtained by the standard analysis.
To conclude this section, we briefly outline the recipe to generalize the methods for multiple-turning-point and linear multiple-boundary-layer problems. (For linear cases, with the help of the transformation (3.45) both problems can be transformed into the canonical form and can be treated in a unified way.) We need only consider the case in which Q(x) in (3.46) has multiple turning points. Without loss of generality, we assume Q has the form: x − x n ), n > 1 is a polynomial of x with n zeros x 1 < x 2 < · · · < x n , and ψ(x) > 0 has no zeros. The general strategy is first to introduce a new independent variable t defined implicitly as dt/ Q/f (t)dx/ǫ, where f is chosen to cancel all the zeros of Q. Then we develop the straightforward perturbation series for the resultant equation, and renormalize the integration constant to absorb the secular divergence. This procedure avoids performing multiple connection formulae matching and leads to a uniformly valid approximation. For higher order WKB problems or linear boundary layer problems, the generalization of the methods given here is straightforward.

Switchback Problems
In previous sections, we have already seen that the RG approach not only has conceptual, but also technical advantages compared with various conventional methods. In this section, we will demonstrate this further, by studying, with the aid of RG more complicated problems which involve the so-called 'switchback'. In switchback problems, as conventionally treated, only through subtle analysis in the course of actually solving the problem is it possible to realize the need for, e.g., unexpected order terms to make asymptotic matching consistent.

Example 1: Stokes-Oseen Caricature
A model example is a caricature of the Stokes-Oseen singular boundary layer problem, which describes the low Reynolds number viscous flow past a sphere of unit radius. The main result of this problem has been presented in ref. 16, and in the following, we will only briefly summarize the final results and make some additional comments. where ǫ, the Reynolds number, is a small non-negative constant. This is a very delicate singular boundary layer problem, with complicated asymptotic expansions and matching, involving unexpected orders such as ǫ log(1/ǫ).
The result from asymptotic matching is given by the following expression. 5 For r fixed, we have, to O ǫ 2 log 2 (1/ǫ) , Accordingly, examining the asymptotic result of (4.3) in the limit ǫ → 0, by expanding both e 2 (ǫr) and e 2 (ǫ) for r fixed, and e 2 (ǫ) only for ρ = ǫr fixed, respectively, it is found that the resulting asymptotic solution using RG is correct to O[ǫ log(1/ǫ)] and agrees with that obtained by asymptotic matching. Note that in our method, the ǫ log ǫ term appears naturally from the asymptotic expansion of e 2 (ǫ), whereas some artistry is required to obtain this term conventionally. This fact may suggest that our RG result is inferior to the conventional one. It is important to notice, however, that neither the asymptotic expansion (4.4) augmented with the (log r)/r term of order ǫ nor (4.5) is uniformly valid in its variable r or ρ, respectively. In contrast, it seems that our full result 1 − e 2 (ǫr)/e 2 (ǫ) to order λ 1 = 1/e 2 (ǫ) is uniformly valid as is clearly seen in Fig.  2.
As discussed in the preceding paragraph, (4.3) is not an asymptotic series in powers of ǫ; thus, one might conclude that our result is not even an asymptotic series in any sense. Recall, however, that the asymptotic expansion of a function is unique only when an asymptotic sequence of functions is fixed. The choice of the sequence is a question of vital importance, if one wishes to have a useful asymptotic series. In the conventional singular perturbation methods, an asymptotic sequence is selected by the matching conditions. However, there is no compelling reason to believe that the selected sequence is practically the best asymptotic sequence (of course, it should be the most convenient one for the matching procedure). As we have seen, the RG approach also produces an asymptotic sequence {λ i (ǫ)} from the requirement to satisfy the boundary condition order by order. Therefore, we propose the point of view that a consistent and presumably better asymptotic expansion (starting with λ 1 = 1/e 2 (ǫ) in the present problem) may be obtained by RG. The standard ǫ expansion may well be an inferior asymptotic expansion to our expansion. In addition, the superiority to the RG approach can also be seen from the fact that a closed expression uniformly valid for the whole (infinite) interval has been obtained for the problem, which is not the case for the standard asymptotic matching method.

Example 2: Difficulty with Asymptotic Matching
To illustrate that the RG method is generally simpler to use, and yields practically better approximants than other methods, let us next consider a 'terrible' problem whose model equation can be written as where ǫ is a small non-negative constant, and α = 0 or 1. For α = 1, the asymptotic matching is notoriously difficult, because an infinite number of terms must be calculated before even the leading order can be matched successfully.
We will see how the RG avoids such difficulties in obtaining the leading order result uniformly valid for the entire interval 1 ≤ r < ∞.
Since there exists a boundary layer of thickness δ = O(ǫ) near r = ∞, setting x = ǫr transforms (4.6) into the following 'inner' equation: As in other boundary layer problems, let us first look for the general form of the solution, and then impose the required boundary conditions to determine the constants of integration left in the solution. To do so, we solve (4.2) as an initialvalue problem, given an initial condition u(x 0 ) = A 0 at some arbitrary point x = x 0 , where A 0 is a finite constant. Assuming a naive expansion u(x; ǫ) = u 0 (x) + λ 1 (ǫ)u 1 (x) + λ 2 (ǫ)u 2 (x) + · · · with initial conditions u 0 (x 0 ) = A 0 , u i (x 0 ) = 0, i = 1, 2, · · ·, where the asymptotic sequence λ i (ǫ), i = 1, 2, · · · are to be determined later, we obtain The finite uniform solution can be guessed as u 0 (x) = A 0 , because the uniform field should not be affected appreciably by the distant disturbance source. Thus, the goal is to find out the small perturbation effect on this uniform field in the presence of a distant disturbance.

Reductive Perturbation Theory and RG
In previous examples, we have already mentioned the idea that amplitude or phase equations are RG equations. We will demonstrate that the RG theory is a general and systematic method to derive slow motion equations, even for those complicated problems for which no explicit analytic zeroth-order solutions are known. In previous reports we already discussed the one-dimensional Swift-Hohenberg equation 16 and the Burgers equation 11 as renormalization group equations. Center manifold theory can be considered from the reductive perturbation point of view, because it also extracts slow motion equations on the manifold. Thus, we may expect that the center manifold theory can also be interpreted as an application of the renormalization approach as well.

Newell-Whitehead equation
The example we consider here is the two-dimensional Swift-Hohenberg equation widely used as a simple model of the Rayleigh-Benard convection, where ǫ is a control parameter or a reduced Rayleigh number, a measure of the degree of convective instability of the stationary state u = 0. For small positive ǫ, the system exhibits a supercritical bifurcation. Since we wish to treat ǫu − u 3 as a perturbative term, to be consistent ǫu and u 3 must be of the same order. We scale u as √ ǫu, and denote the new u with the same symbol. Then, the original equation reads We consider this in the whole plane for all positive t. As a zeroth order solution, we choose the roll solution along the y-axis: Ae ikx + complex conjugate, where A is a complex numerical constant. We expand u around this solution as u = Ae ikx + ǫu 1 + · · · + complex conjugate. The first order correction obeys Here, to study only the singular behavior of u 1 , e 3ikx and similar non-resonant terms are ignored. We rewrite this equation as where the operators are given defined as We must look for space-time secular terms in the solution. Secular terms appear only in the special solution of the equation consistent with the inhomogeneous term. In order to find (space-time secular) special solutions of (5.4) we have only to solve L i u Si = (1 − 3|A| 2 )Ae ikx separately, and to make the linear combination of their solutions as µ i u Si with µ i = 1. This is because all four operators L i commute, and L j e ikx = 0, so that L i L j u Si = 0. A trivial calculation gives That is, Here we do not pay attention to inhomogeneous terms nonresonant with the operator. Hence, the most singular part is Similarly, we get and In this way we get the following perturbation result, Here all the less singular terms (e ikx times 1, x, y, xy, y 2 , y 3 ), higher order terms and nonsecular terms (those terms which do not grow indefinitely far away or in the long future) are omitted. These terms will not contribute to the final result, as shown in the argument below. Now, the secular terms are absorbed into the redefinition of the amplitude A as follows. We introduce regularization points X, Y and T and split, for example, x α as x α − X α + X α (for some exponent α), and absorb X α into A. Thus we get, (5.13) Since u should not depend on X, Y or T , the renormalization group equation, to O(ǫ), reads ∂ α+β+γ u/∂T α ∂X β ∂Y γ = 0 for any positive integers α, β, γ with αβγ = 0, where values of α, β, γ are chosen in such a way that the universal slow motion equation we are seeking is independent of any system details. Thus, we have (5.14) Obviously, µ i are still almost arbitrary and must be fixed by the auxiliary conditions. Therefore, to get an auxiliary condition free equation of motion, we use µ i = 1 to arrive at the following RG equation after equating X, Y, T and x, y, t, respectively Thus, we have arrived at the Newell-Whitehead equation.
Let us compare this derivation with the conventional method, for which a summary may be found in the Appendix to the review article by Cross and Hohenberg. 40 Perhaps the most notable point is that no scaling of spatial variables like x → ǫ 1/2 x, y → ǫ 1/4 y is needed. Furthermore, the expansion is a straightforward one in terms of ǫ instead of ǫ 1/2 . That is, the result is almost automatically obtained from the global well-definedness of the perturbation result.
If there are no spatial degrees of freedom, each step of the standard reductive perturbation 7 using the solvability condition and that in the RG derivation above are in one-to-one correspondence. However, if there are spatial degrees of freedom, the standard reductive perturbation regards the spatial derivatives as a perturbation if the zeroth order solution is space-independent, or uses the multiple scale analysis if the zeroth order solution is spatially varying. In contrast, in our RG approach, spatial and time coordinates are treated on an equal footing, and the correct scalings of variables are given automatically.
As the reader may have realized, kinetic equations are expected to be derivable as slow motion equations from the BBGKY hierarchy. For example, the Boltzmann equation can be derived by an RG method. Thus we suggest that it is a rule that slow motion equations are RG equations.

Center Manifold and RG
In this section, we discuss briefly the general relationship between RG theory and center manifold theory. 6 In the general theory of reduction, we wish to know the slow manifold (e.g., inertial manifold, center manifold) which attracts all the long-time asymptotic solutions, and the equation of motion on the manifold. It is well known that the center manifold reduction and normal form theory 6 have played a significant role in studying instabilities and bifurcations encountered in dynamical systems and fluid dynamics. In many circumstances, this approach provides a greatly simplified picture of complicated dynamics by reducing the dimension of the system without losing essential information concerning the instability and bifurcation. In addition, the local dynamics on the center manifold constructed in this way is invariant or universal, in the sense that the structure of the reduced system is independent of specific physical models under consideration. Thus a variety of different phenomena can have the same type of bifurcation, belonging to the same universality class in the parlance of RG. Although the center manifold fits in the RG picture clearly, the general correspondence between them has not yet been established. In certain cases such as the weakly nonlinear stability of fluid motion, the equivalence of the method of center manifold, the method of multiple scales, and the method of amplitude expansion has been established explicitly by applying these methods to the derivation of the Landau equation from the Navier-Stokes equation to the seventh order. 41 To illustrate the relevance of RG, let us consider the following set of equations: where f and g are higher order in the sense that f (λx, λy) or g(λx, λy) is O(λ 2 ) for small λ. Thus the variable y decays quickly but x does not. Hence, the long time behavior of the system is expected to be confined close to a local 1-manifold near the origin. This local manifold is the center manifold (not unique), and the long time behavior of the system is governed by the equation of motion defined on this manifold. Thus, as discussed at the beginning of this section, the problem of finding a center manifold and the equation on it is a problem of extracting slow motion behavior of the system. In this sense, this problem and the general reductive perturbation can be treated in a unified fashion. Since we are interested in the local center manifold, we may rescale the variables as x → λx and y → λy, and may assume that λ is small. Therefore, instead of the original system (5.16), we study dx dt =λf (x, y), dy dt = − y + λg(x, y).
Our RG program starts with the construction of a power series expansion of the solution for (5.17) in terms of λ as x = x 0 + λx 1 + λ 2 x 2 + · · ·, and y = y 0 + λy 1 + λ 2 y 2 + · · ·. Paquette has also pursued the same line independently. 42 A lengthy but straightforward calculation gives x =A + λf 20 A 2 t + λ 2 (f 2 20 A 3 t 2 + f 11 g 20 A 3 t + f 30 A 3 t) y =λg 20 A 2 + λ 2 [2g 20 f 20 A 3 (t − 1) + g 20 g 11 A 3 + g 30 A 3 ], (5.21) where CT denotes the constant terms and A is the initial condition for x. Here we have discarded all the exponentially decaying terms. For example, to the first order the full solution reads where A 0 , B 0 , B 1 are numerical constants dependent on the initial data. The exponentially decaying terms do not contribute to the secular behavior of perturbation series. We absorb the secular terms proportional to the powers of T into the redefined A by splitting t as t α − T α + T α , where α is an appropriate integer. That this can be achieved consistently must be checked order by order. The simplest way may be to introduce the renormalized counterpart A R of A as A = A R (1+λω 1 +λ 2 ω 2 +λ 3 ω 3 +· · ·), where ω i are determined to remove the powers of T from the perturbation result for x after splitting t. The renormalization condition can be written as A R (1 + λω 1 + λ 2 ω 2 + λ 3 ω 3 + · · ·) + λf 20 (1 + λω 1 + λ 2 ω 2 ) 2 A 2 t +λ 2 (f 2 20 t 2 + f 11 g 20 t + f 30 t)A 3 (1 + λω 1 ) 3 + λ 3 [· · ·] = A R .
(5.23) From this, order by order in powers of λ, we can fix ω i as  This also agrees with the result given above.
The formal solution (5.21) is order by order in λ obtained from the true solution by discarding the transcendentally small terms in the large t limit. Notice that in x n the highest power of t is n (for y n it is less), so that up to a given order n, by choosing λ such that λt = 1, we can make the contribution of the sum of the transcendental terms (such as e −1/λ ) less than any small positive number for sufficiently large t. In this way, locally up to any finite order in λ, the series obtained as the singular (or non-decaying) terms describes the asymptotic behavior of the system. Therefore, if the system has a unique solution to the initial value problem (near the origin), then we can uniquely determine these series, and they give a parametric representation of an approximate center manifold. In the present context, renormalizability means that the motion on the approximate center manifold is autonomous. The renormalization reorganizes the expansion so that dx/dt is not explicitly time-dependent.
The RG procedure given above is actually much more tedious than the conventional approach. However, the obtained center manifold by RG need not be expandable in terms of x. Thus, the RG method works in some cases even when the conventional approach is not applicable.

Summary
In this paper, we have demonstrated that various singular perturbation methods and reductive perturbation methods may be understood in a unified fashion from the renormalization group point of view. Amplitude equations and phase equations describing the slow motion dynamics in nonequilibrium phenomena are RG equations. The RG method seems to be more efficient and simpler to use than standard methods. This is practically meaningful, because RG could yield superior approximations without using often tedious asymptotic matching techniques.
Probably the most outstanding question is to justify mathematically the general renormalized perturbation approach developed in this paper. The rigorous and constructive renormalization group approaches of Bricmont and Kupiainen and our formal perturbative approaches have almost no common technical ground, although their philosophy is identical. Consequently, we do not have even a hint as to how to rigorize, or estimate the errors of our approach.
The Wilson-style RG 44 and Bricmont and Kupiainen's related constructive renormalization group approaches 13 can be implemented numerically, 45 especially with the interpolation-resampling scheme which produces a 'virtual continuum' to allow smooth scaling of any function on discrete grids.